fisica quantistica

Quantum Theory, Groups and Representations:

An Introduction

(under construction)

Peter WoitDepartment of Mathematics, Columbia University

[email protected]

April 21, 2015

c©2015 Peter WoitAll rights reserved.

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Contents

Preface xiii

1 Introduction and Overview 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Basic principles of quantum mechanics . . . . . . . . . . . . . . . 2

1.2.1 Fundamental axioms of quantum mechanics . . . . . . . . 31.2.2 Principles of measurement theory . . . . . . . . . . . . . . 4

1.3 Unitary group representations . . . . . . . . . . . . . . . . . . . . 51.4 Representations and quantum mechanics . . . . . . . . . . . . . . 71.5 Symmetry groups and their representations on function spaces . 81.6 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 The Group U(1) and its Representations 132.1 Some representation theory . . . . . . . . . . . . . . . . . . . . . 142.2 The group U(1) and its representations . . . . . . . . . . . . . . 162.3 The charge operator . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Conservation of charge and U(1) symmetry . . . . . . . . . . . . 212.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.6 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Two-state Systems and SU(2) 233.1 The two-state quantum system . . . . . . . . . . . . . . . . . . . 24

3.1.1 The Pauli matrices: observables of the two-state quantumsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.2 Exponentials of Pauli matrices: unitary transformationsof the two-state system . . . . . . . . . . . . . . . . . . . 26

3.2 Commutation relations for Pauli matrices . . . . . . . . . . . . . 293.3 Dynamics of a two-state system . . . . . . . . . . . . . . . . . . . 313.4 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Linear Algebra Review, Unitary and Orthogonal Groups 334.1 Vector spaces and linear maps . . . . . . . . . . . . . . . . . . . . 334.2 Dual vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 Change of basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

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4.4 Inner products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.5 Adjoint operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.6 Orthogonal and unitary transformations . . . . . . . . . . . . . . 40

4.6.1 Orthogonal groups . . . . . . . . . . . . . . . . . . . . . . 41

4.6.2 Unitary groups . . . . . . . . . . . . . . . . . . . . . . . . 42

4.7 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . 43

4.8 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5 Lie Algebras and Lie Algebra Representations 45

5.1 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2 Lie algebras of the orthogonal and unitary groups . . . . . . . . . 48

5.2.1 Lie algebra of the orthogonal group . . . . . . . . . . . . . 49

5.2.2 Lie algebra of the unitary group . . . . . . . . . . . . . . 50

5.3 Lie algebra representations . . . . . . . . . . . . . . . . . . . . . 51

5.4 Complexification . . . . . . . . . . . . . . . . . . . . . . . . . . . 55


6 The Rotation and Spin Groups in 3 and 4 Dimensions 59

6.1 The rotation group in three dimensions . . . . . . . . . . . . . . 59

6.2 Spin groups in three and four dimensions . . . . . . . . . . . . . 62

6.2.1 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.2.2 Rotations and spin groups in four dimensions . . . . . . . 64

6.2.3 Rotations and spin groups in three dimensions . . . . . . 64

6.2.4 The spin group and SU(2) . . . . . . . . . . . . . . . . . 68

6.3 A summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71


7 Rotations and the Spin 12 Particle in a Magnetic Field 73

7.1 The spinor representation . . . . . . . . . . . . . . . . . . . . . . 73

7.2 The spin 1/2 particle in a magnetic field . . . . . . . . . . . . . . 74

7.3 The Heisenberg picture . . . . . . . . . . . . . . . . . . . . . . . 78

7.4 The Bloch sphere and complex projective space . . . . . . . . . . 79


8 Representations of SU(2) and SO(3) 85

8.1 Representations of SU(2): classification . . . . . . . . . . . . . . 86

8.1.1 Weight decomposition . . . . . . . . . . . . . . . . . . . . 86

8.1.2 Lie algebra representations: raising and lowering operators 88

8.2 Representations of SU(2): construction . . . . . . . . . . . . . . 92

8.3 Representations of SO(3) and spherical harmonics . . . . . . . . 95

8.4 The Casimir operator . . . . . . . . . . . . . . . . . . . . . . . . 101


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9 Tensor Products, Entanglement, and Addition of Spin 1059.1 Tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069.2 Composite quantum systems and tensor products . . . . . . . . . 1089.3 Indecomposable vectors and entanglement . . . . . . . . . . . . . 1099.4 Tensor products of representations . . . . . . . . . . . . . . . . . 110

9.4.1 Tensor products of SU(2) representations . . . . . . . . . 1109.4.2 Characters of representations . . . . . . . . . . . . . . . . 1119.4.3 Some examples . . . . . . . . . . . . . . . . . . . . . . . . 112

9.5 Bilinear forms and tensor products . . . . . . . . . . . . . . . . . 1149.6 Symmetric and antisymmetric multilinear forms . . . . . . . . . . 1159.7 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 116

10 Energy, Momentum and Translation Groups 11710.1 Energy, momentum and space-time translations . . . . . . . . . . 11810.2 Periodic boundary conditions and the group U(1) . . . . . . . . . 12310.3 The group R and the Fourier transform . . . . . . . . . . . . . . 126

10.3.1 Delta functions . . . . . . . . . . . . . . . . . . . . . . . . 12910.4 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 131

11 The Heisenberg group and the Schrodinger Representation 13311.1 The position operator and the Heisenberg Lie algebra . . . . . . 134

11.1.1 Position space representation . . . . . . . . . . . . . . . . 13411.1.2 Momentum space representation . . . . . . . . . . . . . . 13511.1.3 Physical interpretation . . . . . . . . . . . . . . . . . . . . 136

11.2 The Heisenberg Lie algebra . . . . . . . . . . . . . . . . . . . . . 13711.3 The Heisenberg group . . . . . . . . . . . . . . . . . . . . . . . . 13811.4 The Schrodinger representation . . . . . . . . . . . . . . . . . . . 13911.5 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 142

12 The Poisson Bracket and Symplectic Geometry 14312.1 Classical mechanics and the Poisson bracket . . . . . . . . . . . . 14312.2 The Poisson bracket and the Heisenberg Lie algebra . . . . . . . 14612.3 Symplectic geometry . . . . . . . . . . . . . . . . . . . . . . . . . 14812.4 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 151

13 Hamiltonian Vector Fields and the Moment Map 15313.1 Vector fields and the exponential map . . . . . . . . . . . . . . . 15313.2 Hamiltonian vector fields and canonical transformations . . . . . 15513.3 Group actions on M and the moment map . . . . . . . . . . . . . 16013.4 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 164

14 Quadratic Polynomials and the Symplectic Group 16514.1 The symplectic group . . . . . . . . . . . . . . . . . . . . . . . . 165

14.1.1 The symplectic group for d = 1 . . . . . . . . . . . . . . . 16614.1.2 The symplectic group for arbitary d . . . . . . . . . . . . 169

14.2 The symplectic group and automorphisms of the Heisenberg group170

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15 Quantization 177

15.1 Canonical quantization . . . . . . . . . . . . . . . . . . . . . . . . 177

15.2 The Groenewold-van Hove no-go theorem . . . . . . . . . . . . . 179

15.3 Canonical quantization in d dimensions . . . . . . . . . . . . . . 180

15.4 Quantization and symmetries . . . . . . . . . . . . . . . . . . . . 181

15.5 More general notions of quantization . . . . . . . . . . . . . . . . 182


16 Semi-direct Products 185

16.1 An example: the Euclidean group . . . . . . . . . . . . . . . . . . 185

16.2 Semi-direct product groups . . . . . . . . . . . . . . . . . . . . . 186

16.3 Semi-direct product Lie algebras . . . . . . . . . . . . . . . . . . 188


17 The Quantum Free Particle as a Representation of the Eu-clidean Group 191

17.1 The quantum free particle and representations of E(2) . . . . . . 192

17.2 The case of E(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

17.3 Other representations of E(3) . . . . . . . . . . . . . . . . . . . . 199


18 Representations of Semi-direct Products 203

18.1 Intertwining operators and the metaplectic representation . . . . 204

18.2 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

18.2.1 The SO(2) action on the d = 1 phase space . . . . . . . . 206

18.2.2 The SO(2) action by rotations of the plane for d = 2 . . . 208

18.3 Representations of N oK, N commutative . . . . . . . . . . . . 209


19 Central Potentials and the Hydrogen Atom 213

19.1 Quantum particle in a central potential . . . . . . . . . . . . . . 213

19.2 so(4) symmetry and the Coulomb potential . . . . . . . . . . . . 217

19.3 The hydrogen atom . . . . . . . . . . . . . . . . . . . . . . . . . . 221


20 The Harmonic Oscillator 223

20.1 The harmonic oscillator with one degree of freedom . . . . . . . . 224

20.2 Creation and annihilation operators . . . . . . . . . . . . . . . . 226

20.3 The Bargmann-Fock representation . . . . . . . . . . . . . . . . . 229

20.4 The Bargmann transform . . . . . . . . . . . . . . . . . . . . . . 231

20.5 Multiple degrees of freedom . . . . . . . . . . . . . . . . . . . . . 232


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21 The Harmonic Oscillator as a Representation of the HeisenbergGroup 235

21.1 Complex structures and phase space . . . . . . . . . . . . . . . . 236

21.2 Complex structures and quantization . . . . . . . . . . . . . . . . 238

21.3 The positivity condition on J . . . . . . . . . . . . . . . . . . . . 240

21.4 Complex structures for d = 1 and squeezed states . . . . . . . . . 243

21.5 Coherent states and the Heisenberg group action . . . . . . . . . 245


22 The Harmonic Oscillator and the Metaplectic Representation,d = 1 249

22.1 The metaplectic representation for d = 1 . . . . . . . . . . . . . . 249

22.2 Complex structures and the SL(2,R) action on M . . . . . . . . 253

22.3 Normal Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . 256


23 The Harmonic Oscillator as a Representation of U(d) 259

23.1 Complex structures and the Sp(2d,R) action on M . . . . . . . 260

23.2 The metaplectic representation and U(d) ⊂ Sp(2d,R) . . . . . . 262

23.3 Bogoliubov transformations . . . . . . . . . . . . . . . . . . . . . 264

23.4 Examples in d = 2 and 3 . . . . . . . . . . . . . . . . . . . . . . . 266

23.4.1 Two degrees of freedom and SU(2) . . . . . . . . . . . . . 266

23.4.2 Three degrees of freedom and SO(3) . . . . . . . . . . . . 269


24 The Fermionic Oscillator 271

24.1 Canonical anticommutation relations and the fermionic oscillator 271

24.2 Multiple degrees of freedom . . . . . . . . . . . . . . . . . . . . . 273


25 Weyl and Clifford Algebras 277

25.1 The Complex Weyl and Clifford algebras . . . . . . . . . . . . . . 277

25.1.1 One degree of freedom, bosonic case . . . . . . . . . . . . 277

25.1.2 One degree of freedom, fermionic case . . . . . . . . . . . 278

25.1.3 Multiple degrees of freedom . . . . . . . . . . . . . . . . . 280

25.2 Real Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . 281


26 Clifford Algebras and Geometry 285

26.1 Non-degenerate bilinear forms . . . . . . . . . . . . . . . . . . . . 285

26.2 Clifford algebras and geometry . . . . . . . . . . . . . . . . . . . 287

26.2.1 Rotations as iterated orthogonal reflections . . . . . . . . 289

26.2.2 The Lie algebra of the rotation group and quadratic ele-ments of the Clifford algebra . . . . . . . . . . . . . . . . 290


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27 Anticommuting Variables and Pseudo-classical Mechanics 293

27.1 The Grassmann algebra of polynomials on anticommuting gener-ators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

27.2 Pseudo-classical mechanics and the fermionic Poisson bracket . . 296

27.3 Examples of pseudo-classical mechanics . . . . . . . . . . . . . . 299

27.3.1 The pseudo-classical spin degree of freedom . . . . . . . . 299

27.3.2 The pseudo-classical fermionic oscillator . . . . . . . . . . 300


28 Fermionic Quantization and Spinors 303

28.1 Quantization of pseudo-classical systems . . . . . . . . . . . . . . 303

28.1.1 Quantization of the pseudo-classical spin . . . . . . . . . . 307

28.2 The Schrodinger representation for fermions: ghosts . . . . . . . 307

28.3 Spinors and the Bargmann-Fock construction . . . . . . . . . . . 309

28.4 Complex structures, U(d) ⊂ SO(2d) and the spinor representation 311

28.5 An example: spinors for SO(4) . . . . . . . . . . . . . . . . . . . 314


29 A Summary: Parallels Between Bosonic and Fermionic Quan-tization 317

30 Supersymmetry, Some Simple Examples 319

30.1 The supersymmetric oscillator . . . . . . . . . . . . . . . . . . . . 319

30.2 Supersymmetric quantum mechanics with a superpotential . . . . 322

30.3 Supersymmetric quantum mechanics and differential forms . . . . 325


31 The Pauli Equation and the Dirac Operator 327

31.1 The Pauli equation and free spin 12 particles in d = 3 . . . . . . . 327

31.2 The Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . . 333


32 Lagrangian Methods and the Path Integral 335

32.1 Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . . . . . 335

32.2 Quantization and path integrals . . . . . . . . . . . . . . . . . . . 340

32.3 Euclidean path integrals, Brownian motion and statistical me-chanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

32.4 Advantages and disadvantages of the path integral . . . . . . . . 344


33 Correlation Functions 347


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34 Multi-particle Systems 349

34.1 Multi-particle quantum systems as quanta of a harmonic oscillator350

34.1.1 Bosons and the quantum harmonic oscillator . . . . . . . 350

34.1.2 Fermions and the fermionic oscillator . . . . . . . . . . . . 351

34.2 Multi-particle quantum systems of free particles . . . . . . . . . . 353

34.2.1 Box normalization . . . . . . . . . . . . . . . . . . . . . . 353

34.2.2 Continuum normalization . . . . . . . . . . . . . . . . . . 356

34.3 Multi-particle systems and Bargmann-Fock quantization . . . . . 357

34.4 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360


35 Non-relativistic Quantum Fields 363

35.1 Quantum field operators . . . . . . . . . . . . . . . . . . . . . . . 363

35.2 Field quantization . . . . . . . . . . . . . . . . . . . . . . . . . . 367

35.3 Dynamics of the free quantum field . . . . . . . . . . . . . . . . . 368

35.4 Correlation functions and the propagator . . . . . . . . . . . . . 370

35.5 Interacting quantum fields . . . . . . . . . . . . . . . . . . . . . . 371

35.6 The Lagrangian density and the path integral . . . . . . . . . . . 372


36 Symmetries and Non-relativistic Quantum Fields 373

36.1 Internal symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 374

36.1.1 U(1) symmetry . . . . . . . . . . . . . . . . . . . . . . . . 374

36.1.2 U(m) symmetry . . . . . . . . . . . . . . . . . . . . . . . 376

36.2 Spatial symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . 377

36.3 Fermion and spin 1/2 particle systems . . . . . . . . . . . . . . . 380


37 Quantization of Infinite-dimensional Phase Spaces 381

37.1 Inequivalent irreducible representations . . . . . . . . . . . . . . 382

37.2 The anomaly and the Schwinger term . . . . . . . . . . . . . . . 385

37.3 Spontaneous symmetry breaking . . . . . . . . . . . . . . . . . . 386

37.4 Higher order operators and renormalization . . . . . . . . . . . . 387


38 Minkowski Space and the Lorentz Group 389

38.1 Minkowski space . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

38.2 The Lorentz group and its Lie algebra . . . . . . . . . . . . . . . 393

38.3 Spin and the Lorentz group . . . . . . . . . . . . . . . . . . . . . 395


39 Representations of the Lorentz Group 399

39.1 Representations of the Lorentz group . . . . . . . . . . . . . . . . 399

39.2 Dirac γ matrices and Cliff(3, 1) . . . . . . . . . . . . . . . . . . . 404


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40 The Poincare Group and its Representations 40940.1 The Poincare group and its Lie algebra . . . . . . . . . . . . . . . 40940.2 Representations of the Poincare group . . . . . . . . . . . . . . . 411

40.2.1 Positive energy time-like orbits . . . . . . . . . . . . . . . 41340.2.2 Negative energy time-like orbits . . . . . . . . . . . . . . . 41440.2.3 Space-like orbits . . . . . . . . . . . . . . . . . . . . . . . 41440.2.4 The zero orbit . . . . . . . . . . . . . . . . . . . . . . . . 41440.2.5 Positive energy null orbits . . . . . . . . . . . . . . . . . . 41540.2.6 Negative energy null orbits . . . . . . . . . . . . . . . . . 415


41 The Klein-Gordon Equation and Scalar Quantum Fields 41741.1 The Klein-Gordon equation and its solutions . . . . . . . . . . . 41741.2 The complex structure on the space of Klein-Gordon solutions . 42241.3 Quantization of the Klein-Gordon theory in momentum space . . 42341.4 Classical relativistic scalar fields . . . . . . . . . . . . . . . . . . 42541.5 Quantization of the real scalar field . . . . . . . . . . . . . . . . . 42741.6 The propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42841.7 Interacting scalar field theories . . . . . . . . . . . . . . . . . . . 42941.8 Fermionic scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . 42941.9 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 430

42 Symmetries and Relativistic Scalar Quantum Fields 43142.1 Internal symmetries . . . . . . . . . . . . . . . . . . . . . . . . . 431

42.1.1 SO(m) symmetry and real scalar fields . . . . . . . . . . . 43242.1.2 U(1) symmetry and complex scalar fields . . . . . . . . . 434

42.2 Poincare symmetry and scalar fields . . . . . . . . . . . . . . . . 43742.3 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 437

43 U(1) Gauge Symmetry and Electromagnetic Fields 43943.1 U(1) gauge symmetry . . . . . . . . . . . . . . . . . . . . . . . . 43943.2 Electric and magnetic fields . . . . . . . . . . . . . . . . . . . . . 44143.3 Field equations with background electromagnetic fields . . . . . . 44243.4 The non-Abelian case . . . . . . . . . . . . . . . . . . . . . . . . 44443.5 The geometric significance of the gauge potential . . . . . . . . . 44543.6 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 445

44 Quantization of the Electromagnetic Field: the Photon 44744.1 Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . . 44744.2 Hamiltonian formalism for electromagnetic fields . . . . . . . . . 44844.3 Coulomb gauge quantization . . . . . . . . . . . . . . . . . . . . . 45044.4 Gauss’s law as an operator condition . . . . . . . . . . . . . . . . 45244.5 Covariant quantization methods . . . . . . . . . . . . . . . . . . . 45244.6 The Poincare group and the electromagnetic field . . . . . . . . . 45244.7 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 452

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45 The Dirac Equation and Spin-1/2 Fields 45345.1 The Dirac and Weyl Equations . . . . . . . . . . . . . . . . . . . 45345.2 Quantum Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45645.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45645.4 The propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45645.5 Quantum electrodynamics . . . . . . . . . . . . . . . . . . . . . . 45645.6 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 456

46 An Introduction to the Standard Model 45746.1 Non-Abelian gauge fields . . . . . . . . . . . . . . . . . . . . . . . 45746.2 Fundamental fermions . . . . . . . . . . . . . . . . . . . . . . . . 45746.3 Spontaneous symmetry breaking . . . . . . . . . . . . . . . . . . 45746.4 For further reading . . . . . . . . . . . . . . . . . . . . . . . . . . 457

47 Further Topics 459

A Conventions 461

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Preface

This document began as course notes prepared for a class taught at Columbiaduring the 2012-13 academic year. The intent was to cover the basics of quantummechanics, up to and including basic material on relativistic quantum fieldtheory, from a point of view emphasizing the role of unitary representations ofLie groups in the foundations of the subject. It has been significantly rewrittenand extended during the past year and the intent is to continue this processbased upon experience teaching the same material during 2014-5. The currentstate of the document is that of a first draft of a book. As changes are made,the latest version will be available at

http://www.math.columbia.edu/~woit/QM/qmbook.pdf

Corrections, comments, criticism, and suggestions about improvements areencouraged, with the best way to contact me email to [email protected]

The approach to this material is simultaneously rather advanced, using cru-cially some fundamental mathematical structures normally only discussed ingraduate mathematics courses, while at the same time trying to do this in aselementary terms as possible. The Lie groups needed are relatively simple onesthat can be described purely in terms of small matrices. Much of the represen-tation theory will just use standard manipulations of such matrices. The onlyprerequisite for the course as taught was linear algebra and multi-variable cal-culus. My hope is that this level of presentation will simultaneously be useful tomathematics students trying to learn something about both quantum mechan-ics and representation theory, as well as to physics students who already haveseen some quantum mechanics, but would like to know more about the mathe-matics underlying the subject, especially that relevant to exploiting symmetryprinciples.

The topics covered often intentionally avoid overlap with the material ofstandard physics courses in quantum mechanics and quantum field theory, forwhich many excellent textbooks are available. This document is best read inconjunction with such a text. Some of the main differences with standard physicspresentations include:

• The role of Lie groups, Lie algebras, and their unitary representations issystematically emphasized, including not just the standard use of these toderive consequences for the theory of a “symmetry” generated by operatorscommuting with the Hamiltonian.

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mailto:[email protected]

• Symplectic geometry and the role of the Lie algebra of functions on phasespace in Hamiltonian mechanics is emphasized, with quantization just thepassage to a unitary representation of (a subalgebra of) this Lie algebra.

• The role of the metaplectic representation and the subtleties of the pro-jective factor involved are described in detail.

• The parallel role of the Clifford algebra and spinor representation areextensively investigated.

• Some topics usually first encountered in the context of relativistic quan-tum field theory are instead first developed in simpler non-relativistic orfinite-dimensional contexts. Non-relativistic quantum field theory basedon the Schrodinger equation is described in detail before moving on tothe relativistic case. The topic of irreducible representations of space-time symmetry groups is first encountered with the case of the Euclideangroup, where the implications for the non-relativistic theory are explained.The analogous problem for the relativistic case, that of the irreducible rep-resentations of the Poincare group, is then worked out later on.

• The emphasis is on the Hamiltonian formalism and its representation-theoretical implications, with the Lagrangian formalism de-emphasized.In particular, the operators generating symmetry transformations are de-rived using the moment map for the action of such transformations onphase space, not by invoking Noether’s theorem for transformations thatleave invariant a Lagrangian.

• Care is taken to keep track of the distinction between vector spaces andtheir duals, as well as the distinction between real and complex vectorspaces, making clear exactly where complexification and the choice of acomplex structure enters the theory.

• A fully rigorous treatment of the subject is beyond the scope of what iscovered here, but an attempt is made to keep clear the difference betweenwhere a rigorous treatment could be pursued relatively straight-forwardly,and where there are serious problems of principle making a rigorous treat-ment very hard to achieve.

xiv

Chapter 1

Introduction and Overview

1.1 Introduction

A famous quote from Richard Feynman goes “I think it is safe to say that no oneunderstands quantum mechanics.”[18]. In this book we’ll pursue one possibleroute to such an understanding, emphasizing the deep connections of quantummechanics to fundamental ideas and powerful techniques of modern mathemat-ics. The strangeness inherent in quantum theory that Feynman was referringto has two rather different sources. One of them is the inherent disjunction andincommensurability between the conceptual framework of the classical physicswhich governs our everyday experience of the physical world, and the very dif-ferent framework which governs physical reality at the atomic scale. Familiaritywith the powerful formalisms of classical mechanics and electromagnetism pro-vides deep understanding of the world at the distance scales familiar to us.Supplementing these with the more modern (but still “classical” in the senseof “not quantum”) subjects of special and general relativity extends our under-standing into other less accessible regimes, while still leaving atomic physics amystery.

Read in context though, Feynman was pointing to a second source of diffi-culty, contrasting the mathematical formalism of quantum mechanics with thatof the theory of general relativity, a supposedly equally hard to understandsubject. General relativity can be a difficult subject to master, but its math-ematical and conceptual structure involves a fairly straight-forward extensionof structures that characterize 19th century physics. The fundamental physicallaws (Einstein’s equations for general relativity) are expressed as partial differ-ential equations, a familiar if difficult mathematical subject. The state of thesystem is determined by the set of fields satisfying these equations, and observ-able quantities are functionals of these fields. The mathematics is just that ofthe usual calculus: differential equations and their real-valued solutions.

In quantum mechanics, the state of a system is best thought of as a differentsort of mathematical object: a vector in a complex vector space, the so-called

1

state space. One can sometimes interpret this vector as a function, the wave-function, although this comes with the non-classical feature that wavefunctionsare complex-valued. What’s truly completely different is the treatment of ob-servable quantities, which correspond to self-adjoint linear operators on the statespace. This has no parallel in classical physics, and violates our intuitions abouthow physics should work, with observables now often no longer commuting.

During the earliest days of quantum mechanics, the mathematician HermannWeyl quickly recognized that the mathematical structures being used were oneshe was quite familiar with from his work in the field of representation theory.From the point of view that takes representation theory as a fundamental struc-ture, the framework of quantum mechanics looks perfectly natural. Weyl soonwrote a book expounding such ideas [79], but this got a mixed reaction fromphysicists unhappy with the penetration of unfamiliar mathematical structuresinto their subject (with some of them characterizing the situation as the “Grup-penpest”, the group theory plague). One goal of this course will be to try andmake some of this mathematics as accessible as possible, boiling down Weyl’sexposition to its essentials while updating it in the light of many decades ofprogress and better understanding of the subject.

Weyl’s insight that quantum mechanics crucially involves understanding theLie groups that act on the phase space of a physical system and the unitary rep-resentations of these groups has been vindicated by later developments whichdramatically expanded the scope of these ideas. The use of representation the-ory to exploit the symmetries of a problem has become a powerful tool that hasfound uses in many areas of science, not just quantum mechanics. I hope thatreaders whose main interest is physics will learn to appreciate the mathematicalstructures that lie behind the calculations of standard textbooks, helping themunderstand how to effectively exploit them in other contexts. Those whose maininterest is mathematics will hopefully gain some understanding of fundamen-tal physics, at the same time as seeing some crucial examples of groups andrepresentations. These should provide a good grounding for appreciating moreabstract presentations of the subject that are part of the standard mathemat-ical curriculum. Anyone curious about the relation of fundamental physics tomathematics, and what Eugene Wigner described as “The Unreasonable Ef-fectiveness of Mathematics in the Natural Sciences”[80] should benefit from anexposure to this remarkable story at the intersection of the two subjects.

The following sections give an overview of the fundamental ideas behindmuch of the material to follow. In this sketchy and abstract form they willlikely seem rather mystifying to those meeting them for the first time. As wework through basic examples in coming chapters, a better understanding of theoverall picture described here should start to emerge.

1.2 Basic principles of quantum mechanics

We’ll divide the conventional list of basic principles of quantum mechanics intotwo parts, with the first covering the fundamental mathematics structures.

2

1.2.1 Fundamental axioms of quantum mechanics

In classical physics, the state of a system is given by a point in a “phase space”,which one can think of equivalently as the space of solutions of an equationof motion, or as (parametrizing solutions by initial value data) the space ofcoordinates and momenta. Observable quantities are just functions on this space(i.e. functions of the coordinates and momenta). There is one distinguishedobservable, the energy or Hamiltonian, and it determines how states evolve intime through Hamilton’s equations.

The basic structure of quantum mechanics is quite different, with the for-malism built on the following simple axioms:

Axiom (States). The state of a quantum mechanical system is given by a non-zero vector in a complex vector space H with Hermitian inner product 〈·, ·〉.

We’ll review in chapter 4 some linear algebra, including the properties of in-ner products on complex vector spaces. H may be finite or infinite dimensional,with further restrictions required in the infinite-dimensional case (e.g. we maywant to require H to be a Hilbert space). Note two very important differenceswith classical mechanical states:

• The state space is always linear: a linear combination of states is also astate.

• The state space is a complex vector space: these linear combinations canand do crucially involve complex numbers, in an inescapable way. In theclassical case only real numbers appear, with complex numbers used onlyas an inessential calculational tool.

In this course we will sometimes use the notation introduced by Dirac forvectors in the state space H: such a vector with a label ψ is denoted

|ψ〉

Axiom (Observables). The observables of a quantum mechanical system aregiven by self-adjoint linear operators on H.

We’ll also review the notion of self-adjointness in our review of linear algebra.When H is infinite-dimensional, further restrictions will be needed on the classof linear operators to be used.

Axiom (Dynamics). There is a distinguished observable, the Hamiltonian H.Time evolution of states |ψ(t)〉 ∈ H is given by the Schrodinger equation

d

dt|ψ(t)〉 = − i

~H|ψ(t)〉

The Hamiltonian observable H will have a physical interpretation in termsof energy, and one may also want to specify some sort of positivity property onH, in order to assure the existence of a stable lowest energy state.

3

~ is a dimensional constant, the value of which depends on what units oneuses for time and for energy. It has the dimensions [energy] · [time] and itsexperimental values are

1.054571726(47)× 10−34Joule · seconds = 6.58211928(15)× 10−16eV · seconds

(eV is the unit of “electron-Volt”, the energy acquired by an electron movingthrough a one-Volt electric potential). The most natural units to use for quan-tum mechanical problems would be energy and time units chosen so that ~ = 1.For instance one could use seconds for time and measure energies in the verysmall units of 6.6 × 10−16 eV, or use eV for energies, and then the very smallunits of 6.6× 10−16 seconds for time. Schrodinger’s equation implies that if oneis looking at a system where the typical energy scale is an eV, one’s state-vectorwill be changing on the very short time scale of 6.6 × 10−16 seconds. Whenwe do computations, usually we will just set ~ = 1, implicitly going to a unitsystem natural for quantum mechanics. When we get our final result, we caninsert appropriate factors of ~ to allow one to get answers in more conventionalunit systems.

It is sometimes convenient however to carry along factors of ~, since thiscan help make clear which terms correspond to classical physics behavior, andwhich ones are purely quantum mechanical in nature. Typically classical physicscomes about in the limit where

(energy scale)(time scale)

~is large. This is true for the energy and time scales encountered in everydaylife, but it can also always be achieved by taking ~ → 0, and this is what willoften be referred to as the “classical limit”.

1.2.2 Principles of measurement theory

The above axioms characterize the mathematical structure of a quantum theory,but they don’t address the “measurement problem”. This is the question ofhow to apply this structure to a physical system interacting with some sortof macroscopic, human-scale experimental apparatus that “measures” what isgoing on. This is highly thorny issue, requiring in principle the study of twointeracting quantum systems (the one being measured, and the measurementapparatus) in an overall state that is not just the product of the two states,but is highly “entangled” (for the meaning of this term, see chapter 9). Since amacroscopic apparatus will involve something like 1023 degrees of freedom, thisquestion is extremely hard to analyze purely within the quantum mechanicalframework (for one thing, one would need to solve a Schrodinger equation in1023 variables).

Instead of trying to resolve in general this problem of how classical physicsbehavior emerges for macroscopic objects, one can adopt the following two prin-ciples as describing what will happen, and these allow one to make precise sta-tistical predictions using quantum theory:

4

Principle (Observables). States where the value of an observable can be char-acterized by a well-defined number are the states that are eigenvectors for thecorresponding self-adjoint operator. The value of the observable in such a statewill be a real number, the eigenvalue of the operator.

This principle identifies the states we have some hope of sensibly associatinga label to (the eigenvalue), a label which in some contexts corresponds to anobservable quantity characterizing states in classical mechanics. The observablesof most use will turn out to correspond to some group action on the physicalsystem (for instance the energy, momentum, angular momentum, or charge).

Principle (The Born rule). Given an observable O and two unit-norm states|ψ1〉 and |ψ2〉 that are eigenvectors of O with eigenvalues λ1 and λ2 (i.e. O|ψ1〉 =λ1|ψ1〉 and O|ψ2〉 = λ2|ψ2〉), the complex linear combination state

c1|ψ1〉+ c2|ψ2〉

may not have a well-defined value for the observable O. If one attempts tomeasure this observable, one will get either λ1 or λ2, with probabilities

|c21||c21|+ |c22|

and|c22|

|c21|+ |c22|respectively.

The Born rule is sometimes raised to the level of an axiom of the theory, butit is plausible to expect that, given a full understanding of how measurementswork, it can be derived from the more fundamental axioms of the previoussection. Such an understanding though of how classical behavior emerges inexperiments is a very challenging topic, with the notion of “decoherence” playingan important role. See the end of this chapter for some references that discussthese issues in detail.

Note that the state c|ψ〉 will have the same eigenvalues and probabilities asthe state |ψ〉, for any complex number c. It is conventional to work with statesof norm fixed to the value 1, which fixes the amplitude of c, leaving a remainingambiguity which is a phase eiθ. By the above principles this phase will notcontribute to the calculated probabilities of measurements. We will howevernot at all take the point of view that this phase information can be ignored. Itplays an important role in the mathematical structure, and the relative phaseof two different states certainly does affect measurement probabilities.

1.3 Unitary group representations

The mathematical framework of quantum mechanics is closely related to whatmathematicians describe as the theory of “unitary group representations”. We

5

will be examining this notion in great detail and working through many examplesin coming chapters, but here’s a quick summary of the relevant definitions, aswell as an indication of the relationship to the quantum theory formalism.

Definition (Group). A group G is a set with an associative multiplication, suchthat the set contains an identity element, as well as the multiplicative inverseof each element.

Many different kinds of groups are of interest in mathematics, with an ex-ample of the sort that we will be interested in the group of all rotations abouta point in 3-dimensional space. Most of the groups we will consider are “matrixgroups”, i.e. subgroups of the group of n by n invertible matrices (with real orcomplex coefficients).

Definition (Representation). A (complex) representation (π, V ) of a group Gis a homomorphism

π : g ∈ G→ π(g) ∈ GL(V )

where GL(V ) is the group of invertible linear maps V → V , with V a complexvector space.

Saying the map π is a homomorphism means

π(g1)π(g2) = π(g1g2)

for all g1, g2 ∈ G. When V is finite dimensional and we have chosen a basis ofV , then we have an identification of linear maps and matrices

GL(V ) ' GL(n,C)

where GL(n,C) is the group of invertible n by n complex matrices. We willbegin by studying representations that are finite dimensional and will try tomake rigorous statements. Later on we will get to representations on functionspaces, which are infinite dimensional, and from then on will need to consider theserious analytical difficulties that arise when one tries to make mathematicallyprecise statements in the infinite-dimensional case.

One source of confusion is that representations (π, V ) are sometimes referredto by the map π, leaving implicit the vector space V that the matrices π(g) acton, but at other times referred to by specifying the vector space V , leavingimplicit the map π. One reason for this is that the map π may be the identitymap: often G is a matrix group, so a subgroup of GL(n,C), acting on V ' Cn

by the standard action of matrices on vectors. One should keep in mind thoughthat just specifying V is generally not enough to specify the representation,since it may not be the standard one. For example, it could very well carry thetrivial representation, where

π(g) = 1n

i.e. each element of G acts on V as the identity.It turns out that in mathematics the most interesting classes of complex

representations are “unitary”, i.e. preserving the notion of length given by

6

the standard Hermitian inner product in a complex vector space. In physicalapplications, the group representations under consideration typically correspondto physical symmetries, and will preserve lengths in H, since these correspondto probabilities of various observations. We have the definition

Definition (Unitary representation). A representation (π, V ) on a complex vec-tor space V with Hermitian inner product 〈·, ·〉 is a unitary representation if itpreserves the inner product, i.e.

〈π(g)v1, π(g)v2〉 = 〈v1, v2〉

for all g ∈ G and v1, v2 ∈ V .

For a unitary representation, the matrices π(g) take values in a subgroupU(n) ⊂ GL(n,C). In our review of linear algebra we will see that U(n) can becharacterized as the group of n by n complex matrices U such that

U−1 = U†

where U† is the conjugate-transpose of U . Note that we’ll be using the notation“†” to mean the “adjoint” or conjugate-transpose matrix. This notation is prettyuniversal in physics, whereas mathematicians prefer to use “∗” instead of “†”.

1.4 Representations and quantum mechanics

The fundamental relationship between quantum mechanics and representationtheory is that whenever we have a physical quantum system with a group Gacting on it, the space of states H will carry a unitary representation of G (atleast up to a phase factor). For physicists working with quantum mechanics,this implies that representation theory provides information about quantummechanical state spaces. For mathematicians studying representation theory,this means that physics is a very fruitful source of unitary representations tostudy: any physical system with a symmetry group G will provide one.

For a representation π and group elements g that are close to the identity,one can use exponentiation to write π(g) ∈ GL(n,C) as

π(g) = eA

where A is also a matrix, close to the zero matrix.We will study this situation in much more detail and work extensively with

examples, showing in particular that if π(g) is unitary (i.e. in the subgroupU(n) ⊂ GL(n,C)), then A will be skew-adjoint:

A† = −A

where A† is the conjugate-transpose matrix. Defining B = iA, we find that Bis self-adjoint

B† = B

7

We thus see that, at least in the case of finite-dimensional H, the unitaryrepresentation π of G on H coming from a symmetry G of our physical sys-tem gives us not just unitary matrices π(g), but also corresponding self-adjointoperators B on H. Symmetries thus give us quantum mechanical observables,with the fact that these are self-adjoint linear operators corresponding to thefact that symmetries are realized as unitary representations on the state space.

In the following chapters we’ll see many examples of this phenomenon. Afundamental example that we will study in detail is that of time-translationsymmetry. Here the group G = R and we get a unitary representation ofR on the space of states H. The corresponding self-adjoint operator is theHamiltonian operator H. This unitary representation gives the dynamics of thetheory, with the Schrodinger equation just the statement that i

~H∆t is the skew-adjoint operator that gets exponentiated to give the unitary transformation thatmoves states ψ(t) ahead in time by an amount ∆t.

1.5 Symmetry groups and their representationson function spaces

It is conventional to refer to the groups that appear in this subject as “symmetrygroups”, which emphasizes the phenomenon of invariance of properties of objectsunder sets of transformations that form a group. This is a bit misleading though,since we are interested in not just invariance, but the more general phenomenonof groups acting on sets, according to the following definition:

Definition (Group action on a set). An action of a group G on a set M isgiven by a map

(g, x) ∈ G×M → g · x ∈M

such thatg1 · (g2 · x) = (g1g2) · x

ande · x = x

where e is the identity element of G

A good example to keep in mind is that of 3-dimensional space M = R3 withthe standard inner product. This comes with an action of the group G = R3 onX = M by translations, and of the group G′ = O(3) of 3-dimensional orthogonaltransformations (by rotations about the origin). Note that order matters: wewill often be interested in non-commutative groups like G′ where g1g2 6= g2g1

for some group elements g1, g2

A fundamental principle of modern mathematics is that the way to under-stand a space X, given as some set of points, is to look at Fun(M), the set offunctions on this space. This “linearizes” the problem, since the function spaceis a vector space, no matter what the geometrical structure of the original setis. If our original set has a finite number of elements, the function space will be

8

a finite dimensional vector space. In general though it will be infinite dimen-sional and we will need to further specify the space of functions (i.e. continuousfunctions, differentiable functions, functions with finite integral, etc.).

Given a group action of G on M , taking complex functions on M providesa representation (π, Fun(M)) of G, with π defined on functions f by

(π(g)f)(x) = f(g−1 · x)

Note the inverse that is needed to get the group homomorphism property towork since one has

(π(g1)π(g2)f)(x) = (π(g2)f)(g−11 · x)

= f(g−12 · (g−1

1 · x))

= f((g−12 g−1

1 ) · x)

= f((g1g2)−1 · x)

= π(g1g2)f(x)

This calculation would not work out properly for non-commutative G if onedefined (π(g)f)(x) = f(g · x).

One way to construct quantum mechanical state spaces H is as “wavefunc-tions”, meaning complex-valued functions on space-time. The above shows thatgiven any group action on space-time, we get a representation π on the statespace H of such wavefunctions.

Note that only in the case of M a finite set of points will we get a finite-dimensional representation this way, since only then will Fun(M) be a finite-dimensional vector space (C# of points in M). A good example to consider tounderstand this construction is the following:

• Take M to be a set of 3 elements x1, x2, x3. So Fun(M) = C3. Forf ∈ Fun(M), f is a vector in C3, with components (f(x1), f(x2), f(x3)).

• Take G = S3, the group of permutations of 3 elements. This group has3! = 6 elements.

• Take G to act on M by permuting the 3 elements.

(g, xi)→ g · xi

where i = 1, 2, 3 gives the three elements of M .

• Find the representation matrices π(g) for the representation of G onFun(M) as above

(π(g)f)(xi) = f(g−1 · xi)

This construction gives six 3 by 3 complex matrices, which under multiplicationof matrices satisfy the same relations as the elements of the group under groupmultiplication. In this particular case, all the entries of the matrix will be 0 or1, but that is special to the permutation representation.

9

The discussion here has been just a quick sketch of some of the ideas behindthe material we will cover in later chapters. These ideas will be examined inmuch greater detail, beginning with the next two chapters, where they willappear very concretely when we discuss the simplest possible quantum systems,those with one and two-complex dimensional state spaces.

1.6 For further reading

We will be approaching the subject of quantum theory from a different direc-tion than the conventional one, starting with the role of symmetry and with thesimplest possible finite-dimensional quantum systems, systems which are purelyquantum mechanical, with no classical analog. This means that the early dis-cussion one finds in most physics textbooks is rather different than the onehere. They will generally include the same fundamental principles describedhere, but often begin with the theory of motion of a quantized particle, tryingto motivate it from classical mechanics. The state space is then a space of wave-functions, which is infinite-dimensional and necessarily brings some analyticaldifficulties. Quantum mechanics is inherently a quite different conceptual struc-ture than classical mechanics. The relationship of the two subjects is rathercomplicated, but it is clear that quantum mechanics cannot be derived fromclassical mechanics, so attempts to motivate it that way are of necessity un-convincing, although they correspond to the very interesting historical story ofhow the subject evolved. We will come to the topic of the quantized motion ofa particle only in chapter 10, at which point it should become much easier tofollow the standard books.

There are many good physics quantum mechanics textbooks available, aimedat a wide variety of backgrounds, and a reader of this book should look for oneat an appropriate level to supplement the discussions here. One example wouldbe [63], which is not really an introductory text, but it includes the physicist’sversion of many of the standard calculations we will also be considering. Someuseful textbooks on the subject aimed at mathematicians are [14], [31], [32], [44],and [71]. The first few chapters of [21] provide an excellent while very concisesummary of both basic physics and quantum mechanics. One important topicwe won’t discuss is that of the application of the representation theory of finitegroups in quantum mechanics. For this as well as a discussion that overlapsquite a bit with the point of view of this course while emphasizing differentareas, see [65].

For the difficult issue of how measurements work and how classical physicsemerges from quantum theory, an important part of the story is the notion of“decoherence”. Good places to read about this are Wojciech Zurek’s updatedversion of his 1991 Physics Today article [87], as well as his more recent workon “quantum Darwinism” [88]. There is an excellent book on the subject bySchlosshauer [58] and for the details of what happens in real experimental setups,see the book by Haroche and Raimond [33]. For a review of how classicalphysics emerges from quantum written from the mathematical point of view,

10

see Landsman [41]. Finally, to get an idea of the wide variety of points of viewavailable on the topic of the “interpretation” of quantum mechanics, there’s avolume of interviews [59] with experts on the topic.

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Chapter 2

The Group U(1) and itsRepresentations

The simplest example of a Lie group is the group of rotations of the plane,with elements parametrized by a single number, the angle of rotation θ. It isuseful to identify such group elements with unit vectors in the complex plane,given by eiθ. The group is then denoted U(1), since such complex numberscan be thought of as 1 by 1 unitary matrices . We will see in this chapter howthe general picture described in chapter 1 works out in this simple case. Statespaces will be unitary representations of the group U(1), and we will see that anysuch representation decomposes into a sum of one-dimensional representations.These one-dimensional representations will be characterized by an integer q, andsuch integers are the eigenvalues of a self-adjoint operator we will call Q, whichis an observable of the quantum theory.

One motivation for the notation Q is that this is the conventional physicsnotation for electric charge, and this is one of the places where a U(1) groupoccurs in physics. Examples of U(1) groups acting on physical systems include:

• Quantum particles can be described by a complex-valued “wavefunction”,and U(1) acts on such wavefunctions by phase transformations of thevalue of the function. This phenomenon can be used to understand howparticles interact with electromagnetic fields, and in this case the physicalinterpretation of the eigenvalues of the Q operator will be the electriccharge of the particle. We will discuss this in detail in chapter 43.

• If one chooses a particular direction in three-dimensional space, then thegroup of rotations about that axis can be identified with the group U(1).The eigenvalues of Q will have a physical interpretation as the quantumversion of angular momentum in the chosen direction. The fact that sucheigenvalues are not continuous, but integral, shows that quantum angularmomentum has quite different behavior than classical angular momentum.

• When we study the harmonic oscillator we will find that it has a U(1) sym-

13

metry (rotations in the position-momentum plane), and that the Hamil-tonian operator is a multiple of the operator Q for this case. This im-plies that the eigenvalues of the Hamiltonian (which give the energy ofthe system) will be integers times some fixed value. When one describesmulti-particle systems in terms of quantum fields one finds a harmonicoscillator for each momentum mode, and then the Q for that mode countsthe number of particles with that momentum.

We will sometimes refer to the operator Q as a “charge” operator, assigninga much more general meaning to the term than that of the specific example ofelectric charge. U(1) representations are also ubiquitous in mathematics, whereoften the integral eigenvalues of the Q operator will be called “weights”.

In a very real sense, the reason for the “quantum” in “quantum mechanics”is precisely because of the role of U(1) symmetries. Such symmetries implyobservables that characterize states by an integer eigenvalue of an operator Q,and it is this “quantization” of observables that motivates the name of thesubject.

2.1 Some representation theory

Recall the definition of a group representation:

Definition (Representation). A (complex) representation (π, V ) of a group Gon a complex vector space V (with a chosen basis identifying V ' Cn) is ahomomorphism

π : G→ GL(n,C)

This is just a set of n by n matrices, one for each group element, satisfyingthe multiplication rules of the group elements. n is called the dimension of therepresentation.

The groups G we are interested in will be examples of what mathematicianscall “Lie groups”. For those familiar with differential geometry, such groupsare examples of smooth manifolds. This means one can define derivatives offunctions on G and more generally the derivative of maps between Lie groups.We will assume that our representations are given by differentiable maps π.Some difficult general theory shows that considering the more general case ofcontinuous maps gives nothing new since the homomorphism property of thesemaps is highly constraining. In any case, our goal in this course will be to studyquite explicitly certain specific groups and representations which are central inquantum mechanics, and these representations will always be easily seen to bedifferentiable.

Given two representations one can form their direct sum:

Definition (Direct sum representation). Given representations π1 and π2 ofdimensions n1 and n2, one can define another representation, of dimensionn1 + n2 called the direct sum of the two representations, denoted by π1 ⊕ π2.

14

This representation is given by the homomorphism

(π1 ⊕ π2) : g ∈ G→(π1(g) 0

0 π2(g)

)In other words, one just takes as representation matrices block-diagonal

matrices with π1 and π2 giving the blocks.To understand the representations of a group G, one proceeds by first iden-

tifying the irreducible ones, those that cannot be decomposed into two repre-sentations of lower dimension:

Definition (Irreducible representation). A representation π is called irreducibleif it is not of the form π1⊕π2, for π1 and π2 representations of dimension greaterthan zero.

This criterion is not so easy to check, and the decomposition of an arbitraryreducible representation into irreducible components can be a very non-trivialproblem. Recall that one gets explicit matrices for the π(g) of a representation(π, V ) only when a basis for V is chosen. To see if the representation is reducible,one can’t just look to see if the π(g) are all in block-diagonal form. One needsto find out whether there is some basis for V with respect to which they areall in such form, something very non-obvious from just looking at the matricesthemselves.

Digression. Another approach to this would be to check to see if the represen-tation has no proper non-trivial sub-representations (subspaces of V preservedby the π(g)). This is not necessarily equivalent to our definition of irreducibil-ity (which is often called “indecomposability”), since a sub-representation mayhave no complement that is also a sub-representation. A simple example ofthis occurs for the action of upper triangular matrices on column vectors. Suchrepresentations are however non-unitary. In the unitary case indecomposabilityand irreducibility are equivalent. In these notes unless otherwise specified, oneshould assume that all representations are unitary, so the distinction betweenirreducibility and indecomposability will generally not arise.

The following theorem provides a criterion for determining if a representationis irreducible or not:

Theorem (Schur’s lemma). If a complex representation (π, V ) is irreducible,then the only linear maps M : V → V commuting with all the π(g) are λ1,multiplication by a scalar λ ∈ C.

Proof. Since we are working over the field C (this doesn’t work for R), we canalways solve the eigenvalue equation

det(M − λ1) = 0

to find the eigenvalues λ of M . The eigenspaces

Vλ = v ∈ V : Mv = λv

15

are non-zero vector subspaces of V and can also be described as ker(M − λ1),the kernel of the operator M−λ1. Since this operator and all the π(g) commute,we have

v ∈ ker(M − λ1) =⇒ π(g)v ∈ ker(M − λ1)

so ker(M − λ1) ⊂ V is a representation of G. If V is irreducible, we musthave either ker(M − λ1) = V or ker(M − λ1) = 0. Since λ is an eigenvalue,ker(M − λ1) 6= 0, so ker(M − λ1) = V and thus M = λ1 as a linear operatoron V .

More concretely Schur’s lemma says that for an irreducible representation, ifa matrix M commutes with all the representation matrices π(g), then M mustbe a scalar multiple of the unit matrix.

Note that the proof crucially uses the fact that one can solve the eigenvalueequation. This will only be true in general if one works with C and thus withcomplex representations. For the theory of representations on real vector spaces,Schur’s lemma is no longer true.

An important corollary of Schur’s lemma is the following characterization ofirreducible representations of G when G is commutative.

Theorem. If G is commutative, all of its irreducible representations are one-dimensional.

Proof. For G commutative, g ∈ G, any representation will satisfy

π(g)π(h) = π(h)π(g)

for all h ∈ G. If π is irreducible, Schur’s lemma implies that, since they commutewith all the π(g), the matrices π(h) are all scalar matrices, i.e. π(h) = λh1 forsome λh ∈ C. π is then irreducible exactly when it is the one-dimensionalrepresentation given by π(h) = λh.

2.2 The group U(1) and its representations

One can think of the group U(1) as the unit circle, with the multiplication ruleon its points given by addition of angles. More explicitly:

Definition (The group U(1)). The elements of the group U(1) are points onthe unit circle, which can be labeled by the unit complex number eiθ, for θ ∈ R.Note that θ and θ+N2π label the same group element for N ∈ Z. Multiplicationof group elements is just complex multiplication, which by the properties of theexponential satisfies

eiθ1eiθ2 = ei(θ1+θ2)

so in terms of angles the group law is just addition (mod 2π).

16

By our theorem from the last section, since U(1) is a commutative group,all irreducible representations will be one-dimensional. Such an irreducible rep-resentation will be given by a map

π : U(1)→ GL(1,C)

but an invertible 1 by 1 matrix is just an invertible complex number, and we willdenote the group of these as C∗. We will always assume that our representationsare given by differentiable maps, since we will often want to study them in termsof their derivatives. A differentiable map π that is a representation of U(1) mustsatisfy homomorphism and periodicity properties which can be used to show:

Theorem 2.1. All irreducible representations of the group U(1) are unitary,and given by

πk : θ ∈ U(1)→ πk(θ) = eikθ ∈ U(1) ⊂ GL(1,C) ' C∗

for k ∈ Z.

Proof. The given πk satisfy the homomorphism property

πk(θ1 + θ2) = πk(θ1)πk(θ2)

and periodicity property

πk(2π) = πk(0) = 1

We just need to show that any differentiable map

f : U(1)→ C∗

satisfying the homomorphism and periodicity properties is of this form. Com-puting the derivative f ′(θ) = df

dθ we find

f ′(θ) = lim∆θ→0

f(θ + ∆θ)− f(θ)

∆θ

= f(θ) lim∆θ→0

(f(∆θ)− 1)

∆θ(using the homomorphism property)

= f(θ)f ′(0)

Denoting the constant f ′(0) by C, the only solutions to this differential equationsatisfying f(0) = 1 are

f(θ) = eCθ

Requiring periodicity we find

f(2π) = eC2π = f(0) = 1

which implies C = ik for k ∈ Z, and f = πk for some integer k.

17

The representations we have found are all unitary, with πk taking values notjust in C∗, but in U(1) ⊂ C∗. One can check that the complex numbers eikθ

satisfy the condition to be a unitary 1 by 1 matrix, since

(eikθ)−1 = e−ikθ = eikθ

These representations are restrictions to the unit circle U(1) of the irre-ducible representations of the group C∗, which are given by

πk : z ∈ C∗ → πk(z) = zk ∈ C∗

Such representations are not unitary, but they have an extremely simple form,so it sometimes is convenient to work with them, later restricting to the unitcircle, where the representation is unitary.

Digression (Fourier analysis of periodic functions). We’ll discuss Fourier anal-ysis more seriously in chapter 10 when we come to the case of the translationgroups and of state-spaces that are spaces of “wavefunctions” on space-time. Fornow though, it might be worth pointing out an important example of a repre-sentation of U(1): the space Fun(S1) of complex-valued functions on the circleS1. We will evade discussion here of the very non-trivial analysis involved, bynot specifying what class of functions we are talking about (e.g. continuous,integrable, differentiable, etc.). Periodic functions can be studied by rescalingthe period to 2π, thus looking at complex-valued functions of a real variable φsatisfying

f(φ+N2π) = f(φ)

for integer N , which we can think of as functions on a circle, parametrized byangle φ. We have an action of the group U(1) on the circle by rotation, withthe group element eiθ acting as:

φ→ φ+ θ

where φ is the angle parametrizing the circle S1.In chapter 1 we saw that given an action of a group on a space X, we can

“linearize” and get a representation (π, Fun(X)) of the group on the functionson the space, by taking

(π(g)f)(x) = f(g−1 · x)

for f ∈ Fun(X), x ∈ X. Here X = S1, the action is the rotation action and wefind

(π(θ)f)(φ) = f(φ− θ)since the inverse of a rotation by θ is a rotation by −θ.

This representation (π, Fun(S1)) is infinite-dimensional, but one can stillask how it decomposes into the one-dimensional irreducible representations (πk,C)of U(1). What we learn from the subject of Fourier analysis is that each (πk,C)occurs exactly once in the decomposition of Fun(S1) into irreducibles, i.e.

(π, Fun(S1)) =⊕

k∈Z(πk,C)

18

where we have matched the sin of not specifying the class of functions in Fun(S1)on the left-hand side with the sin of not explaining how to handle the infinite

direct sum⊕

on the right-hand side. What can be specified precisely is howthe irreducible sub-representation (πk,C) sits inside Fun(S1). It is the set offunctions f satisfying

(π(θ)f)(φ) = f(φ− θ) = eikθf(φ)

so explicitly given by the one-complex dimensional space of functions propor-tional to e−ikφ.

One part of the relevance of representation theory to Fourier analysis isthat the representation theory of U(1) picks out a distinguished basis of theinfinite-dimensional space of periodic functions by using the decomposition ofthe function space into irreducible representations. One can then effectivelystudy functions by expanding them in terms of their components in this specialbasis, writing an f ∈ Fun(S1) as

f(φ) =∑k∈Z

ckeikφ

for some complex coefficients ck, with analytical difficulties then appearing asquestions about the convergence of this series.

2.3 The charge operator

Recall from chapter 1 the claim of a general principle that, since the state spaceH is a unitary representation of a Lie group, we get an associated self-adjointoperator on H. We’ll now illustrate this for the simple case of G = U(1). For Hirreducible, the representation is one-dimensional, of the form (πq,C) for someq ∈ Z, and the self-adjoint operator will just be multiplication by the integer q.In general, we have

H = Hq1 ⊕Hq2 ⊕ · · · ⊕ Hqnfor some set of integers q1, q2, . . . , qn (n is the dimension of H, the qi may notbe distinct) and can define:

Definition. The charge operator Q for the U(1) representation (π,H) is theself-adjoint linear operator on H that acts by multiplication by qj on the irre-ducible representation Hqj . Taking basis elements in Hqj it acts on H as thematrix

q1 0 · · · 00 q2 · · · 0· · · · · ·0 0 · · · qn

Q is our first example of a quantum mechanical observable, a self-adjoint

operator on H. States in the subspaces Hqj will be eigenvectors for Q and will

19

have a well-defined numerical value for this observable, the integer qj . A generalstate will be a linear superposition of state vectors from different Hqj and therewill not be a well-defined numerical value for the observable Q on such a state.

From the action of Q onH, one can recover the representation, i.e. the actionof the symmetry group U(1) on H, by multiplying by i and exponentiating, toget

π(θ) = eiQθ =

eiq1θ 0 · · · 0

0 eiq2θ · · · 0· · · · · ·0 0 · · · eiqnθ

∈ U(n) ⊂ GL(n,C)

The standard physics terminology is that “Q generates the U(1) symmetrytransformations”.

The general abstract mathematical point of view (which we will discuss inmore detail later) is that the representation π is a map between manifolds,from the Lie group U(1) to the Lie group GL(n,C) that takes the identity ofU(1) to the identity of GL(n,C). As such it has a differential π′, which is amap from the tangent space at the identity of U(1) (which here is iR) to thetangent space at the identity of GL(n,C) (which is the space M(n,C) of n byn complex matrices). The tangent space at the identity of a Lie group is calleda “Lie algebra”. We will later study these in detail in many examples to come,including their role in specifying a representation.

Here the relation between the differential of π and the operator Q is

π′ : iθ ∈ iR→ π′(iθ) = iQθ

One can sketch the situation like this:

20

The right-hand side of the picture is supposed to somehow representGL(n,C),which is the 2n2 dimensional real vector space of n by n complex matrices, mi-nus the locus of matrices with zero determinant, which are those that can’t beinverted. It has a distinguished point, the identity. The derivative π′ of therepresentation map π is the linear operator iQ.

In this very simple example, this abstract picture is over-kill and likely con-fusing. We will see the same picture though occurring in many other examplesin later chapters, examples where the abstract technology is increasingly useful.Keep in mind that, just like in this U(1) case, the maps π will just be exponen-tial maps in the examples we care about, with very concrete incarnations givenby exponentiating matrices.

2.4 Conservation of charge and U(1) symmetry

The way we have defined observable operators in terms a group representationon H, the action of these operators has nothing to do with the dynamics. Ifwe start at time t = 0 in a state in Hqj , with definite numerical value qj forthe observable, there is no reason that time evolution should preserve this.Recall from one of our basic axioms that time evolution of states is given by theSchrodinger equation

d

dt|ψ(t)〉 = −iH|ψ(t)〉

(we have set ~ = 1). We will later more carefully study the relation of thisequation to the symmetry of time translation (basically the Hamiltonian op-erator H generates an action of the group R of time translations, just as theoperator Q generates an action of the group U(1)). For now though, note thatfor time-independent Hamiltonian operators H, the solution to this equation isgiven by exponentiating H, with

|ψ(t)〉 = U(t)|ψ(0)〉

for

U(t) = e−itH

The commutator of two operators O1, O2 is defined by

[O1, O2] := O1O2 −O2O1

and such operators are said to commute if [O1, O2] = 0. If the Hamiltonianoperator H and the charge operator Q commute then Q will also commute withall powers of H

[Hk, Q] = 0

and thus with the exponential of H, so

[U(t), Q] = 0

21

This conditionU(t)Q = QU(t) (2.1)

implies that if a state has a well-defined value qj for the observable Q at timet = 0, it will continue to have the same value at any other time t, since

Q|ψ(t)〉 = QU(t)|ψ(0)〉 = U(t)Q|ψ(0)〉 = U(t)qj |ψ(0)〉 = qj |ψ(t)〉

This will be a general phenomenon: if an observable commutes with the Hamil-tonian observable, we get a conservation law. This conservation law says thatif we start in a state with a well-defined numerical value for the observable (aneigenvector for the observable operator), we will remain in such a state, withthe value not changing, i.e. “conserved”.

In this situation, the group U(1) is said to act as a “symmetry group” of thesystem. Equation 2.1 implies that

U(t)eiQθ = eiQθU(t)

so the action of the U(1) group on the state space of the system commutes withthe time evolution law determined by the choice of Hamiltonian. We see thatthis notion of symmetry implies a corresponding conservation law.

2.5 Summary

To summarize the situation for G = U(1), we have found

• Irreducible representations π are one-dimensional and characterized bytheir derivative π′ at the identity. If G = R, π′ could be any complexnumber. If G = U(1), periodicity requires that π′ must be iq, q ∈ Z, soirreducible representations are labeled by an integer.

• An arbitrary representation π of U(1) is of the form

π(eiθ) = eiθQ

where Q is a matrix with eigenvalues a set of integers qj . For a quan-tum system, Q is the self-adjoint observable corresponding to the U(1)symmetry of the system, and is said to be a “generator” of the symmetry.

• If [Q,H] = 0, the U(1) group acts on the state space as “symmetries”. Inthis case the qj will be “conserved quantities”, numbers that characterizethe quantum states, and do not change as the states evolve in time.


I’ve had trouble finding another source that covers the material here. Mostquantum mechanics books consider it somehow too trivial to mention, startingtheir discussion of symmetries with more complicated examples.

22

Chapter 3

Two-state Systems andSU(2)

The simplest truly non-trivial quantum systems have state spaces that are in-herently two-complex dimensional. This provides a great deal more structurethan that seen in chapter 2, which could be analyzed by breaking up the spaceof states into one-dimensional subspaces of given charge. We’ll study these two-state systems in this section, encountering for the first time the implications ofworking with representations of non-commutative groups. Since they give thesimplest non-trivial realization of many quantum phenomena, such systems arethe fundamental objects of quantum information theory (the “qubit”) and thefocus of attempts to build a quantum computer (which would be built out ofmultiple copies of this sort of fundamental object). Many different possible two-state quantum systems could potentially be used as the physical implementationof a qubit.

One of the simplest possibilities to take would be the idealized situationof a single electron, somehow fixed so that its spatial motion could be ignored,leaving its quantum state described just by its so-called “spin degree of freedom”,which takes values in H = C2. The term “spin” is supposed to call to mindthe angular momentum of an object spinning about about some axis, but suchclassical physics has nothing to do with the qubit, which is a purely quantumsystem.

In this chapter we will analyze what happens for general quantum systemswith H = C2 by first finding the possible observables. Exponentiating thesewill give the group U(2) of unitary 2 by 2 matrices acting on H = C2. Thisis a specific representation of U(2), the “defining” representation. By restrict-ing to the subgroup SU(2) ⊂ U(2) of elements of determinant one, we get arepresentation of SU(2) on C2 often called the “spin 1/2” representation.

Later on, in chapter 8, we will find all the irreducible representations ofSU(2). These are labeled by a natural number

N = 0, 1, 2, 3, . . .

23

and have dimension N + 1. The corresponding quantum systes are said to have“spin N/2”. The case N = 0 is the trivial representation on C and the caseN = 1 is the case of this chapter. In the limit N → ∞ one can make contactwith classical notions of spinning objects and angular momentum, but the spin1/2 case is at the other limit, where the behavior is purely quantum-mechanical.

3.1 The two-state quantum system

3.1.1 The Pauli matrices: observables of the two-statequantum system

For a quantum system with two-dimensional state space H = C2, observablesare self-adjoint linear operators on C2. With respect to a chosen basis of C2,these are 2 by 2 complex matricesM satisfying the conditionM = M† (M† is theconjugate transpose of M). Any such matrix will be a (real) linear combinationof four matrices:

M = c01 + c1σ1 + c2σ2 + c3σ3

with cj ∈ R and the standard choice of basis elements given by

1 =

(1 00 1

), σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

)The σj are called the “Pauli matrices” and are a pretty universal choice of basisin this subject. This choice of basis is a convention, with one aspect of thisconvention that of taking the basis element in the 3-direction to be diagonal.In common physical situations and conventions, the third direction is the dis-tinguished “up-down” direction in space, so often chosen when a distinguisheddirection in R3 is needed.

Recall that the basic principle of how measurements are supposed to workin quantum theory says that the only states that have well-defined values forthese four observables are the eigenvectors for these matrices. The first matrixgives a trivial observable (the identity on every state), whereas the last one, σ3,has the two eigenvectors

σ3

(10

)=

(10

)and

σ3

(01

)= −

(01

)with eigenvalues +1 and −1. In quantum information theory, where this isthe qubit system, these two eigenstates are labeled |0〉 and |1〉 because of theanalogy with a classical bit of information. Later on when we get to the theory ofspin, we will see that 1

2σ3 is the observable corresponding to the SO(2) = U(1)symmetry group of rotations about the third spatial axis, and the eigenvalues

24

− 12 ,+

12 of this operator will be used to label the two eigenstates

|+ 1

2〉 =

(10

)and | − 1

2〉 =

(01

)The two eigenstates |+ 1

2 〉 and | − 12 〉 provide a basis for C2, so an arbitrary

vector in H can be written as

|ψ〉 = α|+ 1

2〉+ β| − 1

2〉

for α, β ∈ C. Only if α or β is 0 does the observable σ3 correspond to a well-defined number that characterizes the state and can be measured. This will beeither 1

2 (if β = 0 so the state is an eigenvector |+ 12 〉), or − 1

2 (if α = 0 so thestate is an eigenvector | − 1

2 〉).An easy to check fact is that |+ 1

2 〉 and | − 12 〉 are NOT eigenvectors for the

operators σ1 and σ2. One can also check that no pair of the three σj commute,which implies that one cannot find vectors that are simultaneous eigenvectors formore than one σj . This non-commutativity of the operators is responsible for thecharacteristic classically paradoxical property of quantum observables: one canfind states with a well defined number for the measured value of one observableσj , but such states will not have a well-defined number for the measured valueof the other two non-commuting observables. The physical description of thisphenomenon in the realization of this system as a spin 1

2 particle is that if oneprepares states with a well-defined spin component in the j-direction, the twoother components of the spin can’t be assigned a numerical value in such astate. Any attempt to prepare states that simultaneously have specific chosennumerical values for the 3 observables corresponding to the σj is doomed. So isany attempt to simultaneously measure such values: if one measures the valuefor a particular observable σj , then going on to measure one of the other twowill ensure that the first measurement is no longer valid (repeating it will notnecessarily give the same thing). There are many subtleties in the theory ofmeasurement for quantum systems, but this simple two-state example alreadyshows some of the main features of how the behavior of observables is quitedifferent than in classical physics.

The choice we have made for the σj corresponds to a choice of basis for Hsuch that the basis vectors are eigenvectors of σ3. σ1 and σ2 take these basisvectors to non-trivial linear combinations of basis vectors. It turns out thatthere are two specific linear combinations of σ1 and σ2 that do something verysimple to the basis vectors, since

(σ1 + iσ2) =

(0 20 0

)and (σ1 − iσ2) =

(0 02 0

)we have

(σ1 + iσ2)

(01

)= 2

(10

)(σ1 + iσ2)

(10

)=

(00

)25

and

(σ1 − iσ2)

(10

)= 2

(01

)(σ1 − iσ2)

(01

)=

(00

)(σ1 + iσ2) is called a “raising operator”: on eigenvectors of σ3 it either

increases the eigenvalue by 2, or annihilates the vector. (σ1 − iσ2) is calleda “lowering operator”: on eigenvectors of σ3 it either decreases the eigenvalueby 2, or annihilates the vector. Note that these linear combinations are notself-adjoint, (σ1 + iσ2) is the adjoint of (σ1 − iσ2) and vice-versa.

3.1.2 Exponentials of Pauli matrices: unitary transforma-tions of the two-state system

We saw in chapter 2 that in the U(1) case, knowing the observable operator Q onH determined the representation of U(1), with the representation matrices foundby exponentiating iθQ. Here we will find the representation corresponding tothe two-state system observables by exponentiating the observables in a similarway.

Taking the the identity matrix first, multiplication by iθ and exponentiationgives the diagonal unitary matrix

eiθ1 =

(eiθ 00 eiθ

)This is just exactly the case studied in chapter 2, for a U(1) group acting onH = C2, with

Q =

(1 00 1

)This matrix commutes with any other 2 by 2 matrix, so we can treat its actionon H independently of the action of the σj .

Turning to the other three basis elements of the space of observables, thePauli matrices, it turns out that since all the σj satisfy σ2

j = 1, their exponentialsalso take a simple form.

eiθσj = 1 + iθσj +1

2(iθ)2σ2

j +1

3!(iθ)3σ3

j + · · ·

= 1 + iθσj −1

2θ21− i 1

3!θ3σj + · · ·

= (1− 1

2!θ2 + · · · )1 + i(θ − 1

3!θ3 + · · · )σj

= (cos θ)1 + iσj(sin θ) (3.1)

As θ goes from θ = 0 to θ = 2π, this exponential traces out a circle in thespace of unitary 2 by 2 matrices, starting and ending at the unit matrix. Thiscircle is a group, isomorphic to U(1). So, we have found three different U(1)

26

subgroups inside the unitary 2 by 2 matrices, but only one of them (the casej = 3) will act diagonally on H, with the U(1) representation determined by

Q =

(1 00 −1

)For the other two cases j = 1 and j = 2, by a change of basis one could puteither one in the same diagonal form, but doing this for one value of j makesthe other two no longer diagonal. All three values of j need to be treatedsimultaneously, and one needs to consider not just the U(1)s but the group onegets by exponentiating general linear combinations of Pauli matrices.

To compute such exponentials, one can check that these matrices satisfy thefollowing relations, useful in general for doing calculations with them instead ofmultiplying out explicitly the 2 by 2 matrices:

[σj , σk]+ = σjσk + σkσj = 2δjk1

Here [·, ·]+ is the anticommutator. This relation says that all σj satisfy σ2j = 1

and distinct σj anticommute (e.g. σjσk = −σkσj for j 6= k).Notice that the anticommutation relations imply that, if we take a vector

v = (v1, v2, v3) ∈ R3 and define a 2 by 2 matrix by

v · σ = v1σ1 + v2σ2 + v3σ3 =

(v3 v1 − iv2

v1 + iv2 −v3

)then taking powers of this matrix we find

(v · σ)2 = (v21 + v2

2 + v23)1 = |v|21

If v is a unit vector, we have

(v · σ)n =

1 n even

(v · σ) n odd

Replacing σj by v · σ, the same calculation as for equation 3.1 gives (for va unit vector)

eiθv·σ = (cos θ)1 + i(sin θ)v · σ

Notice that one can easily compute the inverse of this matrix:

(eiθv·σ)−1 = (cos θ)1− i(sin θ)v · σ

since

((cos θ)1 + i(sin θ)v · σ)((cos θ)1− i(sin θ)v · σ) = (cos2 θ + sin2 θ)1 = 1

We’ll review linear algebra and the notion of a unitary matrix in chapter 4, butone form of the condition for a matrix M to be unitary is

M† = M−1

27

so the self-adjointness of the σj implies unitarity of eiθv·σ since

(eiθv·σ)† = ((cos θ)1 + i(sin θ)v · σ)†

= ((cos θ)1− i(sin θ)v · σ†)= ((cos θ)1− i(sin θ)v · σ)

= (eiθv·σ)−1

One can also easily compute the determinant of eiθv·σ, finding

det(eiθv·σ) = det((cos θ)1 + i(sin θ)v · σ)

= det

(cos θ + i sin θv3 i sin θ(v1 − iv2)i sin θ(v1 + iv2) cos θ − i sin θv3

)= cos2 θ + sin2 θ(v2

1 + v22 + v2

3)

= 1

So, we see that by exponentiating i times linear combinations of the self-adjoint Pauli matrices (which all have trace zero), we get unitary matrices ofdeterminant one. These are invertible, and form the group named SU(2), thegroup of unitary 2 by 2 matrices of determinant one. If we exponentiated notjust iθv · σ, but i(φ1 + θv · σ) for some real constant φ (such matrices will nothave trace zero unless φ = 0), we would get a unitary matrix with determinantei2φ. The group of unitary 2 by 2 matrices with arbitrary determinant is calledU(2). It contains as subgroups SU(2) as well as the U(1) described at thebeginning of this section. U(2) is slightly different than the product of thesetwo subgroups, since the group element(

−1 00 −1

)is in both subgroups. In our review of linear algebra to come we will encounterthe generalization to SU(n) and U(n), groups of unitary n by n complex ma-trices.

To get some more insight into the structure of the group SU(2), consider anarbitrary 2 by 2 complex matrix (

α βγ δ

)Unitarity implies that the rows are orthonormal. One can see this explicitlyfrom the condition that the matrix times its conjugate-transpose is the identity(

α βγ δ

)(α γ

β δ

)=

(1 00 1

)Orthogonality of the two rows gives the relation

γα+ δβ = 0 =⇒ δ = −γαβ

28

The condition that the first row has length one gives

αα+ ββ = |α|2 + |β|2 = 1

Using these two relations and computing the determinant (which has to be 1)gives

αδ − βγ = −ααγβ− βγ = −γ

β(αα+ ββ) = −γ

β= 1

so one must haveγ = −β, δ = α

and an SU(2) matrix will have the form(α β

−β α

)where (α, β) ∈ C2 and

|α|2 + |β|2 = 1

So, the elements of SU(2) are parametrized by two complex numbers, withthe sum of their length-squareds equal to one. Identifying C2 = R4, these arejust vectors of length one in R4. Just as U(1) could be identified as a space withthe unit circle S1 in C = R2, SU(2) can be identified with the unit three-sphereS3 in R4.

3.2 Commutation relations for Pauli matrices

An important set of relations satisfied by Pauli matrices are their commutationrelations:

[σj , σk] = σjσk − σkσj = 2i

3∑l=1

εjklσl

where εjkl satisfies ε123 = 1, is antisymmetric under permutation of two of itssubscripts, and vanishes if two of the subscripts take the same value. Moreexplicitly, this says:

[σ1, σ2] = 2iσ3, [σ2, σ3] = 2iσ1, [σ3, σ1] = 2iσ2

One can easily check these relations by explicitly computing with the matrices.Putting together the anticommutation and commutation relations, one gets aformula for the product of two Pauli matrices:

σjσk = δjk1 + i

3∑l=1

εjklσl

While physicists prefer to work with self-adjoint Pauli matrices and theirreal eigenvalues, one can work instead with the following skew-adjoint matrices

Xj = −iσj2

29

which satisfy the slightly simpler commutation relations

[Xj , Xk] =

3∑l=1

εjklXl

or more explicitly

[X1, X2] = X3, [X2, X3] = X1, [X3, X1] = X2

If these commutators were zero, the SU(2) elements one gets by exponentiat-ing linear combinations of the Xj would be commuting group elements. Thenon-triviality of the commutators reflects the non-commutativity of the group.Group elements U ∈ SU(2) near the identity satisfy

U ' 1 + ε1X1 + ε2X2 + ε3X2

for εj small and real, just as group elements z ∈ U(1) near the identity satisfy

z ' 1 + iε

One can think of the Xj and their commutation relations as an infinites-imal version of the full group and its group multiplication law, valid nearthe identity. In terms of the geometry of manifolds, recall that SU(2) is thespace S3. The Xj give a basis of the tangent space R3 to the identity ofSU(2), just as i gives a basis of the tangent space to the identity of U(1).

30

3.3 Dynamics of a two-state system

Recall that the time dependence of states in quantum mechanics is given by theSchrodinger equation

d

dt|ψ(t)〉 = −iH|ψ(t)〉

where H is a particular self-adjoint linear operator on H, the Hamiltonian op-erator. The most general such operator on C2 will be given by

H = h01 + h1σ1 + h2σ2 + h3σ3

for four real parameters h0, h1, h2, h3. The solution to the Schrodinger equationis just given by exponentiation:

|ψ(t)〉 = U(t)|ψ(0)〉

whereU(t) = e−itH

The h01 term in H just contributes an overall phase factor e−ih0t, with theremaining factor of U(t) an element of the group SU(2) rather than the largergroup U(2) of all 2 by 2 unitaries.

Using our earlier equation

eiθv·σ = (cos θ)1 + i(sin θ)v · σ

valid for a unit vector v, our U(t) is given by taking h = (h1, h2, h3), v = h|h|

and θ = −t|h|, so we find

U(t) =e−ih0t(cos(−t|h|)1 + i sin(−t|h|)h1σ1 + h2σ2 + h3σ3

|h|)

=e−ih0t(cos(t|h|)1− i sin(t|h|)h1σ1 + h2σ2 + h3σ3

|h|)

=

(e−ih0t(cos(t|h|)− i h3

|h| sin(t|h|)) −i sin(t|h|)h1−ih2

|h|−i sin(t|h|)h1+ih2

|h| e−ih0t(cos(t|h|) + i h3

|h| sin(t|h|))

)In the special case h = (0, 0, h3) we have

U(t) =

(e−it(h0+h3) 0

0 e−it(h0−h3)

)so if our initial state is

|ψ(0)〉 = α|+ 1

2〉+ β| − 1

2〉

for α, β ∈ C, at later times the state will be

|ψ(t)〉 = αe−it(h0+h3)|+ 1

2〉+ βe−it(h0−h3)| − 1

2〉

31

In this special case, one can see that the eigenvalues of the Hamiltonian areh0 ± h3.

In the physical realization of this system by a spin 1/2 particle (ignoring itsspatial motion), the Hamiltonian is given by

H =ge

4mc(B1σ1 +B2σ2 +B3σ3)

where the Bj are the components of the magnetic field, and the physical con-stants are the gyromagnetic ratio (g), the electric charge (e), the mass (m) andthe speed of light (c), so we have solved the problem of the time evolution ofsuch a system, setting hj = ge

4mcBj . For magnetic fields of size |B| in the 3-direction, we see that the two different states with well-defined energy (| + 1

2 〉and | − 1

2 〉) will have an energy difference between them of

ge

2mc|B|

This is known as the Zeeman effect and is readily visible in the spectra of atomssubjected to a magnetic field. We will consider this example in more detail inchapter 7, seeing how the group of rotations of R3 appears. Much later, inchapter 43, we will derive this Hamiltonian term from general principles of howelectromagnetic fields couple to such spin 1/2 particles.


Many quantum mechanics textbooks now begin with the two-state system, giv-ing a much more detailed treatment than the one given here, including muchmore about the physical interpretation of such systems (see for example [75]).Volume III of Feynman’s Lectures on Physics [17] is a quantum mechanics textwith much of the first half devoted to two-state systems. The field of “QuantumInformation Theory” gives a perspective on quantum theory that puts such sys-tems (in this context called the “qubit”) front and center. One possible referencefor this material is John Preskill’s notes on quantum computation [53].

32

Chapter 4

Linear Algebra Review,Unitary and OrthogonalGroups

A significant background in linear algebra will be assumed in later chapters,and we’ll need a range of specific facts from that subject. These will includesome aspects of linear algebra not emphasized in a typical linear algebra course,such as the role of the dual space and the consideration of various classes ofinvertible matrices as defining a group. For now our vector spaces will be finite-dimensional. Later on we will come to state spaces that are infinite dimensional,and will address the various issues that this raises at that time.

4.1 Vector spaces and linear maps

A vector space V over a field k is just a set such that one can consistently takelinear combinations of elements with coefficients in k. We will only be usingthe cases k = R and k = C, so such finite-dimensional V will just be Rn orCn. Choosing a basis (set of n linearly independent vectors) ej, an arbitraryvector v ∈ V can be written as

v = v1e1 + v2e2 + · · ·+ vnen

giving an explicit identification of V with n-tuples vj of real or complex numberswhich we will usually write as column vectors

v =

v1

v2

...vn

33

The choice of a basis ej also allows us to express the action of a linearoperator Ω on V

Ω : v ∈ V → Ωv ∈ Vas multiplication by an n by n matrix:

v1

v2

...vn

→

Ω11 Ω12 . . . Ω1n

Ω21 Ω22 . . . Ω2n

......

......

Ωn1 Ωn2 . . . Ωnn

v1

v2

...vn

The invertible linear operators on V form a group under composition, a

group we will sometimes denote GL(V ). Choosing a basis identifies this groupwith the group of invertible matrices, with group law matrix multiplication.For V n-dimensional, we will denote this group by GL(n,R) in the real case,GL(n,C) in the complex case.

Note that when working with vectors as linear combinations of basis vectors,we can use matrix notation to write a linear transformation as

v → Ωv =(e1 · · · en

)

Ω11 Ω12 . . . Ω1n

Ω21 Ω22 . . . Ω2n

......

......

Ωn1 Ωn2 . . . Ωnn

v1

v2

...vn

One sees from this that we can think of the transformed vector as we did

above in terms of transformed coefficients vj with respect to fixed basis vectors,but also could leave the vj unchanged and transform the basis vectors. At timeswe will want to use matrix notation to write formulas for how the basis vectorstransform in this way, and then will write

e1

e2

...en

→

Ω11 Ω21 . . . Ωn1

Ω12 Ω22 . . . Ωn2

......

......

Ω1n Ω2n . . . Ωnn

e1

e2

...en

Note that putting the basis vectors ej in a column vector like this causes thematrix for Ω to act on them by the transposed matrix. This is not a group actionsince in general the product of two transposed matrices is not the transpose ofthe product.

4.2 Dual vector spaces

To any vector space V one can associate a new vector space, its dual:

Definition (Dual vector space). Given a vector space V over a field k, the dualvector space V ∗ is the set of all linear maps V → k, i.e.

V ∗ = l : V → k such that l(αv + βw) = αl(v) + βl(w)

34

for α, β ∈ k, v, w ∈ V .

Given a linear transformation Ω acting on V , one can define:

Definition (Transpose transformation). The transpose of Ω is the linear trans-formation

ΩT : V ∗ → V ∗

that satisfies(ΩT l)(v) = l(Ωv)

for l ∈ V ∗, v ∈ V .

For any representation (π, V ) of a group G on V , one can define a corre-sponding representation on V ∗

Definition (Dual or contragredient representation). The dual or contragredientrepresentation on V ∗ is given by taking as linear operators

(πT )−1(g) : V ∗ → V ∗

These satisfy the homomorphism property since

(πT (g1))−1(πT (g2))−1 = (πT (g2)πT (g1))−1 = ((π(g1)π(g2))T )−1

For any choice of basis ej of V , one has a dual basis e∗j of V ∗ thatsatisfies

e∗j (ek) = δjk

Coordinates on V with respect to a basis are linear functions, and thus elementsof V ∗. One can identify the coordinate function vj with the dual basis vectore∗j since

e∗j (v1e1 + v2e2 + · · ·+ vnen) = vj

One can easily show that the elements of the matrix for Ω in the basis ejare given by

Ωjk = e∗j (Ωek)

and that the matrix for the transpose map (with respect to the dual basis) isjust the matrix transpose

(ΩT )jk = Ωkj

One can use matrix notation to write elements

l = l1e∗1 + l2e

∗2 + · · ·+ lne∗n ∈ V ∗

of V ∗ as row vectors (l1 l2 · · · ln

)of coordinates on V ∗. Then evaluation of l on a vector v given by matrixmultiplication

l(v) =(l1 l2 · · · ln

)v1

v2

...vn

= l1v1 + l2v2 + · · ·+ lnvn

35

4.3 Change of basis

Any invertible transformation A on V can be used to change the basis ej of Vto a new basis e′j by taking

ej → e′j = Aej

The matrix for a linear transformation Ω transforms under this change of basisas

Ωjk = e∗j (Ωek)→ (e′j)∗(Ωe′k) =(Aej)

∗(ΩAek)

=(AT )−1(e∗j )(ΩAek)

=e∗j (A−1ΩAek)

=(A−1ΩA)jk

In the second step we are using the fact that elements of the dual basis transformas the dual representation. One can check that this is what is needed to ensurethe relation

(e′j)∗(e′k) = δjk

The change of basis formula shows that if two matrices Ω1 and Ω2 are relatedby conjugation by a third matrix A

Ω2 = A−1Ω1A

then one can think of them as both representing the same linear transforma-tion, just with respect to two different choices of basis. Recall that a finite-dimensional representation is given by a set of matrices π(g), one for each groupelement. If two representations are related by

π2(g) = A−1π1(g)A

(for all g, A does not depend on g), then we can think of them as being thesame representation, with different choices of basis. In such a case the represen-tations π1 and π2 are called “equivalent”, and we will often implicitly identifyrepresentations that are equivalent.

4.4 Inner products

An inner product on a vector space V is an additional structure that providesa notion of length for vectors, of angle between vectors, and identifies V ∗ ' V .One has, in the real case:

Definition (Inner Product, real case). An inner product on a real vector spaceV is a map

〈·, ·〉 : V × V → R

that is linear in both variables and symmetric (〈v, w〉 = 〈w, v〉).

36

Our inner products will usually be positive-definite (〈v, v〉 ≥ 0 and 〈v, v〉 =0 =⇒ v = 0), with indefinite inner products only appearing in the con-text of special or general relativity, where an indefinite inner product on four-dimensional space-time is used.

In the complex case, one has

Definition (Inner Product, complex case). An Hermitian inner product on acomplex vector space V is a map

〈·, ·〉 : V × V → C

that is conjugate symmetric

〈v, w〉 = 〈w, v〉

as well as linear in the second variable, and antilinear in the first variable: forα ∈ C and u, v, w ∈ V

〈u+ v, w〉 = 〈u,w〉+ 〈v, w〉, 〈αu, v〉 = α〈u, v〉

An inner product gives a notion of length || · || for vectors, with

||v||2 = 〈v, v〉

Note that whether to specify antilinearity in the first or second variable is amatter of convention. The choice we are making is universal among physicists,with the opposite choice common among mathematicians.

An inner product also provides an (antilinear in the complex case) isomor-phism V ' V ∗ by the map

v ∈ V → lv ∈ V ∗

where lv is defined bylv(w) = 〈v, w〉

Physicists have a useful notation for elements of vector space and their duals,for the case when V is a complex vector space with an Hermitian inner product(such as the state space for a quantum theory). An element of such a vectorspace V is written as a “ket vector”

|v〉

where v is a label for a vector. An element of the dual vector space V ∗ is writtenas a “bra vector”

〈l|

Evaluating l ∈ V ∗ on v ∈ V gives an element of C, written

〈l|v〉

37

If Ω : V → V is a linear map

〈l|Ω|v〉 = 〈l|Ωv〉 = l(Ωv)

In the bra-ket notation, one denotes the dual vector lv by 〈v|. Note that inthe inner product the angle bracket notation means something different than inthe bra-ket notation. The similarity is intentional though, since in the bra-ketnotation one has

〈v|w〉 = 〈v, w〉Note that our convention of linearity in the second variable of the inner product,antilinearity in the first, implies

|αv〉 = α|v〉, 〈αv| = α〈v|

for α ∈ C.For a choice of orthonormal basis ej, i.e. satisfying

〈ej , ek〉 = δjk

a useful notation is|j〉 = ej

Because of orthonormality, coefficients of vectors can be calculated as

vj = 〈ej , v〉

In bra-ket notation we havevj = 〈j|v〉

and

|v〉 =

n∑j=1

|j〉〈j|v〉

For corresponding elements of V ∗, one has (using antilinearity)

〈v| =n∑j=1

vj〈j| =n∑j=1

〈v|j〉〈j|

With respect to the chosen orthonormal basis ej, one can represent vectorsv as column vectors and the operation of taking a vector |v〉 to a dual vector 〈v|corresponds to taking a column vector to the row vector that is its conjugate-transpose.

〈v| =(v1 v2 · · · vn

)Then one has

〈v|w〉 =(v1 v2 · · · vn

)w1

w2

...wn

= v1w1 + v2w2 + · · ·+ vnwn

38

If Ω is a linear operator Ω : V → V , then with respect to the chosen basis itbecomes a matrix with matrix elements

Ωkj = 〈k|Ωj〉

The decomposition of a vector v in terms of coefficients

|v〉 =

n∑j=1

|j〉〈j|v〉

can be interpreted as a matrix multiplication by the identity matrix

1 =

n∑j=1

|j〉〈j|

and this kind of expression is referred to by physicists as a “completeness rela-tion”, since it requires that the set of |j〉 be a basis with no missing elements.The operator

Pj = |j〉〈j|

is called the projection operator onto the j’th basis vector, it corresponds to thematrix that has 0s everywhere except in the jj component.

Digression. In this course, all our indices will be lower indices. One way tokeep straight the difference between vectors and dual vectors is to use upperindices for components of vectors, lower indices for components of dual vectors.This is quite useful in Riemannian geometry and general relativity, where theinner product is given by a metric that can vary from point to point, causingthe isomorphism between vectors and dual vectors to also vary. For quantummechanical state spaces, we will be using a single, standard, fixed inner product,so there will be a single isomorphism between vectors and dual vectors. Thebra-ket notation will take care of the notational distinction between vectors anddual vectors as necessary.

4.5 Adjoint operators

When V is a vector space with inner product, one can define the adjoint of Ωby

Definition (Adjoint Operator). The adjoint of a linear operator Ω : V → V isthe operator Ω† satisfying

〈Ωv, w〉 = 〈v,Ω†w〉

or, in bra-ket notation〈Ωv|w〉 = 〈v|Ω†w〉

for all v, w ∈ V .

39

Generalizing the fact that

〈αv| = α〈v|

for α ∈ C, one can write〈Ωv| = 〈v|Ω†

Note that mathematicians tend to favor Ω∗ as notation for the adjoint of Ω,as opposed to the physicist’s notation Ω† that we are using.

In terms of explicit matrices, since 〈Ωv| is the conjugate-transpose of |Ωv〉,the matrix for Ω† will be given by the conjugate transpose ΩT of the matrix forΩ:

Ω†jk = Ωkj

In the real case, the matrix for the adjoint is just the transpose matrix. Wewill say that a linear transformation is self-adjoint if Ω† = Ω, skew-adjoint ifΩ† = −Ω.

4.6 Orthogonal and unitary transformations

A special class of linear transformations will be invertible transformations thatpreserve the inner product, i.e. satisfying

〈Ωv,Ωw〉 = 〈Ωv|Ωw〉 = 〈v, w〉 = 〈v|w〉

for all v, w ∈ V . Such transformations take orthonormal bases to orthonormalbases, so they will appear in one role as change of basis transformations.

In terms of adjoints, this condition becomes

〈Ωv,Ωw〉 = 〈v,Ω†Ωw〉 = 〈v, w〉

soΩ†Ω = 1

or equivalentlyΩ† = Ω−1

In matrix notation this first condition becomesn∑k=1

(Ω†)jkΩkl =

n∑k=1

ΩkjΩkl = δjl

which says that the column vectors of the matrix for Ω are orthonormal vectors.Using instead the equivalent condition

ΩΩ† = 1

one finds that the row vectors of the matrix for Ω are also orthornormal.Since such linear transformations preserving the inner product can be com-

posed and are invertible, they form a group, and some of the basic examples ofLie groups are given by these groups for the cases of real and complex vectorspaces.

40

4.6.1 Orthogonal groups

We’ll begin with the real case, where these groups are called orthogonal groups:

Definition (Orthogonal group). The orthogonal group O(n) in n-dimensionsis the group of invertible transformations preserving an inner product on a realn-dimensional vector space V . This is isomorphic to the group of n by n realinvertible matrices Ω satisfying

Ω−1 = ΩT

The subgroup of O(n) of matrices with determinant 1 (equivalently, the subgrouppreserving orientation of orthonormal bases) is called SO(n).

Recall that for a representation π of a group G on V , one has a dual repre-sentation on V ∗ given by taking the transpose-inverse of π. If G is an orthogonalgroup, then π and its dual are the same matrices, with V identified by V ∗ bythe inner product.

Since the determinant of the transpose of a matrix is the same as the deter-minant of the matrix, we have

Ω−1Ω = 1 =⇒ det(Ω−1)det(Ω) = det(ΩT )det(Ω) = (det(Ω))2 = 1

sodet(Ω) = ±1

O(n) is a continuous Lie group, with two components: SO(n), the subgroup oforientation-preserving transformations, which include the identity, and a com-ponent of orientation-changing transformations.

The simplest non-trivial example is for n = 2, where all elements of SO(2)are given by matrices of the form(

cos θ − sin θsin θ cos θ

)These matrices give counter-clockwise rotations in R2 by an angle θ. The othercomponent of O(2) will be given by matrices of the form(

cos θ sin θsin θ − cos θ

)Note that the group SO(2) is isomorphic to the group U(1) by(


)⇔ eiθ

so the representation theory of SO(2) is just as for U(1), with irreducible com-plex representations one-dimensional and classified by an integer.

In chapter 6 we will consider in detail the case of SO(3), which is crucialfor physical applications because it is the group of rotations in the physicalthree-dimensional space.

41

4.6.2 Unitary groups

In the complex case, groups of invertible transformations preserving the Hermi-tian inner product are called unitary groups:

Definition (Unitary group). The unitary group U(n) in n-dimensions is thegroup of invertible transformations preserving an Hermitian inner product on acomplex n-dimensional vector space V . This is isomorphic to the group of n byn complex invertible matrices satisfying

Ω−1 = ΩT

= Ω†

The subgroup of U(n) of matrices with determinant 1 is called SU(n).

In the unitary case, the dual of a representation π has representation matricesthat are transpose-inverses of those for π, but

(π(g)T )−1 = π(g)

so the dual representation is given by conjugating all elements of the matrix.The same calculation as in the real case here gives

det(Ω−1)det(Ω) = det(Ω†)det(Ω) = det(Ω)det(Ω) = |det(Ω)|2 = 1

so det(Ω) is a complex number of modulus one. The map

Ω ∈ U(n)→ det(Ω) ∈ U(1)

is a group homomorphism.We have already seen the examples U(1), U(2) and SU(2). For general

values of n, the case of U(n) can be split into the study of its determinant,which lies in U(1) so is easy to deal with, and the subgroup SU(n), which is amuch more complicated story.

Digression. Note that is not quite true that the group U(n) is the productgroup SU(n) × U(1). If one tries to identify the U(1) as the subgroup of U(n)of elements of the form eiθ1, then matrices of the form

eimn 2π1

for m an integer will lie in both SU(n) and U(1), so U(n) is not a product ofthose two groups.

We saw at the end of section 3.1.2 that SU(2) can be identified with the three-sphere S3, since an arbitrary group element can be constructed by specifying onerow (or one column), which must be a vector of length one in C2. For the casen = 3, the same sort of construction starts by picking a row of length one in C3,which will be a point in S5. The second row must be orthornormal, and one canshow that the possibilities lie in a three-sphere S3. Once the first two rows arespecified, the third row is uniquely determined. So as a manifold, SU(3) is eight-dimensional, and one might think it could be identified with S5×S3. It turns out

42

that this is not the case, since the S3 varies in a topologically non-trivial wayas one varies the point in S5. As spaces, the SU(n) are topologically “twisted”products of odd-dimensional spheres, providing some of the basic examples ofquite non-trivial topological manifolds.

4.7 Eigenvalues and eigenvectors

We have seen that that the matrix for a linear transformation Ω of a vectorspace V changes by conjugation when we change our choice of basis of V . Toget basis-independent information about Ω, one considers the eigenvalues of thematrix. Complex matrices behave in a much simpler fashion than real matrices,since in the complex case the eigenvalue equation

det(λ1− Ω) = 0

can always be factored into linear factors, and solved for the eigenvalues λ. Foran arbitrary n by n complex matrix there will be n solutions (counting repeatedeigenvalues with multiplicity). One can always find a basis for which the matrixwill be in upper triangular form.

The case of self-adjoint matrices Ω is much more constrained, since transpo-sition relates matrix elements. One has:

Theorem (Spectral theorem for self-adjoint matrices). Given a self-adjointcomplex n by n matrix Ω, one can always find a unitary matrix U such that

UΩU−1 = D

where D is a diagonal matrix with entries Djj = λj , λj ∈ R.

Given Ω, one finds the eigenvalues λj by solving the eigenvalue equation. Onecan then go on to solve for the eigenvectors and use these to find U . For distincteigenvalues one finds that the corresponding eigenvectors are orthogonal.

This theorem is of crucial importance in quantum mechanics, where for Ω anobservable, the eigenvectors are the states in the state space with well-definednumerical values characterizing the state, and these numerical values are theeigenvalues. The theorem also tells us that given an observable, we can useit to choose distinguished orthonormal bases for the state space by picking abasis of eigenvectors, normalized to length one. This is a theorem about finite-dimensional vector spaces, but later on in the course we will see that somethingsimilar will be true even in the case of infinite-dimensional state spaces.

One can also diagonalize unitary matrices themselves by conjugation byanother unitary. The diagonal entries will all be complex numbers of unit length,so of the form eiλj , λj ∈ R.

For the simplest examples, consider the cases of the groups SU(2) and U(2).Any matrix in U(2) can be conjugated by a unitary matrix to the diagonalmatrix (

eiλ1 00 eiλ2

)43

which is the exponential of a corresponding diagonalized skew-adjoint matrix(iλ1 00 iλ2

)For matrices in the subgroup SU(2), one has λ2 = −λ1 = λ so in diagonal forman SU(2) matrix will be (

eiλ 00 e−iλ

)which is the exponential of a corresponding diagonalized skew-adjoint matrixthat has trace zero (

iλ 00 −iλ

)


Almost any of the more advanced linear algebra textbooks should cover thematerial of this chapter.

44

Chapter 5

Lie Algebras and LieAlgebra Representations

For a groupG we have defined unitary representations (π, V ) for finite-dimensionalvector spaces V of complex dimension n as homomorphisms

π : G→ U(n)

Recall that in the case of G = U(1), we could use the homomorphism propertyof π to determine π in terms of its derivative at the identity. This turns out tobe a general phenomenon for Lie groups G: we can study their representationsby considering the derivative of π at the identity, which we will call π′. Becauseof the homomorphism property, knowing π′ is often sufficient to characterizethe representation π it comes from. π′ is a linear map from the tangent spaceto G at the identity to the tangent space of U(n) at the identity. The tangentspace to G at the identity will carry some extra structure coming from the groupmultiplication, and this vector space with this structure will be called the Liealgebra of G.

The subject of differential geometry gives many equivalent ways of definingthe tangent space at a point of manifolds like G, but we do not want to enterhere into the subject of differential geometry in general. One of the standarddefinitions of the tangent space is as the space of tangent vectors, with tangentvectors defined as the possible velocity vectors of parametrized curves g(t) inthe group G.

More advanced treatments of Lie group theory develop this point of view(see for example [77]) which applies to arbitrary Lie groups, whether or notthey are groups of matrices. In our case though, since we are interested inspecific groups that are explicitly given as groups of matrices, we can give amore concrete definition, just using the exponential map on matrices. For amore detailed exposition of this subject, using the same concrete definition ofthe Lie algebra in terms of matrices, see Brian Hall’s book [29] or the abbreviatedon-line version [30].

45

Note that the material of this chapter is quite general, and may be hardto make sense of until one has some experience with basic examples. The nextchapter will discuss in detail the groups SU(2) and SO(3) and their Lie algebras,as well as giving some examples of their representations, and this may be helpfulin making sense of the general theory of this chapter.

5.1 Lie algebras

We’ll work with the following definition of a Lie algebra:

Definition (Lie algebra). For G a Lie group of n by n invertible matrices, theLie algebra of G (written Lie(G) or g) is the space of n by n matrices X suchthat etX ∈ G for t ∈ R.

Notice that while the group G determines the Lie algebra g, the Lie algebradoes not determine the group. For example, O(n) and SO(n) have the sametangent space at the identity, and thus the same Lie algebra, but elements inO(n) not in the component of the identity can’t be written in the form etX

(since then you could make a path of matrices connecting such an element tothe identity by shrinking t to zero). Note also that, for a given X, differentvalues of t may give the same group element, and this may happen in differentways for different groups sharing the same Lie algebra. For example, considerG = U(1) and G = (R,+), which both have the same Lie algebra g = R. In thefirst case an infinity of values of t give the same group element, in the second,only one does. In the next chapter we’ll see a more subtle example of this:SU(2) and SO(3) are different groups with the same Lie algebra.

We have G ⊂ GL(n,C), and X ∈ M(n,C), the space of n by n complexmatrices. For all t ∈ R, the exponential etX is an invertible matrix (with inversee−tX), so in GL(n,C). For each X, we thus have a path of elements of GL(n,C)going through the identity matrix at t = 0, with velocity vector

d

dtetX = XetX

which takes the value X at t = 0:

d

dt(etX)|t=0 = X

To calculate this derivative, just use the power series expansion for the expo-nential, and differentiate term-by-term.

For the case G = GL(n,C), we just have gl(n,C) = M(n,C), which is alinear space of the right dimension to be the tangent space to G at the identity,so this definition is consistent with our general motivation. For subgroups G ⊂GL(n,C) given by some condition (for example that of preserving an innerproduct), we will need to identify the corresponding condition on X ∈M(n,C)and check that this defines a linear space.

The existence of such a linear space g ⊂ M(n,C) will provide us with adistinguished representation, called the “adjoint representation”

46

Definition (Adjoint representation). The adjoint representation (Ad, g) is givenby the homomorphism

Ad : g ∈ G→ Ad(g) ∈ GL(g)

where Ad(g) acts on X ∈ g by

(Ad(g))(X) = gXg−1

To show that this is well-defined, one needs to check that gXg−1 ∈ g whenX ∈ g, but this can be shown using the identity

etgXg−1

= getXg−1

which implies that etgXg−1 ∈ G if etX ∈ G. To check this, just expand the

exponential and use

(gXg−1)k = (gXg−1)(gXg−1) · · · (gXg−1) = gXkg−1

It is also easy to check that this is a homomorphism, with

Ad(g1)Ad(g2) = Ad(g1g2)

A Lie algebra g is not just a real vector space, but comes with an extrastructure on the vector space

Definition (Lie bracket). The Lie bracket operation on g is the bilinear anti-symmetric map given by the commutator of matrices

[·, ·] : (X,Y ) ∈ g× g→ [X,Y ] = XY − Y X ∈ g

We need to check that this is well-defined, i.e. that it takes values in g.

Theorem. If X,Y ∈ g, [X,Y ] = XY − Y X ∈ g.

Proof. Since X ∈ g, we have etX ∈ G and we can act on Y ∈ g by the adjointrepresentation

Ad(etX)Y = etXY e−tX ∈ g

As t varies this gives us a parametrized curve in g. Its velocity vector will alsobe in g, so

d

dt(etXY e−tX) ∈ g

One has (by the product rule, which can easily be shown to apply in this case)

d

dt(etXY e−tX) = (

d

dt(etXY ))e−tX + etXY (

d

dte−tX)

= XetXY e−tX − etXY Xe−tX

Evaluating this at t = 0 givesXY − Y X

which is thus shown to be in g.

47

The relationd

dt(etXY e−tX)|t=0 = [X,Y ] (5.1)

used in this proof will be continually useful in relating Lie groups and Lie alge-bras.

To do calculations with a Lie algebra, one can just choose a basisX1, X2, . . . , Xn

for the vector space g, and use the fact that the Lie bracket can be written interms of this basis as

[Xj , Xk] =

n∑l=1

cjklXl

where cjkl is a set of constants known as the “structure constants” of the Liealgebra. For example, in the case of su(2), the Lie algebra of SU(2) one has abasis X1, X2, X3 satisfying

[Xj , Xk] =

3∑l=1

εjklXl

so the structure constants of su(2) are just the totally antisymmetric εjkl.

5.2 Lie algebras of the orthogonal and unitarygroups

The groups we are most interested in are the groups of linear transformationspreserving an inner product: the orthogonal and unitary groups. We have seenthat these are subgroups of GL(n,R) or GL(n,C), consisting of those elementsΩ satisfying the condition

ΩΩ† = 1

In order to see what this condition becomes on the Lie algebra, write Ω = etX ,for some parameter t, and X a matrix in the Lie algebra. Since the transpose ofa product of matrices is the product (order-reversed) of the transposed matrices,i.e.

(XY )T = Y TXT

and the complex conjugate of a product of matrices is the product of the complexconjugates of the matrices, one has

(etX)† = etX†

The condition

ΩΩ† = 1

thus becomes

etX(etX)† = etXetX†

= 1

48

Taking the derivative of this equation gives

etXX†etX†

+XetXetX†

= 0

Evaluating this at t = 0 one finds

X +X† = 0

so the matrices we want to exponentiate are skew-adjoint, satisfying

X† = −X

Note that physicists often choose to define the Lie algebra in these casesas self-adjoint matrices, then multiplying by i before exponentiating to get agroup element. We will not use this definition, with one reason that we want tothink of the Lie algebra as a real vector space, so want to avoid an unnecessaryintroduction of complex numbers at this point.

5.2.1 Lie algebra of the orthogonal group

Recall that the orthogonal group O(n) is the subgroup of GL(n,R) of matricesΩ satisfying ΩT = Ω−1. We will restrict attention to the subgroup SO(n) ofmatrices with determinant 1 which is the component of the group containingthe identity, and thus elements that can be written as

Ω = etX

These give a path connecting Ω to the identity (taking esX , s ∈ [0, t]). Wesaw above that the condition ΩT = Ω−1 corresponds to skew-symmetry of thematrix X

XT = −XSo in the case of G = SO(n), we see that the Lie algebra so(n) is the space ofskew-symmetric (XT = −X) n by n real matrices, together with the bilinear,antisymmetric product given by the commutator:

(X,Y ) ∈ so(n)× so(n)→ [X,Y ] ∈ so(n)

The dimension of the space of such matrices will be

1 + 2 + · · ·+ (n− 1) =n2 − n

2

and a basis will be given by the matrices εjk, with j, k = 1, . . . , n, j < k definedas

(εjk)lm =

−1 if j = l, k = m

+1 if j = m, k = l

0 otherwise

(5.2)

In chapter 6 we will examine in detail the n = 3 case, where the Lie algebraso(3) is R3, realized as the space of antisymmetric real 3 by 3 matrices.

49

5.2.2 Lie algebra of the unitary group

For the case of the group U(n), the group is connected and one can write allgroup elements as etX , where now X is a complex n by n matrix. The unitaritycondition implies that X is skew-adjoint (also called skew-Hermitian), satisfying

X† = −X

So the Lie algebra u(n) is the space of skew-adjoint n by n complex matrices,together with the bilinear, antisymmetric product given by the commutator:

(X,Y ) ∈ u(n)× u(n)→ [X,Y ] ∈ u(n)

Note that these matrices form a subspace of Cn2

of half the dimension,so of real dimension n2. u(n) is a real vector space of dimension n2, but itis NOT a space of real n by n matrices. It is the space of skew-Hermitianmatrices, which in general are complex. While the matrices are complex, onlyreal linear combinations of skew-Hermitian matrices are skew-Hermitian (recallthat multiplication by i changes a skew-Hermitian matrix into a Hermitianmatrix). Within this space of complex matrices, if one looks at the subspace ofreal matrices one gets the sub-Lie algebra so(n) of antisymmetric matrices (theLie algebra of SO(n) ⊂ U(n)).

Given any complex matrix Z ∈M(n,C), one can write it as a sum of

Z =1

2(Z + Z†) +

1

2(Z − Z†)

where the first term is self-adjoint, the second skew-Hermitian. This secondterm can also be written as i times a self-adjoint matrix

1

2(Z − Z†) = i(

1

2i(Z − Z†))

so we see that we can get all of M(n,C) by taking all complex linear combina-tions of self-adjoint matrices..

There is an identity relating the determinant and the trace of a matrix

det(eX) = etrace(X)

which can be proved by conjugating the matrix to upper-triangular form andusing the fact that the trace and the determinant of a matrix are conjugation-invariant. Since the determinant of an SU(n) matrix is 1, this shows that theLie algebra su(n) of SU(n) will consist of matrices that are not only skew-Hermitian, but also of trace zero. So in this case su(n) is again a real vectorspace, of dimension n2 − 1.

One can show that U(n) and u(n) matrices can be diagonalized by conju-gation by a unitary matrix to show that any U(n) matrix can be written as anexponential of something in the Lie algebra. The corresponding theorem is alsotrue for SO(n) but requires looking at diagonalization into 2 by 2 blocks. It isnot true for O(n) (you can’t reach the disconnected component of the identityby exponentiation). It also turns out to not be true for the groups GL(n,R)and GL(n,C) for n ≥ 2.

50

5.3 Lie algebra representations

We have defined a group representation as a homomorphism (a map of groupspreserving group multiplication)

π : G→ GL(n,C)

We can similarly define a Lie algebra representation as a map of Lie algebraspreserving the Lie bracket:

Definition (Lie algebra representation). A (complex) Lie algebra representation(φ, V ) of a Lie algebra g on an n-dimensional complex vector space V is givenby a linear map

φ : X ∈ g→ φ(X) ∈ gl(n,C) = M(n,C)

satisfying

φ([X,Y ]) = [φ(X), φ(Y )]

Such a representation is called unitary if its image is in u(n), i.e. it satisfies

φ(X)† = −φ(X)

More concretely, given a basis X1, X2, . . . , Xd of a Lie algebra g of dimensiond with structure constants cjkl, a representation is given by a choice of d complexn-dimensional matrices φ(Xj) satisfying the commutation relations

[φ(Xj), φ(Xk)] =

d∑l=1

cjklφ(Xl)

The representation is unitary when the matrices are skew-adjoint.

The notion of a Lie algebra is motivated by the fact that the homomorphismproperty causes the map π to be largely determined by its behavior infinitesi-mally near the identity, and thus by the derivative π′. One way to define thederivative of such a map is in terms of velocity vectors of paths, and this sort ofdefinition in this case associates to a representation π : G→ GL(n,C) a linearmap

π′ : g→M(n,C)

where

π′(X) =d

dt(π(etX))|t=0

51

In the case of U(1) we classified all irreducible representations (homomor-phisms U(1) → GL(1,C) = C∗) by looking at the derivative of the map atthe identity. For general Lie groups G, one can do something similar, show-ing that a representation π of G gives a representation of the Lie algebra (bytaking the derivative at the identity), and then trying to classify Lie algebrarepresentations.

Theorem. If π : G→ GL(n,C) is a group homomorphism, then

π′ : X ∈ g→ π′(X) =d

dt(π(etX))|t=0 ∈ gl(n,C) = M(n,C)

satisfies

1.

π(etX) = etπ′(X)

2. For g ∈ G

π′(gXg−1) = π(g)π′(X)(π(g))−1

3. π′ is a Lie algebra homomorphism:

π′([X,Y ]) = [π′(X), π′(Y )]

52

Proof. 1. We have

d

dtπ(etX) =

d

dsπ(e(t+s)X)|s=0

=d

dsπ(etXesX)|s=0

= π(etX)d

dsπ(esX)|s=0

= π(etX)π′(X)

So f(t) = π(etX) satisfies the differential equation ddtf = fπ′(X) with

initial condition f(0) = 1. This has the unique solution f(t) = etπ′(X)

2. We have

etπ′(gXg−1) = π(etgXg

−1

)

= π(getXg−1)

= π(g)π(etX)π(g)−1

= π(g)etπ′(X)π(g)−1

Differentiating with respect to t at t = 0 gives

π′(gXg−1) = π(g)π′(X)(π(g))−1

3. Recall that (5.1)

[X,Y ] =d

dt(etXY e−tX)|t=0

so

π′([X,Y ]) = π′(d

dt(etXY e−tX)|t=0)

=d

dtπ′(etXY e−tX)|t=0 (by linearity)

=d

dt(π(etX)π′(Y )π(e−tX))|t=0 (by 2.)

=d

dt(etπ

′(X)π′(Y )e−tπ′(X))|t=0 (by 1.)

= [π′(X), π′(Y )]

This theorem shows that we can study Lie group representations (π, V )by studying the corresponding Lie algebra representation (π′, V ). This willgenerally be much easier since the π′(X) are just linear maps. We will proceedin this manner in chapter 8 when we construct and classify all SU(2) and SO(3)representations, finding that the corresponding Lie algebra representations aremuch simpler to analyze.

53

For any Lie group G, we have seen that there is a distinguished representa-tion, the adjoint representation (Ad, g). The corresponding Lie algebra represen-tation is also called the adjoint representation, but written as (Ad′, g) = (ad, g).From the fact that

Ad(etX)(Y ) = etXY e−tX

we can differentiate with respect to t to get the Lie algebra representation

ad(X)(Y ) =d

dt(etXY e−tX)|t=0 = [X,Y ] (5.3)

From this we see that one can define

Definition (Adjoint Lie algebra representation). ad is the Lie algebra repre-sentation given by

X ∈ g→ ad(X)

where ad(X) is defined as the linear map from g to itself given by

Y → [X,Y ]

Note that this linear map ad(X), which one can write as [X, ·], can bethought of as the infinitesimal version of the conjugation action

(·)→ etX(·)e−tX

The Lie algebra homomorphism property of ad says that

ad([X,Y ]) = ad(X) ad(Y )− ad(Y ) ad(X)

where these are linear maps on g, with composition of linear maps, so operatingon Z ∈ g we have

ad([X,Y ])(Z) = (ad(X) ad(Y )((Z)− (ad(Y ) ad(X))(Z)

Using our expression for ad as a commutator, we find

[[X,Y ], Z] = [X, [Y,Z]]− [Y, [X,Z]]

This is called the Jacobi identity. It could have been more simply derived asan identity about matrix multiplication, but here we see that it is true for amore abstract reason, reflecting the existence of the adjoint representation. Itcan be written in other forms, rearranging terms using antisymmetry of thecommutator, with one example the sum of cyclic permutations

[[X,Y ], Z] + [[Z,X], Y ] + [[Y,Z], X] = 0

One can define Lie algebras much more abstractly as follows

54

Definition (Abstract Lie algebra). An abstract Lie algebra over a field k is avector space A over k, with a bilinear operation

[·, ·] : (X,Y ) ∈ A×A→ [X,Y ] ∈ A

satisfying

1. Antisymmetry:[X,Y ] = −[Y,X]

2. Jacobi identity:

[[X,Y ], Z] + [[Z,X], Y ] + [[Y, Z], X] = 0

Such Lie algebras do not need to be defined as matrices, and their Lie bracketoperation does not need to be defined in terms of a matrix commutator (al-though the same notation continues to be used). Later on in this course wewill encounter important examples of Lie algebras that are defined in this moreabstract way.

5.4 Complexification

The way we have defined a Lie algebra g, it is a real vector space, not a complexvector space. Even if G is a group of complex matrices, when it is not GL(n,C)itself but some subgroup, its tangent space at the identity will not necessarilybe a complex vector space. Consider for example the cases G = U(1) andG = SU(2), where u(1) = R and su(2) = R3. While the tangent space to thegroup of all invertible complex matrices is a complex vector space, imposingsome condition such as unitarity picks out a subspace which generally is just areal vector space, not a complex one. So the adjoint representation (Ad, g) is ingeneral not a complex representation, but a real representation, with

Ad(g) ∈ GL(g) = GL(dim g,R)

The derivative of this is the Lie algebra representation

ad : X ∈ g→ ad(X) ∈ gl(dim g,R)

and once we pick a basis of g, we can identify gl(dim g,R) = M(dim g,R). So,for each X ∈ g we get a real linear operator on a real vector space.

We would however often like to work with not real representations, butcomplex representations, since it is for these that Schur’s lemma applies, andrepresentation operators can be diagonalized. To get from a real Lie algebrarepresentation to a complex one, we can “complexify”, extending the action ofreal scalars to complex scalars. If we are working with real matrices, complex-ification is nothing but allowing complex entries and using the same rules formultiplying scalars as before.

More generally, for any real vector space we can define:

55

Definition. The complexification VC of a real vector space V is the space ofpairs (v1, v2) of elements of V with multiplication by a+ bi ∈ C given by

(a+ ib)(v1, v2) = (av1 − bv2, av2 + bv1)

One should think of the complexification of V as

VC = V + iV

with v1 in the first copy of V , v2 in the second copy. Then the rule for mul-tiplication by a complex number comes from the standard rules for complexmultiplication. In the cases we will be interested in this level of abstraction isnot really needed, since V will be given as a subspace of a complex space, and VCwill just be the larger subspace you get by taking complex linear combinationsof elements of V .

Given a real Lie algebra g, the complexification gC is pairs of elements (X,Y )of g, with the above rule for multiplication by complex scalars. The Lie bracketon g extends to a Lie bracket on gC by the rule

[(X1, Y1), (X2, Y2)] = ([X1, X2]− [Y1, Y2], [X1, Y2] + [Y1, Y2])

and gC is a Lie algebra over the complex numbers. In many cases this definitionis isomorphic to something just defined in terms of complex matrices, with thesimplest case

gl(n,R)C = gl(n,C)

Recalling our discussion from section 5.2.2 of u(n), a real Lie algebra, withelements certain (skew-Hermitian) complex matrices, one can see that complex-ifying will just give all complex matrices so

u(n)C = gl(n,C)

This example shows that two different real Lie algebras may have the same com-plexification. For yet another example, since so(n) is the Lie algebra of all realantisymmetric matrices, so(n)C is the Lie algebra of all complex antisymmetricmatrices.

We can extend the operators ad(X) on g by complex linearity to turn adinto a complex representation of gC on the vector space gC itself

ad : Z ∈ gC → ad(Z)

Here Z is now a complex linear combination of elements of Xj ∈ g, and ad(Z)is the corresponding complex linear combination of the real matrices ad(Xj).


The material of this section is quite conventional mathematics, with many goodexpositions, although most aimed at a higher level than this course. An example

56

at the level of this course is the book Naive Lie Theory [66]. It covers basics ofLie groups and Lie algebras, but without representations. The notes [30] andbook [29] of Brian Hall are a good source to study from. Some parts of theproofs given here are drawn from those notes.

57

Chapter 6

The Rotation and SpinGroups in 3 and 4Dimensions

Among the basic symmetry groups of the physical world is the orthogonal groupSO(3) of rotations about a point in three-dimensional space. The observablesone gets from this group are the components of angular momentum, and under-standing how the state space of a quantum system behaves as a representationof this group is a crucial part of the analysis of atomic physics examples andmany others. This is a topic one will find in some version or other in everyquantum mechanics textbook.

Remarkably, it turns out that the quantum systems in nature are oftenrepresentations not of SO(3), but of a larger group called Spin(3), one that hastwo elements corresponding to every element of SO(3). Such a group exists inany dimension n, always as a “doubled” version of the orthogonal group SO(n),one that is needed to understand some of the more subtle aspects of geometryin n dimensions. In the n = 3 case it turns out that Spin(3) ' SU(2) and wewill study in detail the relationship of SO(3) and SU(2). This appearance ofthe unitary group SU(2) is special to geometry in 3 and 4 dimensions, and wewill see that quaternions provide an explanation for this.

6.1 The rotation group in three dimensions

In R2 rotations about the origin are given by elements of SO(2), with a counter-clockwise rotation by an angle θ given by the matrix

R(θ) =

(cos θ − sin θsin θ cos θ

)59

This can be written as an exponential, R(θ) = eθL = cos θ1 + L sin θ for

L =

(0 −11 0

)Here SO(2) is a commutative Lie group with Lie algebra so(2) = R (it is one-dimensional, with trivial Lie bracket, all elements of the Lie algebra commute).Note that we have a representation on V = R2 here, but it is a real representa-tion, not one of the complex ones we have when we have a representation on aquantum mechanical state space.

In three dimensions the group SO(3) is 3-dimensional and non-commutative.Choosing a unit vector w and angle θ, one gets an element R(θ,w) of SO(3),rotation by θ about the w axis. Using standard basis vectors ej , rotations aboutthe coordinate axes are given by

R(θ, e1) =

1 0 00 cos θ − sin θ0 sin θ cos θ

, R(θ, e2) =

cos θ 0 sin θ0 1 0

− sin θ 0 cos θ

R(θ, e3) =

cos θ − sin θ 0sin θ cos θ 0

0 0 1

A standard parametrization for elements of SO(3) is in terms of 3 “Euler angles”φ, θ, ψ with a general rotation given by

R(φ, θ, ψ) = R(ψ, e3)R(θ, e1)R(φ, e3) (6.1)

i.e. first a rotation about the z-axis by an angle φ, then a rotation by an angleθ about the new x-axis, followed by a rotation by ψ about the new z-axis.Multiplying out the matrices gives a rather complicated expression for a rotationin terms of the three angles, and one needs to figure out what range to choosefor the angles to avoid multiple counting.

The infinitesimal picture near the identity of the group, given by the Liealgebra structure on so(3), is much easier to understand. Recall that for orthog-onal groups the Lie algebra can be identified with the space of antisymmetricmatrices, so one in this case has a basis

l1 =

0 0 00 0 −10 1 0

l2 =

0 0 10 0 0−1 0 0

l3 =

0 −1 01 0 00 0 0

which satisfy the commutation relations

[l1, l2] = l3, [l2, l3] = l1, [l3, l1] = l2

Note that these are exactly the same commutation relations satisfied bythe basis vectors X1, X2, X3 of the Lie algebra su(2), so so(3) and su(2) are

60

isomorphic Lie algebras. They both are the vector space R3 with the same Liebracket operation on pairs of vectors. This operation is familiar in yet anothercontext, that of the cross-product of standard basis vectors ej in R3:

e1 × e2 = e3, e2 × e3 = e1, e3 × e1 = e2

We see that the Lie bracket operation

(X,Y ) ∈ R3 ×R3 → [X,Y ] ∈ R3

that makes R3 a Lie algebra so(3) is just the cross-product on vectors in R3.So far we have three different isomorphic ways of putting a Lie bracket on

R3, making it into a Lie algebra:

1. Identify R3 with antisymmetric real 3 by 3 matrices and take the matrixcommutator as Lie bracket.

2. Identify R3 with skew-adjoint, traceless, complex 2 by 2 matrices and takethe matrix commutator as Lie bracket.

3. Use the vector cross-product on R3 to get a Lie bracket, i.e. define

[v,w] = v ×w

Something very special that happens for orthogonal groups only in three di-mensions is that the vector representation (the defining representation of SO(n)matrices on Rn) is isomorphic to the adjoint representation. Recall that any Liegroup G has a representation (Ad, g) on its Lie algebra g. so(n) can be identified

with the antisymmetric n by n matrices, so is of (real) dimension n2−n2 . Only

for n = 3 is this equal to n, the dimension of the representation on vectors inRn. This corresponds to the geometrical fact that only in 3 dimensions is aplane (in all dimensions rotations are built out of rotations in various planes)determined uniquely by a vector (the vector perpendicular to the plane). Equiv-alently, only in 3 dimensions is there a cross-product v × w which takes twovectors determining a plane to a unique vector perpendicular to the plane.

The isomorphism between the vector representation (πvector,R3) on column

vectors and the adjoint representation (Ad, so(3)) on antisymmetric matrices isgiven by v1

v2

v3

↔ v1l1 + v2l2 + v3l3 =

0 −v3 v2

v3 0 −v1

−v2 v1 0

or in terms of bases by

ej ↔ lj

For the vector representation on column vectors, πvector(g) = g and π′vector(X) =X, where X is an antisymmetric 3 by 3 matrix, and g = eX is an orthogonal 3by 3 matrix. Both act on column vectors by the usual multiplication.

61

For the adjoint representation on antisymmetric matrices, one has

Ad(g)

0 −v3 v2

v3 0 −v1

−v2 v1 0

= g

0 −v3 v2

v3 0 −v1

−v2 v1 0

g−1

The corresponding Lie algebra representation is given by

ad(X)

0 −v3 v2

v3 0 −v1

−v2 v1 0

= [X,

0 −v3 v2

v3 0 −v1

−v2 v1 0

]

where X is a 3 by 3 antisymmetric matrix.One can explicitly check that these representations are isomorphic, for in-

stance by calculating how basis elements lj ∈ so(3) act. On vectors, these lj actby matrix multiplication, giving for instance, for j = 1

l1e1 = 0, l1e2 = e3, l1e3 = −e2

On antisymmetric matrices one has instead the isomorphic relations

(ad(l1))(l1) = 0, (ad(l1))(l2) = l3, (ad(l1))(l3) = −l2

6.2 Spin groups in three and four dimensions

A remarkable property of the orthogonal groups SO(n) is that they come with anassociated group, called Spin(n), with every element of SO(n) correspondingto two distinct elements of Spin(n). If you have seen some topology, whatis at work here is that (for n > 2) the fundamental group of SO(n) is non-trivial, with π1(SO(n)) = Z2 (this means there is a non-contractible loop inSO(n), contractible if you go around it twice). Spin(n) is topologically thesimply-connected double-cover of SO(n), and one can choose the covering mapΦ : Spin(n)→ SO(n) to be a group homomorphism. Spin(n) is a Lie group ofthe same dimension as SO(n), with an isomorphic tangent space at the identity,so the Lie algebras of the two groups are isomorphic: so(n) ' spin(n).

In chapter 26 we will explicitly construct the groups Spin(n) for any n buthere we will just do this for n = 3 and n = 4, using methods specific to these twocases. In the cases n = 5 (where Spin(5) = Sp(2), the 2 by 2 norm-preservingquaternionic matrices) and n = 6 (where Spin(6) = SU(4)) one can also usespecial methods to identify Spin(n) with other matrix groups. For n > 6 thegroup Spin(n) will be something truly distinct.

Given such a construction of Spin(n), we also need to explicitly constructthe homomorphism Φ, and show that its derivative Φ′ is an isomorphism ofLie algebras. We will see that the simplest construction of the spin groupshere uses the group Sp(1) of unit-length quaternions, with Spin(3) = Sp(1)and Spin(4) = Sp(1)× Sp(1). By identifying quaternions and pairs of complexnumbers, we can show that Sp(1) = SU(2) and thus work with these spin groupsas either 2 by 2 complex matrices (for Spin(3)), or pairs of such matrices (forSpin(4)).

62

6.2.1 Quaternions

The quaternions are a number system (denoted by H) generalizing the complexnumber system, with elements q ∈ H that can be written as

q = q0 + q1i + q2j + q3k, qi ∈ R

with i, j,k ∈ H satisfying

i2 = j2 = k2 = −1, ij = −ji = k,ki = −ik = j, jk = −kj = i

and a conjugation operation that takes

q → q = q0 − q1i− q2j− q3k

This operation satisfies (for u, v ∈ H)

uv = vu

As a vector space over R, H is isomorphic with R4. The length-squaredfunction on this R4 can be written in terms of quaternions as

|q|2 = qq = q20 + q2

1 + q22 + q2

3

and is multiplicative since

|uv|2 = uvuv = uvvu = |u|2|v|2

Usingqq

|q|2= 1

one has a formula for the inverse of a quaternion

q−1 =q

|q|2

The length one quaternions thus form a group under multiplication, calledSp(1). There are also Lie groups called Sp(n) for larger values of n, consistingof invertible matrices with quaternionic entries that act on quaternionic vectorspreserving the quaternionic length-squared, but these play no significant role inquantum mechanics so we won’t study them further. Sp(1) can be identifiedwith the three-dimensional sphere since the length one condition on q is

q20 + q2

1 + q22 + q2

3 = 1

the equation of the unit sphere S3 ⊂ R4.

63

6.2.2 Rotations and spin groups in four dimensions

Pairs (u, v) of unit quaternions give the product group Sp(1) × Sp(1). Anelement of this group acts on H = R4 by

q → uqv

This action preserves lengths of vectors and is linear in q, so it must correspondto an element of the group SO(4). One can easily see that pairs (u, v) and(−u,−v) give the same linear transformation of R4, so the same element ofSO(4). One can show that SO(4) is the group Sp(1) × Sp(1), with the twoelements (u, v) and (−u,−v) identified. The name Spin(4) is given to the Liegroup Sp(1)× Sp(1) that “double covers” SO(4) in this manner.

6.2.3 Rotations and spin groups in three dimensions

Later on in the course we’ll encounter Spin(4) and SO(4) again, but for nowwe’re interested in the subgroup Spin(3) that only acts non-trivially on 3 of thedimensions, and double-covers not SO(4) but SO(3). To find this, consider thesubgroup of Spin(4) consisting of pairs (u, v) of the form (u, u−1) (a subgroupisomorphic to Sp(1), since elements correspond to a single unit length quaternionu). This subgroup acts on quaternions by conjugation

q → uqu−1

an action which is trivial on the real quaternions, but nontrivial on the “pureimaginary” quaternions of the form

q = ~v = v1i + v2j + v3k

An element u ∈ Sp(1) acts on ~v ∈ R3 ⊂ H as

~v → u~vu−1

This is a linear action, preserving the length |~v|, so corresponds to an elementof SO(3). We thus have a map (which can easily be checked to be a homomor-phism)

Φ : u ∈ Sp(1)→ ~v → u~vu−1 ∈ SO(3)

64

Both u and −u act in the same way on ~v, so we have two elements inSp(1) corresponding to the same element in SO(3). One can show that Φ is asurjective map (one can get any element of SO(3) this way), so it is what is calleda “covering” map, specifically a two-fold cover. It makes Sp(1) a double-cover ofSO(3), and we give this the name “Spin(3)”. This also allows us to characterizemore simply SO(3) as a geometrical space. It is S3 = Sp(1) = Spin(3) withopposite points on the three-sphere identified. This space is known as RP(3),real projective 3-space, which can also be thought of as the space of lines throughthe origin in R4 (each such line intersects S3 in two opposite points).

65

For those who have seen some topology, note that the covering map Φ isan example of a topologically non-trivial cover. It is just not true that topo-logically S3 ' RP3 × (+1,−1). S3 is a connected space, not two disconnectedpieces. This topological non-triviality implies that globally there is no possiblehomomorphism going in the opposite direction from Φ (i.e. SO(3)→ Spin(3)).One can do this locally, picking a local patch in SO(3) and taking the inverseof Φ to a local patch in Spin(3), but this won’t work if we try and extend itglobally to all of SO(3).

The identification R2 = C allowed us to represent elements of the unit circlegroup U(1) as exponentials eiθ, where iθ was in the Lie algebra u(1) = R ofU(1). For Sp(1) one can do much the same thing, with the Lie algebra sp(1)now the space of all pure imaginary quaternions, which one can identify withR3 by

w =

w1

w2

w3

∈ R3 ↔ ~w = w1i + w2j + w3k ∈ H

Unlike the U(1) case, there’s a non-trivial Lie bracket, just the commutator ofquaternions.

Elements of the group Sp(1) are given by exponentiating such Lie algebraelements, which we will write in the form

u(θ,w) = eθ ~w = cos θ + ~w sin θ

66

where θ ∈ R and ~w is a purely imaginary quaternion of unit length. Taking θas a parameter, these give paths in Sp(1) going through the identity at θ = 0,with velocity vector ~w since

d

dθu(θ,w)|θ=0 = (− sin θ + ~w cos θ)|θ=0 = ~w

We can explicitly evaluate the homomorphism Φ on such elements u(θ,w) ∈Sp(1), with the result that Φ takes u(θ,w) to a rotation by an angle 2θ aroundthe axis w:

Theorem 6.1.

Φ(u(θ,w)) = R(2θ,w)

Proof. First consider the special case w = e3 of rotations about the 3-axis.

u(θ, e3) = eθk = cos θ + k sin θ

and

u(θ, e3)−1 = e−θk = cos θ − k sin θ

so Φ(u(θ, e3)) is the rotation that takes v (identified with the quaternion ~v =v1i + v2j + v3k) to

u(θ, e3)~vu(θ, e3)−1 =(cos θ + k sin θ)(v1i + v2j + v3k)(cos θ − k sin θ)

=(v1(cos2 θ − sin2 θ)− v2(2 sin θ cos θ))i

+ (2v1 sin θ cos θ + v2(cos2 θ − sin2 θ))j + v3k

=(v1 cos 2θ − v2 sin 2θ)i + (v1 sin 2θ + v2 cos 2θ)j + v3k

This is the orthogonal transformation of R3 given by

v =

v1

v2

v3

→cos 2θ − sin 2θ 0

sin 2θ cos 2θ 00 0 1

v1

v2

v3

(6.2)

One can readily do the same calculation for the case of e1, then use theEuler angle parametrization of equation 6.1 to show that a general u(θ,w) canbe written as a product of the cases already worked out.

Notice that as θ goes from 0 to 2π, u(θ,w) traces out a circle in Sp(1). Thehomomorphism Φ takes this to a circle in SO(3), one that gets traced out twiceas θ goes from 0 to 2π, explicitly showing the nature of the double coveringabove that particular circle in SO(3).

The derivative of the map Φ will be a Lie algebra homomorphism, a linearmap

Φ′ : sp(1)→ so(3)

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It takes the Lie algebra sp(1) of pure imaginary quaternions to the Lie algebraso(3) of 3 by 3 antisymmetric real matrices. One can compute it easily on basisvectors, using for instance equation 6.2 above to find for the case ~w = k

Φ′(k) =d

dθΦ(cos θ + k sin θ)|θ=0

=

−2 sin 2θ −2 cos 2θ 02 cos 2θ −2 sin 2θ 0

0 0 0

|θ=0

=

0 −2 02 0 00 0 0

= 2l3

Repeating this on other basis vectors one finds that

Φ′(i) = 2l1,Φ′(j) = 2l2,Φ

′(k) = 2l3

Thus Φ′ is an isomorphism of sp(1) and so(3) identifying the bases

i

2,

j

2,k

2and l1, l2, l3

Note that it is the i2 ,

j2 ,

k2 that satisfy simple commutation relations

[i

2,

j

2] =

k

2, [

j

2,k

2] =

i

2, [

k

2,

i

2] =

j

2

6.2.4 The spin group and SU(2)

Instead of doing calculations using quaternions with their non-commutativityand special multiplication laws, it is more conventional to choose an isomorphismbetween quaternions H and a space of 2 by 2 complex matrices, and work justwith matrix multiplication and complex numbers. The Pauli matrices can beused to gives such an isomorphism, taking

1→ 1 =

(1 00 1

), i→ −iσ1 =

(0 −i−i 0

), j→ −iσ2 =

(0 −11 0

)

k→ −iσ3 =

(−i 00 i

)The correspondence between H and 2 by 2 complex matrices is then given

by

q = q0 + q1i + q2j + q3k↔(q0 − iq3 −q2 − iq1

q2 − iq1 q0 + iq3

)Since

det

(q0 − iq3 −q2 − iq1

q2 − iq1 q0 + iq3

)= q2

0 + q21 + q2

2 + q23

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we see that the length-squared function on quaternions corresponds to the de-terminant function on 2 by 2 complex matrices. Taking q ∈ Sp(1), so of lengthone, the corresponding complex matrix is in SU(2).

Under this identification of H with 2 by 2 complex matrices, we have anidentification of Lie algebras sp(1) = su(2) between pure imaginary quaternionsand skew-Hermitian trace-zero 2 by 2 complex matrices

~w = w1i + w2j + w3k↔(−iw3 −w2 − iw1

w2 − iw1 iw3

)= −iw · σ

The basis i2 ,

j2 ,

k2 gets identified with a basis for the Lie algebra su(2) which

written in terms of the Pauli matrices is

Xj = −iσj2

with the Xj satisfying the commutation relations

[X1, X2] = X3, [X2, X3] = X1, [X3, X1] = X2

which are precisely the same commutation relations as for so(3)

[l1, l2] = l3, [l2, l3] = l1, [l3, l1] = l2

We now have no less than three isomorphic Lie algebras sp(1) = su(2) =so(3), with elements that get identified as follows

(w1

i2 + w2

j2 + w3

k2

)↔ − i

2

(w3 w1 − iw2

w1 + iw2 −w3

)↔

0 −w3 w2

w3 0 −w1

−w2 w1 0

This isomorphism identifies basis vectors by

i

2↔ −iσ1

2↔ l1

etc. The first of these identifications comes from the way we chose to identifyH with 2 by 2 complex matrices. The second identification is Φ′, the derivativeat the identity of the covering map Φ.

On each of these isomorphic Lie algebras we have adjoint Lie group (Ad)and Lie algebra (ad) representations. Ad si given by conjugation with the cor-responding group elements in Sp(1), SU(2) and SO(3). ad is given by takingcommutators in the respective Lie algebras of pure imaginary quaternions, skew-Hermitian trace-zero 2 by 2 complex matrices and 3 by 3 real antisymmetricmatrices.

Note that these three Lie algebras are all three-dimensional real vectorspaces, so these are real representations. If one wants a complex representa-tion, one can complexify and take complex linear combinations of elements.This is less confusing in the case of su(2) than for sp(1) since taking complex

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linear combinations of skew-Hermitian trace-zero 2 by 2 complex matrices justgives all trace-zero 2 by 2 matrices (the Lie algebra sl(2,C)).

In addition, recall that there is a fourth isomorphic version of this repre-sentation, the representation of SO(3) on column vectors. This is also a realrepresentation, but can straightforwardly be complexified. Since so(3) and su(2)are isomorphic Lie algebras, their complexifications so(3)C and sl(2,C) will alsobe isomorphic.

In terms of 2 by 2 complex matrices, one can exponentiate Lie algebra ele-ments to get group elements in SU(2) and define

Ω(θ,w) = eθ(w1X1+w2X2+w3X3) = e−iθ2w·σ (6.3)

= cos(θ

2)1− i(w · σ) sin(

θ

2) (6.4)

Transposing the argument of theorem 6.1 from H to complex matrices, one findsthat, identifying

v↔ v · σ =

(v3 v1 − iv2

v1 + iv2 −v3

)one has

Φ(Ω(θ,w)) = R(θ,w)

with Ω(θ,w) acting by conjugation, taking

v · σ → Ω(θ,w)(v · σ)Ω(θ,w)−1 = (R(θ,w)v) · σ (6.5)

Note that in changing from the quaternionic to complex case, we are treatingthe factor of 2 differently, since in the future we will want to use Ω(θ,w) toperform rotations by an angle θ. In terms of the identification SU(2) = Sp(1),we have Ω(θ,w) = u( θ2 ,w).

Recall that any SU(2) matrix can be written in the form(α β

−β α

)α = q0 − iq3, β = −q2 − iq1

with α, β ∈ C arbitrary complex numbers satisfying |α|2+|β|2 = 1. One can alsowrite down a somewhat unenlightening formula for the map Φ : SU(2)→ SO(3)in terms of such explicit SU(2) matrices, getting

Φ(

(α β

−β α

)) =

Im(β2 − α2) Re(α2 + β2) 2 Im(αβ)Re(β2 − α2) Im(α2 + β2) 2 Re(αβ)

2 Re(αβ) −2 Im(αβ) |α|2 − |β|2

See [64], page 123-4, for a derivation.

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6.3 A summary

To summarize, we have shown that for three dimensions we have two distinctLie groups:

• Spin(3), which geometrically is the space S3. Its Lie algebra is R3 withLie bracket the cross-product. We have seen two different explicit con-structions of Spin(3), in terms of unit quaternions (Sp(1)), and in termsof 2 by 2 unitary matrices of determinant 1 (SU(2)).

• SO(3), with the same Lie algebra R3 with the same Lie bracket.

There is a group homomorphism Φ that takes the first group to the second,which is a two-fold covering map. Its derivative Φ′ is an isomorphism of the Liealgebras of the two groups.

We can see from these constructions two interesting irreducible representa-tions of these groups:

• A representation on R3 which can be constructed in two different ways: asthe adjoint representation of either of the two groups, or as the definingrepresentation of SO(3). This is known to physicists as the “spin 1”representation.

• A representation of the first group on C2, which is most easily seen asthe defining representation of SU(2). It is not a representation of SO(3),since going once around a non-contractible loop starting at the identitytakes one to minus the identity, not back to the identity as required. Thisis called the “spin 1/2 or “spinor” representation and will be studied inmore detail in chapter 7.


For another discussion of the relationship of SO(3) and SU(2) as well as aconstruction of the map Φ, see [64], sections 4.2 and 4.3, as well as [3], chapter8, and [66] Chapters 2 and 4.

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Chapter 7

Rotations and the Spin 12

Particle in a Magnetic Field

The existence of a non-trivial double-cover Spin(3) of the three-dimensionalrotation group may seem to be a somewhat obscure mathematical fact. Re-markably though, the existence of fundamental spin- 1

2 particles shows that itis Spin(3) rather than SO(3) that is the symmetry group corresponding torotations of fundamental quantum systems. Ignoring the degrees of freedom de-scribing their motion in space, which we will examine in later chapters, states ofelementary particles such as the electron are described by a state space H = C2,with rotations acting on this space by the two-dimensional irreducible represen-tation of SU(2) = Spin(3).

This is the same two-state system studied in chapter 3, with the SU(2)action found there now acquiring an interpretation as corresponding to thedouble-cover of rotations of physical space. In this chapter we will revisit thatexample, emphasizing the relation to rotations.

7.1 The spinor representation

In chapter 6 we examined in great detail various ways of looking at a particularthree-dimensional irreducible real representation of the groups SO(3), SU(2)and Sp(1). This was the adjoint representation for those three groups, andisomorphic to the vector representation for SO(3). In the SU(2) and Sp(1)cases, there is an even simpler non-trivial irreducible representation than theadjoint: the representation of 2 by 2 complex matrices in SU(2) on columnvectors C2 by matrix multiplication or the representation of unit quaternions inSp(1) on H by scalar multiplication. Choosing an identification C2 = H theseare isomorphic representations on C2 of isomorphic groups, and for calculationalconvenience we will use SU(2) and its complex matrices rather than dealing withquaternions. This irreducible representation is known as the “spinor” or “spin”

73

representation of Spin(3) = SU(2). The homomorphism πspinor defining therepresentation is just the identity map from SU(2) to itself.

The spin representation of SU(2) is not a representation of SO(3). Thedouble cover map Φ : SU(2) → SO(3) is a homomorphism, so given a rep-resentation (π, V ) of SO(3) one gets a representation (π Φ, V ) of SU(2) bycomposition. One cannot go in the other direction: there is no homomorphismSO(3) → SU(2) that would allow us to make the standard representation ofSU(2) on C2 into an SO(3) representation.

One could try and define a representation of SO(3) by

π : g ∈ SO(3)→ π(g) = πspinor(g) ∈ SU(2)

where g is some choice of one of the elements g ∈ SU(2) satisfying Φ(g) = g.The problem with this is that we won’t quite get a homomorphism. Changingour choice of g will introduce a minus sign, so π will only be a homomorphismup to sign

π(g1)π(g2) = ±π(g1g2)

The nontrivial nature of the double-covering ensures that there is no way tocompletely eliminate all minus signs, no matter how we choose g. Exampleslike this, which satisfy the representation property only one up to a sign ambi-guity, are known as “projective representations”. So, the spinor representationof SU(2) = Spin(3) is only a projective representation of SO(3), not a truerepresentation of SO(3).

Quantum mechanics texts sometimes deal with this phenomenon by notingthat physically there is an ambiguity in how one specifies the space of states H,with multiplication by an overall scalar not changing the eigenvalues of operatorsor the relative probabilities of observing these eigenvalues. As a result, the signambiguity has no physical effect. It seems more straightforward though to nottry and work with projective representations, but just use the larger groupSpin(3), accepting that this is the correct symmetry group reflecting the actionof rotations on three-dimensional quantum systems.

The spin representation is more fundamental than the vector representation,in the sense that the spin representation cannot be found just knowing thevector representation, but the vector representation of SO(3) can be constructedknowing the spin representation of SU(2). We have seen this in the identificationof R3 with 2 by 2 complex matrices, where rotations become conjugation by spinrepresentation matrices. Another way of seeing this uses the tensor product, andis explained in section 9.4.3. Note that taking spinors as fundamental entailsabandoning the descriptions of three-dimensional geometry purely in terms ofreal numbers. While the vector representation is a real representation of SO(3)or Spin(3), the spinor representation is a complex representation.

7.2 The spin 1/2 particle in a magnetic field

In chapter 3 we saw that a general quantum system with H = C2 could beunderstood in terms of the action of U(2) on C2. The self-adjoint observables

74

correspond (up to a factor of i) to the corresponding Lie algebra representation.The U(1) ⊂ U(2) subgroup commutes with everything else and can be analyzedseparately, so we will just consider the SU(2) subgroup. For an arbitrary suchsystem, the group SU(2) has no particular geometric significance. When itoccurs in its role as double-cover of the rotational group, the quantum systemis said to carry “spin”, in particular “spin one-half” (in chapter 8 will discussstate spaces of higher spin values).

As before, we take as a standard basis for the Lie algebra su(2) the operatorsXj , j = 1, 2, 3, where

Xj = −iσj2


[X1, X2] = X3, [X2, X3] = X1, [X3, X1] = X2

To make contact with the physics formalism, we’ll define self-adjoint operators

Sj = iXj =σj2

We could have chosen the other sign, but this is the standard convention ofthe physics literature. In general, to a skew-adjoint operator (which is whatone gets from a unitary Lie algebra representation and what exponentiates tounitary operators) we will associate a self-adjoint operator by multiplying byi. These self-adjoint operators have real eigenvalues (in this case ± 1

2 ), so arefavored by physicists as observables since such eigenvalues will be related toexperimental results. In the other direction, given a physicist’s observable self-adjoint operator, we will multiply by −i to get a skew-adjoint operator that canbe exponentiated to get a unitary representation.

Note that the conventional definition of these operators in physics textsincludes a factor of ~:

Sphysj = i~Xj =~σj2

A compensating factor of 1/~ is then introduced when exponentiating to getgroup elements

Ω(θ,w) = e−iθ~w·Sphys ∈ SU(2)

which do not depend on ~. The reason for this convention has to do with theaction of rotations on functions on R3 (see chapter 17) and the appearance of~ in the definition of the momentum operator. Our definitions of Sj and ofrotations using (see equation 6.3)

Ω(θ,w) = e−iθw·S = eθw·X

will not include these factors of ~, but in any case they will be equivalent tothe physics text definitions when we make our standard choice of working withunits such that ~ = 1.

75

States in H = C2 that have a well-defined value of the observable Sj will bethe eigenvectors of Sj , with value for the observable the corresponding eigen-value, which will be ± 1

2 . Measurement theory postulates that if we perform themeasurement corresponding to Sj on an arbitrary state |ψ〉, then we will

• with probability c+ get a value of +12 and leave the state in an eigenvector

|j,+ 12 〉 of Sj with eigenvalue + 1

2

• with probability c− get a value of − 12 and leave the state in an eigenvector

|j,− 12 〉 of Sj with eigenvalue − 1

2

where if

|ψ〉 = α|j,+1

2〉+ β|j,−1

2〉

we have

c+ =|α|2

|α|2 + |β|2, c− =

|β|2

|α|2 + |β|2

After such a measurement, any attempt to measure another Sk, k 6= j will give± 1

2 with equal probability and put the system in a corresponding eigenvectorof Sk.

If a quantum system is in an arbitrary state |ψ〉 it may not have a well-defined value for some observable A, but one can calculate the “expected value”of A. This is the sum over a basis of H consisting of eigenvectors (which willall be orthogonal) of the corresponding eigenvalues, weighted by the probabilityof their occurrence. The calculation of this sum in this case (A = Sj) usingexpansion in eigenvectors of Sj gives

〈ψ|A|ψ〉〈ψ|ψ〉

=(α〈j,+ 1

2 |+ β〈j,− 12 |)A(α|j,+ 1

2 〉+ β|j,− 12 〉)

(α〈j,+ 12 |+ β〈j,− 1

2 |)(α|j,+12 〉+ β|j,− 1

2 〉)

=|α|2(+ 1

2 ) + |β|2(− 12 )

|α|2 + |β|2

=c+(+1

2) + c−(−1

2)

One often chooses to simplify such calculations by normalizing states so thatthe denominator 〈ψ|ψ〉 is 1. Note that the same calculation works in generalfor the probability of measuring the various eigenvalues of an observable A, aslong as one has orthogonality and completeness of eigenvectors.

In the case of a spin one-half particle, the group Spin(3) = SU(2) acts onstates by the spinor representation with the element Ω(θ,w) ∈ SU(2) acting as

|ψ〉 → Ω(θ,w)|ψ〉

As we saw in chapter 6, the Ω(θ,w) also act on self-adjoint matrices by conju-gation, an action that corresponds to rotation of vectors when one makes theidentification

v↔ v · σ

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(see equation 6.5). Under this identification the Sj correspond (up to a factorof 2) to the basis vectors ej . One can write their transformation rule as

Sj → S′j = Ω(θ,w)SjΩ(θ,w)−1

and S′1S′2S′3

= R(θ,w)T

S1

S2

S3

Note that, recalling the discussion in section 4.1, rotations on sets of basisvectors like this involve the transpose R(θ,w)T of the matrix R(θ,w) that actson coordinates.

In chapter 43 we will get to the physics of electromagnetic fields and howparticles interact with them in quantum mechanics, but for now all we need toknow is that for a spin one-half particle, the spin degree of freedom that we aredescribing by H = C2 has a dynamics described by the Hamiltonian

H = −µ ·B (7.1)

Here B is the vector describing the magnetic field, and

µ = g−e

2mcS

is an operator called the magnetic moment operator. The constants that appearare: −e the electric charge, c the speed of light, m the mass of the particle, and g,a dimensionless number called the “gyromagnetic ratio”, which is approximately2 for an electron, about 5.6 for a proton.

The Schrodinger equation is

d

dt|ψ(t)〉 = −i(−µ ·B)|ψ(t)〉

with solution

|ψ(t)〉 = U(t)|ψ(0)〉

where

U(t) = eitµ·B = eit−ge2mcS·B = et

ge2mcX·B = et

ge|B|2mc X· B

|B|

The time evolution of a state is thus given at time t by the same SU(2) elementthat, acting on vectors, gives a rotation about the axis w = B

|B| by an angle

ge|B|t2mc

so is a rotation about w taking place with angular velocity ge|B|2mc .

The amount of non-trivial physics that is described by this simple system isimpressive, including:

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• The Zeeman effect: this is the splitting of atomic energy levels that occurswhen an atom is put in a constant magnetic field. With respect to theenergy levels for no magnetic field, where both states in H = C2 have thesame energy, the term in the Hamiltonian given above adds

±ge|B|4mc

to the two energy levels, giving a splitting between them proportional tothe size of the magnetic field.

• The Stern-Gerlach experiment: here one passes a beam of spin one-halfquantum systems through an inhomogeneous magnetic field. We have notyet discussed particle motion, so more is involved here than the simpletwo-state system. However, it turns out that one can arrange this in sucha way as to pick out a specific direction w, and split the beam into twocomponents, of eigenvalue + 1

2 and − 12 for the operator w · S.

• Nuclear magnetic resonance spectroscopy: one can subject a spin one-halfsystem to a time-varying magnetic field B(t), and such a system will bedescribed by the same Schrodinger equation, although now the solutioncannot be found just by exponentiating a matrix. Nuclei of atoms providespin one-half systems that can be probed with time and space-varyingmagnetic fields, allowing imaging of the material that they make up.

• Quantum computing: attempts to build a quantum computer involve try-ing to put together multiple systems of this kind (qubits), keeping themisolated from perturbations by the environment, but still allowing inter-action with the system in a way that preserves its quantum behavior.The 2012 Physics Nobel prize was awarded for experimental work makingprogress in this direction.

7.3 The Heisenberg picture

So far in this course we’ve been describing what is known as the Schrodingerpicture of quantum mechanics. States in H are functions of time, obeyingthe Schrodinger equation determined by a Hamiltonian observable H, whileobservable self-adjoint operators A are time-independent. Time evolution isgiven by a unitary transformation

U(t) = e−itH , |ψ(t)〉 = U(t)|ψ(0)〉

One can instead use U(t) to make a unitary transformation that puts thetime-dependence in the observables, removing it from the states, as follows:

|ψ(t)〉 → |ψ(t)〉H = U−1(t)|ψ(t)〉 = |ψ(0)〉, A→ AH(t) = U−1(t)AU(t)

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where the “H” subscripts for “Heisenberg” indicate that we are dealing with“Heisenberg picture” observables and states. One can easily see that the physi-cally observable quantities given by eigenvalues and expectations values remainthe same:

H〈ψ(t)|AH |ψ(t)〉H = 〈ψ(t)|U(t)(U−1(t)AU(t))U−1(t)|ψ(t)〉 = 〈ψ(t)|A|ψ(t)〉

In the Heisenberg picture the dynamics is given by a differential equationnot for the states but for the operators. Recall from our discussion of the adjointrepresentation (see equation 5.1) the formula

d

dt(etXY e−tX) = (

d

dt(etXY ))e−tX + etXY (

d

dte−tX)

= XetXY e−tX − etXY e−tXX

Using this withY = A, X = iH

we findd

dtAH(t) = [iH,AH(t)] = i[H,AH(t)]

and this equation determines the time evolution of the observables in the Heisen-berg picture.

Applying this to the case of the spin one-half system in a magnetic field, andtaking for our observable S (the Sj , taken together as a column vector) we find

d

dtSH(t) = i[H,SH(t)] = i

eg

2mc[SH(t) ·B,SH(t)] (7.2)

We know from the discussion above that the solution will be

SH(t) = U(t)SH(0)U(t)−1

for

U(t) = e−itge|B|2mc S· B

|B|

and thus the spin vector observable evolves in the Heisenberg picture by rotating

about the magnetic field vector with angular velocity ge|B|2mc .

7.4 The Bloch sphere and complex projectivespace

There is a different approach one can take to characterizing states of a quantumsystem with H = C2. Multiplication of vectors in H by a non-zero complexnumber does not change eigenvectors, eigenvalues or expectation values, so ar-guably has no physical effect. Multiplication by a real scalar just correspondsto a change in normalization of the state, and we will often use this freedomto work with normalized states, those satisfying 〈ψ|ψ〉 = 1. With normalized

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states, one still has the freedom to multiply states by a phase eiθ without chang-ing eigenvectors, eigenvalues or expectation values. In terms of group theory,the overall U(1) in the unitary group U(2) acts on H by a representation ofU(1), which can be characterized by an integer, the corresponding “charge”,but this decouples from the rest of the observables and is not of much interest.One is mainly interested in the SU(2) part of the U(2), and the observablesthat correspond to its Lie algebra.

Working with normalized states in this case corresponds to working withunit-length vectors in C2, which are given by points on the unit sphere S3. Ifwe don’t care about the overall U(1) action, we can imagine identifying all statesthat are related by a phase transformation. Using this equivalence relation wecan define a new set, whose elements are the “cosets”, elements of S3 ⊂ C2,with elements that differ just by multiplication by eiθ identified. The set of theseelements forms a new geometrical space, called the “coset space”, often writtenS3/U(1). This structure is called a “fibering” of S3 by circles, and is knownas the “Hopf fibration”. Try an internet search for various visualizations of thegeometrical structure involved, a surprising decomposition of three-dimensionalspace into non-intersecting curves.

The same space can be represented in a different way, as C2/C∗, by takingall elements of C2 and identifying those related by muliplication by a non-zerocomplex number. If we were just using real numbers, R2/R∗ can be thought ofas the space of all lines in the plane going through the origin.

One sees that each such line hits the unit circle in two opposite points, so

80

this set could be parametrized by a semi-circle, identifying the points at thetwo ends. This space is given the name RP 1, the “real projective line”, andthe analog space of lines through the origin in Rn is called RPn−1. What weare interested in is the complex analog CP 1, which is often called the “complexprojective line”.

To better understand CP 1, one would like to put coordinates on it. Astandard way to choose such a coordinate is to associate to the vector(

z1

z2

)∈ C2

the complex number z1/z2. Overall multiplication by a complex number willdrop out in this ratio, so one gets different values for the coordinate z1/z2 foreach different coset element, and it appears that elements of CP 1 correspond topoints on the complex plane. There is however one problem with this coordinate:the coset of (

10

)does not have a well-defined value: as one approaches this point one moves offto infinity in the complex plane. In some sense the space CP 1 is the complexplane, but with a “point at infinity” added.

It turns out that CP 1 is best thought of not as a plane together with apoint, but as a sphere, with the relation to the plane and the point at infinitygiven by stereographic projection. Here one creates a one-to-one mapping byconsidering the lines that go from a point on the sphere to the north pole ofthe sphere. Such lines will intersect the plane in a point, and give a one-to-onemapping between points on the plane and points on the sphere, except for thenorth pole. Now, one can identify the north pole with the “point at infinity”,and thus the space CP 1 can be identified with the space S2. The picture lookslike this

81

and the equations relating coordinates (X1, X2, X3) on the sphere and the com-plex coordinate z1/z2 = z = x+ iy on the plane are given by

x =X1

1−X3, y =

X2

1−X3

and

X1 =2x

x2 + y2 + 1, X2 =

2y

x2 + y2 + 1, X3 =

x2 + y2 − 1

x2 + y2 + 1

The action of SU(2) on H by(z1

z2

)→(α β

−β α

)(z1

z2

)takes

z =z1

z2→ αz + β

−βz + α

Such transformations of the complex plane are conformal (angle-preserving)transformations known as “Mobius transformations”. One can check that thecorresponding transformation on the sphere is the rotation of the sphere in R3

corresponding to this SU(2) = Spin(3) transformation.To mathematicians, this sphere identified with CP 1 is known as the “Rie-

mann sphere”, whereas physicists often instead use the terminology of “Blochsphere”. It provides a useful parametrization of the states of the qubit system,up to scalar multiplication, which is supposed to be physically irrelevant. The

82

North pole is the “spin-up” state, the South pole is the “spin-down” state, andalong the equator one finds the two states that have definite values for S1, aswell as the two that have definite values for S2.

Notice that the inner product on vectors in H does not correspond at allto the inner product of unit vectors in R3. The North and South poles of theBloch sphere correspond to orthogonal vectors in H, but they are not at allorthogonal thinking of the corresponding points on the Bloch sphere as vectorsin R3. Similarly, eigenvectors for S1 and S2 are orthogonal on the Bloch sphere,but not at all orthogonal in H.

Many of the properties of the Bloch sphere parametrization of states inH arespecial to the fact that H = C2. In the next class we will study systems of spinn2 , where H = Cn. In these cases there is still a two-dimensional Bloch sphere,but only certain states in H are parametrized by it. We will see other examplesof systems with “coherent states” analogous to the states parametrized by theBloch sphere, but the case H has the special property that all states (up toscalar multiplication) are such “coherent states”.

83


Just about every quantum mechanics textbook works out this example of a spin1/2 particle in a magnetic field. For one example, see chapter 14 of [63]. Foran inspirational discussion of spin and quantum mechanics, together with moreabout the Bloch sphere, see chapter 22 of [50].

84

Chapter 8

Representations of SU(2)and SO(3)

For the case of G = U(1), in chapter 2 we were able to classify all complexirreducible representations by an element of Z and explicitly construct eachirreducible representation. We would like to do the same thing here for repre-sentations of SU(2) and SO(3). The end result will be that irreducible repre-sentations of SU(2) are classified by a non-negative integer n = 0, 1, 2, 3, · · · ,and have dimension n+ 1, so we’ll (hoping for no confusion with the irreduciblerepresentations (πn,C) of U(1)) denote them (πn,C

n+1). For even n these willalso be irreducible representations of SO(3), but this will not be true for oddn. It is common in physics to label these representations by s = n

2 = 0, 12 , 1, · · ·

and call the representation labeled by s the “spin s representation”. We alreadyknow the first three examples:

• Spin 0: (π0,C) is the trivial representation on V, with

π0(g) = 1 ∀g ∈ SU(2)

This is also a representation of SO(3). In physics, this is sometimes calledthe “scalar representation”. Saying that something transforms under ro-tations as the “scalar representation” just means that it is invariant underrotations.

• Spin 12 : Taking

π1(g) = g ∈ SU(2) ⊂ U(2)

gives the defining representation on C2. This is the spinor representationdiscussed in chapter 7. It is not a representation of SO(3).

• Spin 1: Since SO(3) is a group of 3 by 3 matrices, it acts on vectors in R3.This is just the standard action on vectors by rotation. In other words,the representation is (ρ,R3), with ρ the identity homomorphism

g ∈ SO(3)→ ρ(g) = g ∈ SO(3)

85

One can complexify to get a representation on C3, which in this case justmeans acting with SO(3) matrices on column vectors, replacing the realcoordinates of vectors by complex coordinates. This is sometimes calledthe “vector representation”, and we saw in chapter 6 that it is isomorphicto the adjoint representation.

One gets a representation (π2,C3) of SU(2) by just composing the homo-

morphisms Φ and ρ:

π2 = ρ Φ : SU(2)→ SO(3)

This is the adjoint representation of SU(2).

8.1 Representations of SU(2): classification

8.1.1 Weight decomposition

If we make a choice of a U(1) ⊂ SU(2), then given any representation (π, V ) ofSU(2) of dimension m, we get a representation (π|U(1), V ) of U(1) by restrictionto the U(1) subgroup. Since we know the classification of irreducibles of U(1),we know that

(π|U(1), V ) = Cq1 ⊕Cq2 ⊕ · · · ⊕Cqm

for q1, q2, · · · , qm ∈ Z, where Cq denotes the one-dimensional representationof U(1) corresponding to the integer q. These are called the “weights” of therepresentation V . They are exactly the same thing we discussed earlier as“charges”, but here we’ll favor the mathematician’s terminology since the U(1)here occurs in a context far removed from that of electromagnetism and itselectric charges.

Since our standard choice of coordinates (the Pauli matrices) picks out thez-direction and diagonalizes the action of the U(1) subgroup corresponding torotation about this axis, this is the U(1) subgroup we will choose to define theweights of the SU(2) representation V . This is the subgroup of elements ofSU(2) of the form (

eiθ 00 e−iθ

)Our decomposition of an SU(2) representation (π, V ) into irreducible represen-tations of this U(1) subgroup equivalently means that we can choose a basis ofV so that

π

(eiθ 00 e−iθ

)=

eiθq1 0 · · · 0

0 eiθq2 · · · 0· · · · · ·0 0 · · · eiθqm

An important property of the set of integers qj is the following:

Theorem. If q is in the set qj, so is −q.

86

Proof. Recall that if we diagonalize a unitary matrix, the diagonal entries arethe eigenvalues, but their order is undetermined: acting by permutations onthese eigenvalues we get different diagonalizations of the same matrix. In thecase of SU(2) the matrix

P =

(0 1−1 0

)has the property that conjugation by it permutes the diagonal elements, inparticular

P

(eiθ 00 e−iθ

)P−1 =

(e−iθ 0

0 eiθ

)So

π(P )π(

(eiθ 00 e−iθ

))π(P )−1 = π(

(e−iθ 0

0 eiθ

))

and we see that π(P ) gives a change of basis of V such that the representationmatrices on the U(1) subgroup are as before, with θ → −θ. Changing θ → −θin the representation matrices is equivalent to changing the sign of the weightsqj . The elements of the set qj are independent of the basis, so the additionalsymmetry under sign change implies that for each non-zero element in the setthere is another one with the opposite sign.

Looking at our three examples so far, we see that the scalar or spin 0 repre-sentation of course is one-dimensional of weight 0

(π0,C) = C0

and the spinor or spin 12 representation decomposes into U(1) irreducibles of

weights −1,+1:(π1,C

2) = C−1 ⊕C+1

For the spin 1 representation, recall that our double-cover homomorphismΦ takes (

eiθ 00 e−iθ

)∈ SU(2)→

cos 2θ − sin 2θ 0sin 2θ cos 2θ 0

0 0 1

∈ SO(3)

Acting with the SO(3) matrix on the right on C3 will give a unitary transforma-tion of C3, so in the group U(3). One can show that the upper left diagonal 2 by2 block acts on C2 with weights −2,+2, whereas the bottom right element actstrivially on the remaining part of C3, which is a one-dimensional representationof weight 0. So, the spin 1 representation decomposes as

(π2,C3) = C−2 ⊕C0 ⊕C+2

Recall that the spin 1 representation of SU(2) is often called the “vector” rep-resentation, since it factors in this way through the representation of SO(3) byrotations on three-dimensional vectors.

87

8.1.2 Lie algebra representations: raising and lowering op-erators

To proceed further in characterizing a representation (π, V ) of SU(2) we needto use not just the action of the chosen U(1) subgroup, but the action ofgroup elements in the other two directions away from the identity. The non-commutativity of the group keeps us from simultaneously diagonalizing thoseactions and assigning weights to them. We can however work instead with thecorresponding Lie algebra representation (π′, V ) of su(2). As in the U(1) case,the group representation is determined by the Lie algebra representation. Wewill see that for the Lie algebra representation, we can exploit the complexifica-tion (recall section 5.4) sl(2,C) of su(2) to further analyze the possible patternsof weights.

Recall that the Lie algebra su(2) can be thought of as the tangent space R3

to SU(2) at the identity element, with a basis given by the three skew-adjoint2 by 2 matrices

Xj = −i12σj


[X1, X2] = X3, [X2, X3] = X1, [X3, X1] = X2

We will often use the self-adjoint versions Sj = iXj that satisfy

[S1, S2] = iS3, [S2, S3] = iS1, [S3, S1] = iS2

A unitary representation (π, V ) of SU(2) of dimension m is given by a homo-morphism

π : SU(2)→ U(m)

and we can take the derivative of this to get a map between the tangent spacesof SU(2) and of U(m), at the identity of both groups, and thus a Lie algebrarepresentation

π′ : su(2)→ u(m)

which takes skew-adjoint 2 by 2 matrices to skew-adjoint m by m matrices,preserving the commutation relations.

We have seen in section 8.1.1 that restricting the representation (π, V ) tothe diagonal U(1) subgroup of SU(2) and decomposing into irreducibles tells usthat we can choose a basis of V so that

(π, V ) = (πq1 ,C)⊕ (πq2 ,C)⊕ · · · ⊕ (πqm ,C)

For our choice of U(1) as all matrices of the form

ei2θS3 =

(eiθ 00 e−iθ

)88

with eiθ going around U(1) once as θ goes from 0 to 2π, this means we canchoose a basis of V so that

π(ei2θS3) =

eiθq1 0 · · · 0

0 eiθq2 · · · 0· · · · · ·0 0 · · · eiθqm

Taking the derivative of this representation to get a Lie algebra representation,using

π′(X) =d

dθπ(eθX)|θ=0

we find for X = i2S3

π′(i2S3) =d

dθ

eiθq1 0 · · · 0

0 eiθq2 · · · 0· · · · · ·0 0 · · · eiθqm

|θ=0

=

iq1 0 · · · 00 iq2 · · · 0· · · · · ·0 0 · · · iqm

Recall that π′ is a real-linear map from a real vector space (su(2) = R3) to

another real vector space (u(n), the skew-Hermitian m by m complex matrices).We can use complex linearity to extend any such map to a complex-linear mapfrom su(2)C (the complexification of su(2)) to u(m)C (the complexification ofu(m)). su(2)C is all complex linear combinations of the skew-adjoint, trace-free 2 by 2 matrices: the Lie algebra sl(2,C) of all complex, trace-free 2 by 2matrices. u(m)C is M(m,C) = gl(m,C), the Lie algebra of all complex m bym matrices.

As an example, mutiplying X = i2S3 ∈ su(2) by −i2 , we have S3 ∈ sl(2,C)

and the diagonal elements in the matrix π′(i2S3) get also multiplied by −i2 (sinceπ′ is a linear map), giving

π′(S3) =

q12 0 · · · 00 q2

2 · · · 0· · · · · ·0 0 · · · qm

2

We see that π′(S3) will have half-integral values, and make the following

definitions

Definition (Weights and Weight Spaces). If π′(S3) has an eigenvalue k2 , we

say that k is a weight of the representation (π, V ).The subspace Vk ⊂ V of the representation V satisfying

v ∈ Vk =⇒ π′(S3)v =k

2v

is called the k’th weight space of the representation. All vectors in it are eigen-vectors of π′(S3) with eigenvalue k

2 .The dimension dim Vk is called the multiplicity of the weight k in the rep-

resentation (π, V ).

89

S1 and S2 don’t commute with S3, so they may not preserve the subspacesVk and we can’t diagonalize them simultaneously with S3. We can howeverexploit the fact that we are in the complexification sl(2,C) to construct twocomplex linear combinations of S1 and S2 that do something interesting:

Definition (Raising and lowering operators). Let

S+ = S1 + iS2 =

(0 10 0

), S− = S1 − iS2 =

(0 01 0

)We have S+, S− ∈ sl(2,C). These are neither self-adjoint nor skew-adjoint, butsatisfy

(S±)† = S∓

and similarly we haveπ′(S±)† = π′(S∓)

We call π′(S+) a “raising operator” for the representation (π, V ), and π′(S−)a “lowering operator”.

The reason for this terminology is the following calculation:

[S3, S+] = [S3, S1 + iS2] = iS2 + i(−iS1) = S1 + iS2 = S+

which implies (since π′ is a Lie algebra homomorphism)

π′(S3)π′(S+)− π′(S+)π′(S3) = π′([S3, S+]) = π′(S+)

For any v ∈ Vk, we have

π′(S3)π′(S+)v = π′(S+)π′(S3)v + π′(S+)v = (k

2+ 1)π′(S+)v

sov ∈ Vk =⇒ π′(S+)v ∈ Vk+2

The linear operator π′(S+) takes vectors with a well-defined weight to vectorswith the same weight, plus 2 (thus the terminology “raising operator”). Asimilar calculation shows that π′(S−) takes Vk to Vk−2, lowering the weight by2.

We’re now ready to classify all finite dimensional irreducible unitary repre-sentations (π, V ) of SU(2). We define

Definition (Highest weights and highest weight vectors). A non-zero vectorv ∈ Vn ⊂ V such that

π′(S+)v = 0

is called a highest weight vector, with highest weight n.

Irreducible representations will be characterized by a highest weight vector,as follows

90

Theorem (Highest weight theorem). Finite dimensional irreducible represen-tations of SU(2) have weights of the form

−n,−n+ 2, · · · , n− 2, n

for n a non-negative integer, each with multiplicity 1, with n a highest weight.

Proof. Finite dimensionality implies there is a highest weight n, and we canchoose any highest weight vector vn ∈ Vn. Repeatedly applying π′(S−) to vnwill give new vectors

vn−2j = π′(S−)jvn ∈ Vn−2j

with weights n− 2j.Consider the span of the vn−2j , j ≥ 0. To show that this is a representation

one needs to show that the π′(S3) and π′(S+) leave it invariant. For π′(S3) thisis obvious, for π′(S+) one can show that

π′(S+)vn−2j = j(n− j + 1)vn−2(j−1) (8.1)

by an induction argument. For j = 0 this is just the highest weight conditionon vn. Assuming validity for j, one can check validity for j + 1 by

π′(S+)vn−2(j+1) =π′(S+)π′(S−)vn−2j

=(π′([S+, S−] + π′(S−)π′(S+))vn−2j

=(π′(2S3) + π′(S−)π′(S+))vn−2j

=((n− 2j)vn−2j + π′(S−)j(n− j + 1)vn−2(j−1)

=((n− 2j) + j(n− j + 1))vn−2j

=(j + 1)(n− (j + 1) + 1)vn−2((j+1)−1)

where we have used the commutation relation

[S+, S−] = 2S3

The span of the vn−2j is not just a representation, but an irreducible one,since all the non-zero vn−2j arise by repeated application of π′(S−) to vn andequation 8.1 shows that (up to a constant) π′(S+) is an inverse to π′(S−) forall j up to the value j = n+ 1. In the sequence of vn−2j for increasing j, finite-dimensionality of V n implies that at some point one one must hit a “lowestweight vector”, one annihilated by π′(S−). From that point on, the vn−2j forhigher j will be zero. Taking into account the fact that the pattern of weightsis invariant under change of sign, one finds that the only possible pattern ofweights is

−n,−n+ 2, · · · , n− 2, n

This is consistent with equation 8.1, which shows that it is at j = n that π′(S−)will act on vn−2j without having an inverse proportional to π′(S+) (which wouldact on vn−2(j+1)).

91

Since we saw in section 8.1.1 that representations can be studied by lookingat by the set of their weights under the action of our chosen U(1) ⊂ SU(2), wecan label irreducible representations of SU(2) by a non-negative integer n, thehighest weight. Such a representation will be of dimension n+ 1, with weights

−n,−n+ 2, · · · , n− 2, n

Each weight occurs with multiplicity one, and we have

(π, V ) = C−n ⊕C−n+2 ⊕ · · ·Cn−2 ⊕Cn

Starting with a highest-weight or lowest-weight vector, one can generate abasis for the representation by repeatedly applying raising or lowering operators.The picture to keep in mind is this

where all the vector spaces are copies of C, and all the maps are isomorphisms(multiplications by various numbers).

In summary, we see that all irreducible finite dimensional unitary SU(2)representations can be labeled by a non-negative integer, the highest weight n.These representations have dimension n+ 1 and we will denote them (πn, V

n =Cn+1). Note that Vn is the n’th weight space, V n is the representation withhighest weight n. The physicist’s terminology for this uses not n, but n

2 andcalls this number the “spin”of the representation. We have so far seen the lowestthree examples n = 0, 1, 2, or spin s = n

2 = 0, 12 , 1, but there is an infinite class

of larger irreducibles, with dim V = n+ 1 = 2s+ 1.

8.2 Representations of SU(2): construction

The argument of the previous section only tells us what properties possiblefinite dimensional irreducible representations of SU(2) must have. It showshow to construct such representations given a highest-weight vector, but doesnot provide any way to construct such highest weight vectors. We would like tofind some method to explicitly construct an irreducible (πn, V

n) for each highestweight n. There are several possible constructions, but perhaps the simplest oneis the following, which gives a representation of highest weight n by looking atpolynomials in two complex variables, homogeneous of degree n.

92

Recall from our early discussion of representations that if one has an actionof a group on a space M , one can get a representation on functions f on M bytaking

(π(g)f)(x) = f(g−1 · x)

For SU(2), we have an obvious action of the group on M = C2 (by matricesacting on column vectors), and we look at a specific class of functions on thisspace, the polynomials. We can break up the infinite-dimensional space ofpolynomials on C2 into finite-dimensional subspaces as follows:

Definition (Homogeneous polynomials). The complex vector space of homoge-neous polynomials of degree m in two complex variables z1, z2 is the space offunctions on C2 of the form

f(z1, z2) = a0zn1 + a1z

n−11 z2 + · · ·+ an−1z1z

n−12 + anz

n2

The space of such functions is a complex vector space of dimension n+ 1.

Using the action of SU(2) on C2, we will see that this space of functions isexactly the representation space V n that we need. More explicitly, for

g =

(α β

−β α

), g−1 =

(α −ββ α

)we can construct the representation as follows:

(πn(g)f)(z1, z2) =f(g−1

(z1

z2

))

=f(αz1 − βz2, βz1 + αz2)

=

n∑k=0

ak(αz1 − βz2)n−k(βz1 + αz2)k

Taking the derivative, the Lie algebra representation is given by

π′n(X)f =d

dtπn(etX)f|t=0 =

d

dtf(e−tX

(z1

z2

))|t=0

By the chain rule this is

π′n(X)f =(∂f

∂z1,∂f

∂z2)(d

dte−tX

(z1

z2

))|t=0

=− ∂f

∂z1(X11z1 +X12z2)− ∂f

∂z2(X21z1 +X22z2)

where the Xij are the components of the matrix X.Computing what happens for X = S3, S+, S−, we get

(π′n(S3)f)(z1, z2) =1

2(− ∂f∂z1

z1 +∂f

∂z2z2)

93

so

π′n(S3) =1

2(−z1

∂

∂z1+ z2

∂

∂z2)

and similarly

π′n(S+) = −z2∂

∂z1, π′n(S−) = −z1

∂

∂z2

The zk1zn−k2 are eigenvectors for S3 with eigenvalue 1

2 (n− 2k) since

π′n(S3)zk1zn−k2 =

1

2(−kzk1zn−k2 + (n− k)zk1z

n−k2 ) =

1

2(n− 2k)zk1z

n−k2

zn2 will be an explicit highest weight vector for the representation (πn, Vn).

An important thing to note here is that the formulas we have found for π′

are not in terms of matrices. Instead we have seen that when we construct ourrepresentations using functions on C2, for any X ∈ su(2) (or its complexificationsl(2,C)), π′n(X) is given by a differential operator. Note that these differentialoperators are independent of n: one gets the same operator π′(X) on all theV n. This is because the original definition of the representation

(π(g)f)(x) = f(g−1 · x)

is on the full infinite dimensional space of polynomials on C2. While this spaceis infinite-dimensional, issues of analysis don’t really come into play here, sincepolynomial functions are essentially an algebraic construction. Later on in thecourse we will need to work with function spaces that require much more seriousconsideration of issues in analysis.

Restricting the differential operators π′(X) to a finite dimensional irreduciblesubspace V n, the homogeneous polynomials of degree n, if one chooses a basisof V n, then the linear operator π′(X) will be given by a n+ 1 by n+ 1 matrix.Clearly though, the expression as a simple first-order differential operator ismuch easier to work with. In the examples we will be studying in much of therest of the course, the representations under consideration will also be on func-tion spaces, with Lie algebra representations appearing as differential operators.Instead of using linear algebra techniques to find eigenvalues and eigenvectors,the eigenvector equation will be a partial differential equation, wtih our focuson using Lie groups and their representation theory to solve such equations.

One issue we haven’t addressed yet is that of unitarity of the representation.We need Hermitian inner products on the spaces V n, inner products that willbe preserved by the action of SU(2) that we have defined on these spaces. Astandard way to define a Hermitian inner product on functions on a space Mis to define them using an integral: for f , g functions on M , take their innerproduct to be

〈f, g〉 =

∫M

fg

While for M = C2 this gives an SU(2) invariant inner product on functions, itis useless for f, g polynomial, since such integrals diverge. What one can do in

94

this case is define an inner product on polynomial functions on C2 by

〈f, g〉 =1

π2

∫C2

f(z1, z2)g(z1, z2)e−(|z1|2+|z2|2)dx1dy1dx2dy2 (8.2)

Here z1 = x1 + iy1, z2 = x2 + iy2. One can do integrals of this kind fairly easilysince they factorize into separate integrals over z1 and z2, each of which can betreated using polar coordinates and standard calculus methods. One can checkby explicit computation that the polynomials

zj1zk2√

j!k!

will be an orthornormal basis of the space of polynomial functions with respectto this inner product, and the operators π′(X), X ∈ su(2) will be skew-adjoint.

Working out what happens for the first few examples of irreducible SU(2)representations, one finds orthonormal bases for the representation spaces V n

of homogeneous polynomials as follows

• For n = s = 01

• For n = 1, s = 12

z1, z2

• For n = 2, s = 11√2z2

1 , z1z2,1√2z2

2

• For n = 3, s = 32

1√6z3

1 ,1√2z2

1z2,1√2z1z

22 ,

1√6z3

2

8.3 Representations of SO(3) and spherical har-monics

We would like to now use the classification and construction of representations ofSU(2) to study the representations of the closely related group SO(3). For anyrepresentation (ρ, V ) of SO(3), we can use the double-covering homomorphismΦ : SU(2)→ SO(3) to get a representation

π = ρ Φ

of SU(2). It can be shown that if ρ is irreducible, π will be too, so we musthave π = ρ Φ = πn, one of the irreducible representations of SU(2) found inthe last section. Using the fact that Φ(−1) = 1, we see that

πn(−1) = ρ Φ(−1) = 1

95

From knowing that the weights of πn are −n,−n+2, · · · , n−2, n, we know that

πn(−1) = πn

(eiπ 00 e−iπ

)=

einπ 0 · · · 0

0 ei(n−2)π · · · 0· · · · · ·0 0 · · · e−inπ

= 1

which will only be true for n even, not for n odd. Since the Lie algebra of SO(3)is isomorphic to the Lie algebra of SU(2), the same Lie algebra argument usingraising and lowering operators as in the last section also applies. The irreduciblerepresentations of SO(3) will be (ρl, V = C2l+1) for l = 0, 1, 2, · · · , of dimension2l + 1 and satisfying

ρl Φ = π2l

Just like in the case of SU(2), we can explicitly construct these representa-tions using functions on a space with an SO(3) action. The obvious space tochoose is R3, with SO(3) matrices acting on x ∈ R3 as column vectors, by theformula we have repeatedly used

(ρ(g)f)(x) = f(g−1 · x) = f(g−1

x1

x2

x3

)

Taking the derivative, the Lie algebra representation is given by

ρ′(X)f =d

dtρ(etX)f|t=0 =

d

dtf(e−tX

x1

x2

x3

)|t=0

where X ∈ so(3). Recall that a basis for so(3) is given by

l1 =

0 0 00 0 −10 1 0

l2 =

0 0 10 0 0−1 0 0

l3 =

0 −1 01 0 00 0 0


[l1, l2] = l3, [l2, l3] = l1, [l3, l1] = l2

Digression. A Note on ConventionsWe’re using the notation lj for the real basis of the Lie algebra so(3) = su(2).

For a unitary representation ρ, the ρ′(lj) will be skew-adjoint linear operators.For consistency with the physics literature, we’ll use the notation Lj = iρ′(lj)for the self-adjoint version of the linear operator corresponding to lj in thisrepresentation on functions. The Lj satisfy the commutation relations

[L1, L2] = iL3, [L2, L3] = iL1, [L3, L1] = iL2

We’ll also use elements l± = l1 ± il2 of the complexified Lie algebra to createraising and lowering operators L± = iρ′(l±).

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As with the SU(2) case, we won’t include a factor of ~ as is usual in physics(e.g. the usual convention is Lj = i~ρ′(lj)), since for considerations of the actionof the rotation group it would just cancel out (physicists define rotations using

ei~ θLj ). The factor of ~ is only of significance when Lj is expressed in terms of

the momentum operator, a topic discussed in chapter 17.

In the SU(2) case, the π′(Sj) had half-integral eigenvalues, with the eigen-values of π′(2S3) the integral weights of the representation. Here the Lj willhave integer eigenvalues, the weights will be the eigenvalues of 2L3, which willbe even integers.

Computing ρ′(l1) we find

ρ′(l1)f =d

dtf(e

−t

0 0 00 0 −10 1 0

x1

x2

x3

)|t=0 (8.3)

=d

dtf(

0 0 00 cos t sin t0 − sin t cos t

x1

x2

x3

)|t=0 (8.4)

=d

dtf(

0x2 cos t+ x3 sin t−x2 sin t+ x3 cos t

)|t=0 (8.5)

=(∂f

∂x1,∂f

∂x2,∂f

∂x3) ·

0x3

−x2

(8.6)

=x3∂f

∂x2− x2

∂f

∂x3(8.7)

so

ρ′(l1) = x3∂

∂x2− x2

∂

∂x3

and similar calculations give

ρ′(l2) = x1∂

∂x3− x3

∂

∂x1, ρ′(l3) = x2

∂

∂x1− x1

∂

∂x2

The space of all functions on R3 is much too big: it will give us an infinity ofcopies of each finite dimensional representation that we want. Notice that whenSO(3) acts on R3, it leaves the distance to the origin invariant. If we work inspherical coordinates (r, θ, φ) (see picture)

97

we will have

x1 =r sin θ cosφ

x2 =r sin θ sinφ

x3 =r cos θ

Acting on f(r, φ, θ), SO(3) will leave r invariant, only acting non-trivially onθ, φ. It turns out that we can cut down the space of functions to somethingthat will only contain one copy of the representation we want in various ways.One way to do this is to restrict our functions to the unit sphere, i.e. just lookat functions f(θ, φ). We will see that the representations we are looking for canbe found in simple trigonometric functions of these two angular variables.

We can construct our irreducible representations ρ′l by explicitly constructinga function we will call Y ll (θ, φ) that will be a highest weight vector of weightl. The weight l condition and the highest weight condition give two differentialequations for Y ll (θ, φ):

L3Yll = lY ll , L+Y

ll = 0

These will turn out to have a unique solution (up to scalars).

We first need to change coordinates from rectangular to spherical in ourexpressions for L3, L±. Using the chain rule to compute expressions like

∂

∂rf(x1(r, θ, φ), x2(r, θ, φ), x3(r, θ, φ))

we find ∂∂r∂∂θ∂∂φ

=

sin θ cosφ sin θ sinφ cos θr cos θ cosφ r cos θ sinφ − sin θ−r sin θ sinφ r sin θ cosφ 0

∂∂x1∂∂x2∂∂x3

98

so ∂∂r

1r∂∂θ

1r sin θ

∂∂φ

=

sin θ cosφ sin θ sinφ cos θcos θ cosφ cos θ sinφ − sin θ− sinφ cosφ 0

∂∂x1∂∂x2∂∂x3

This is an orthogonal matrix, so one can invert it by taking its transpose, to get ∂

∂x1∂∂x2∂∂x3

=

sin θ cosφ cos θ cosφ − sinφsin θ sinφ cos θ sinφ cosφ

cos θ − sin θ 0

∂∂r

1r∂∂θ

1r sin θ

∂∂φ

So we finally have

L1 = iρ′(l1) = i(x3∂

∂x2− x2

∂

∂x3) = i(sinφ

∂

∂θ+ cot θ cosφ

∂

∂φ)

L2 = iρ′(l2) = i(x1∂

∂x3− x3

∂

∂x1) = i(− cosφ

∂

∂θ+ cot θ sinφ

∂

∂φ)

L3 = iρ′(l3) = i(x1∂

∂x3− x3

∂

∂x1) = −i ∂

∂φ

and

L+ = iρ′(l+) = eiφ(∂

∂θ+ i cot θ

∂

∂φ), L− = iρ′(l−) = e−iφ(− ∂

∂θ+ i cot θ

∂

∂φ)

Now that we have expressions for the action of the Lie algebra on functions inspherical coordinates, our two differential equations saying our function Y ll (θ, φ)is of weight l and in the highest-weight space are

L3Yll (θ, φ) = −i ∂

∂φY ll (θ, φ) = lY ll (θ, φ)

and

L+Yll (θ, φ) = eiφ(

∂

∂θ+ i cot θ

∂

∂φ)Y ll (θ, φ) = 0

The first of these tells us that

Y ll (θ, φ) = eilφFl(θ)

for some function Fl(θ), and using the second we get

(∂

∂θ− l cot θ)Fl(θ)

with solutionFl(θ) = Cll sin

l θ

for an arbitrary constant Cll. Finally

Y ll (θ, φ) = Clleilφ sinl θ

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This is a function on the sphere, which is also a highest weight vector in a2l + 1 dimensional irreducible representation of SO(3). To get functions whichgive vectors spanning the rest of the weight spaces, one just repeatedly appliesthe lowering operator L−, getting functions

Y ml (θ, φ) =Clm(L−)l−mY ll (θ, φ)

=Clm(e−iφ(− ∂

∂θ+ i cot θ

∂

∂φ))l−meilφ sinl θ

for m = l, l − 1, l − 2 · · · ,−l + 1,−lThe functions Y ml (θ, φ) are called “spherical harmonics”, and they span the

space of complex functions on the sphere in much the same way that the einθ

span the space of complex valued functions on the circle. Unlike the case ofpolynomials on C2, for functions on the sphere, one gets finite numbers byintegrating such functions over the sphere. So one can define an inner producton these representations for which they are unitary by simply setting

〈f, g〉 =

∫S2

fg sin θdθdφ =

∫ 2π

φ=0

∫ π

θ=0

f(θ, φ)g(θ, φ) sin θdθdφ

We will not try and show this here, but for the allowable values of l,m theY ml (θ, φ) are mutually orthogonal with respect to this inner product.

One can derive various general formulas for the Y ml (θ, φ) in terms of Leg-endre polynomials, but here we’ll just compute the first few examples, withthe proper constants that give them norm 1 with respect to the chosen innerproduct.

• For the l = 0 representation

Y 00 (θ, φ) =

√1

4π

• For the l = 1 representation

Y 11 = −

√3

8πsin θeiφ, Y 0

1 =

√3

4πcos θ, Y −1

1 =

√3

8πsin θe−iφ

(one can easily see that these have the correct eigenvalues for ρ′(L3) =−i ∂∂φ ).

• For the l = 2 representation one has

Y 22 =

√15

32πsin2 θei2φ, Y 1

2 = −√

15

8πsin θ cos θeiφ

Y 02 =

√5

16π(3 cos2 θ − 1)

Y −12 =

√15

8πsin θ cos θe−iφ, Y −2

2 =

√15

32πsin2 θe−i2φ

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We will see later that these functions of the angular variables in spheri-cal coordinates are exactly the functions that give the angular dependence ofwavefunctions for the physical system of a particle in a spherically symmetricpotential. In such a case the SO(3) symmetry of the system implies that thestate space (the wavefunctions) will provide a unitary representation π of SO(3),and the action of the Hamiltonian operator H will commute with the action ofthe operators L3, L±. As a result all of the states in an irreducible representa-tion component of π will have the same energy. States are thus organized into“orbitals”, with singlet states called “s” orbitals (l = 0), triplet states called“p” orbitals (l = 1), multiplicity 5 states called “d” orbitals (l = 2), etc.

8.4 The Casimir operator

For both SU(2) and SO(3), we have found that all representations can beconstructed out of function spaces, with the Lie algebra acting as first-orderdifferential operators. It turns out that there is also a very interesting second-order differential operator that comes from these Lie algebra representations,known as the Casimir operator. For the case of SO(3)

Definition (Casimir operator for SO(3)). The Casimir operator for the repre-sentation of SO(3) on functions on S2 is the second-order differential operator

L2 ≡ L21 + L2

2 + L23

(the symbol L2 is not intended to mean that this is the square of an operator L)

A straightforward calculation using the commutation relations satisfied bythe Lj shows that

[L2, ρ′(X)] = 0

for anyX ∈ so(3). Knowing this, a version of Schur’s lemma says that L2 will acton an irreducible representation as a scalar (i.e. all vectors in the representationare eigenvectors of L2, with the same eigenvalue). This eigenvalue can be usedto characterize the irreducible representation.

The easiest way to compute this eigenvalue turns out to be to act with L2 ona highest weight vector. First one rewrites L2 in terms of raising and loweringoperators.

L−L+ =(L1 − iL2)(L1 + iL2)

=L21 + L2

2 + i[L1, L2]

=L21 + L2

2 − L3

soL2 = L2

1 + L22 + L2

3 = L−L+ + L3 + L23

For the representation ρ of SO(3) on functions on S2 constructed above,we know that on a highest weight vector of the irreducible representation ρl

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(restriction of ρ to the 2l+ 1 dimensional irreducible subspace of functions thatare linear combinations of the Y ml (θ, φ)), we have the two eigenvalue equations

L+f = 0, L3f = lf

with solution the functions proportional to Y ll (θ, φ). Just from these conditionsand our expression for L2 we can immediately find the scalar eigenvalue of L2

sinceL2f = L−L+f + (L3 + L2

3)f = 0 + l + l2 = l(l + 1)

We have thus shown that our irreducible representation ρl can be characterizedas the representation on which L2 acts by the scalar l(l + 1).

In summary, we have two different sets of partial differential equations whosesolutions provide a highest weight vector for and thus determine the irreduciblerepresentation ρl:

•L+f = 0, L3f = lf

which are first order equations, with the first using complexification andsomething like a Cauchy-Riemann equation, and

•L2f = l(l + 1)f, L3f = lf

where the first equation is a second order equation, something like aLaplace equation.

That a solution of the first set of equations gives a solution of the second setis obvious. Much harder to show is that a solution of the second set gives asolution of the first set. The space of solutions to

L2f = l(l + 1)f

for l a non-negative integer includes as we have seen the 2l + 1-dimensionalvector space of linear combinations of the Y ml (θ, φ) (there are no other solu-tions, although we will not show that). Since the action of SO(3) on functionscommutes with the operator L2, this 2l + 1-dimensional space will provide arepresentation, the irreducible one of spin l.

One can compute the explicit second order differential operator L2 in the ρrepresentation on functions, it is

L2 =L21 + L2

2 + L23

=(i(sinφ∂

∂θ+ cot θ cosφ

∂

∂φ))2 + (i(− cosφ

∂

∂θ+ cot θ sinφ

∂

∂φ))2 + (−i ∂

∂φ)2

=− (1

sin θ

∂

∂θ(sin θ

∂

∂θ) +

1

sin2 θ

∂2

∂φ2) (8.8)

We will re-encounter this operator later on in the course as the angular part ofthe Laplace operator on R3.

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For the group SU(2) we can also find irreducible representations as solutionspaces of differential equations on functions on C2. In that case, the differentialequation point of view is much less useful, since the solutions we are looking forare just the homogeneous polynomials, which are more easily studied by purelyalgebraic methods.


The classification of SU(2) representations is a standard topic in all textbooksthat deal with Lie group representations. A good example is [30], which coversthis material well, and from which the discussion here of the construction ofrepresentations as homogeneous polynomials is drawn (see pages 77-79). Thecalculation of the Lj and the derivation of expressions for spherical harmonicsas Lie algebra representations of so(3) appears in most quantum mechanicstextbooks in one form or another (for example, see Chapter 12 of [63]). Anothersource used here for the explicit constructions of representations is [14], Chapters27-30.

103

Chapter 9

Tensor Products,Entanglement, andAddition of Spin

If one has two independent quantum systems, with state spaces H1 and H2,the combined quantum system has a description that exploits the mathematicalnotion of a “tensor product”, with the combined state space the tensor productH1 ⊗ H2. Because of the ability to take linear combinations of states, thiscombined state space will contain much more than just products of independentstates, including states that are described as “entangled”, and responsible forsome of the most counter-intuitive behavior of quantum physical systems.

This same tensor product construction is a basic one in representation the-ory, allowing one to construct a new representation (πW1⊗W2

,W1 ⊗W2) out ofrepresentations (πW1 ,W1) and (πW2 ,W2). When we take the tensor product ofstates corresponding to two irreducible representations of SU(2) of spins s1, s2,we will get a new representation (πV 2s1⊗V 2s2 , V 2s1 ⊗V 2s2). It will be reducible,a direct sum of representations of various spins, a situation we will analyze indetail.

Starting with a quantum system with state space H that describes a singleparticle, one can describe a system of N particles by taking an N -fold tensorproduct H⊗N = H ⊗H ⊗ · · · ⊗ H. It turns out that for identical particles, wedon’t get the full tensor product space, but only the subspaces either symmetricor antisymmetric under the action of the permutation group by permutationsof the factors, depending on whether our particles are “bosons” or “fermions”.This is a separate postulate in quantum mechanics, but finds an explanationwhen particles are treated as quanta of quantum fields.

Digression. When physicists refer to “tensors”, they generally mean the “ten-sor fields” used in general relativity or other geometry-based parts of physics,not tensor products of state spaces. A tensor field is a function on a manifold,

105

taking values in some tensor product of copies of the tangent space and its dualspace. The simplest tensor fields are just vector fields, functions taking valuesin the tangent space. A more non-trivial example is the metric tensor, whichtakes values in the dual of the tensor product of two copies of the tangent space.

9.1 Tensor products

Given two vector spaces V and W (over R or C), one can easily constructthe direct sum vector space V ⊕W , just by taking pairs of elements (v, w) forv ∈ V,w ∈W , and giving them a vector space structure by the obvious additionand multiplication by scalars. This space will have dimension

dim(V ⊕W ) = dimV + dimW

If e1, e2, . . . , edimV is a basis of V , and f1, f2, . . . , fdimW a basis of W , the

e1, e2, . . . , edimV , f1, f2, . . . , fdimW

will be a basis of V ⊕W .A less trivial construction is the tensor product of the vector spaces V and

W . This will be a new vector space called V ⊗W , of dimension

dim(V ⊗W ) = (dimV )(dimW )

One way to motivate the tensor product is to think of vector spaces as vectorspaces of functions. Elements

v = v1e1 + v2e2 + · · ·+ vdimV edimV ∈ V

can be thought of as functions on the dimV points ei, taking values vi at ei. Ifone takes functions on the union of the sets ei and fj one gets elements ofV ⊕W . The tensor product V ⊗W will be what one gets by taking all functionson not the union, but the product of the sets ei and fj. This will be theset with (dimV )(dimW ) elements, which we will write ei ⊗ fj , and elementsof V ⊗W will be functions on this set, or equivalently, linear combinations ofthese basis vectors.

This sort of definition is less than satisfactory, since it is tied to an explicitchoice of bases for V and W . We won’t however pursue more details of thisquestion or a better definition here. For this, one can consult pretty much anyadvanced undergraduate text in abstract algebra, but here we will take as giventhe following properties of the tensor product that we will need:

• Given vectors v ∈ V,w ∈W we get an element v⊗w ∈ V ⊗W , satisfyingbilinearity conditions (for c1, c2 constants)

v ⊗ (c1w1 + c2w2) = c1(v ⊗ w1) + c2(v ⊗ w2)

(c1v1 + c2v2)⊗ w = c1(v1 ⊗ w) + c2(v2 ⊗ w)

106

• There are natural isomorphisms

C⊗ V ' V, V ⊗W 'W ⊗ V

and

U ⊗ (V ⊗W ) ' (U ⊗ V )⊗W

for vector spaces U, V,W

• Given a linear operator A on V and another linear operator B on W , wecan define a linear operator A⊗B on V ⊗W by

(A⊗B)(v ⊗ w) = Av ⊗Bw

for v ∈ V,w ∈W .

With respect to the bases ei, fj of V and W , A will be a (dimV ) by(dimV ) matrix, B will be a (dimW ) by (dimW ) matrix and A⊗ B willbe a (dimV )(dimW ) by (dimV )(dimW ) matrix (which one can think ofas a (dimV ) by (dimV ) matrix of blocks of size (dimW )).

• One often wants to consider tensor products of vector spaces and dual vec-tor spaces. An important fact is that there is an isomorphism between thetensor product V ∗⊗W and linear maps from V to W given by identifyingl ⊗ w (l ∈ V ∗) with the linear map

v ∈ V → l(v)w ∈W

• Given the motivation in terms of functions on a product of sets, for func-tion spaces in general we should have an idenfitication of the tensor prod-uct of function space with functions on the product set. For instance, forsquare-integrable functions on R we expect

L2(R)⊗ L2(R)⊗ · · · ⊗ L2(R))︸︷︷︸N times

= L2(RN ) (9.1)

For V a real vector space, its complexification VC (the vector space onegets by allowing multiplication by both real and imaginary numbers) can beidentified with the tensor product

VC = V ⊗R C

Here the notation ⊗R indicates a tensor product of two real vector spaces: Vof dimension dimV with basis e1, e2, . . . , edimV and C = R2 of dimension 2with basis 1, i.

107

9.2 Composite quantum systems and tensor prod-ucts

Consider two quantum systems, one defined by a state space H1 and a set ofoperators O1 on it, the second given by a state spaceH2 and set of operators O2.One can describe the composite quantum system corresponding to consideringthe two quantum systems as a single one, with no interaction between them, byjust taking as a new state space

HT = H1 ⊗H2

with operators of the form

A⊗ Id + Id⊗B

with A ∈ O1, B ∈ O2. To describe an interacting quantum system, one can usethe state space HT , but with a more general class of operators.

If H is the state space of a quantum system, one can think of this as de-scribing a single particle, and then to describe a system of N such particles, oneuses the multiple tensor product

H⊗N = H⊗H⊗ · · · ⊗ H ⊗H︸︷︷︸N times

The symmetric group SN acts on this state space, and one has a repre-sentation (π,H⊗N ) of SN as follows. For σ ∈ SN a permutation of the set1, 2, . . . , N of N elements, on a tensor product of vectors one has

π(σ)(v1 ⊗ v2 ⊗ · · · ⊗ vN ) = vσ(1) ⊗ vσ(2) ⊗ · · · ⊗ vσ(N)

The representation of SN that this gives is in general reducible, containingvarious components with different irreducible representations of the group SN .

A fundamental axiom of quantum mechanics is that ifH⊗N describesN iden-tical particles, then all physical states occur as one-dimensional representationsof SN , which are either symmetric (“bosons”) or antisymmetric (“fermions”)where

Definition. A state v ∈ H⊗N is called

• symmetric, or bosonic if ∀σ ∈ SN

π(σ)v = v

The space of such states is denoted SN (H).

• antisymmetric, or fermionic if ∀σ ∈ SN

π(σ)v = (−1)|σ|v

The space of such states is denoted ΛN (H). Here |σ| is the minimal num-ber of transpositions that by composition give σ.

108

Note that in the fermionic case, for σ a transposition interchanging twoparticles, the antisymmetric representation π acts on the factor H ⊗ H by in-terchanging vectors, taking

w ⊗ w ∈ H ⊗H

to itself. Antisymmetry requires that this state go to its negative, so the statecannot be non-zero. So one cannot have non-zero states in H⊗N describing twoidentical particles in the same state w ∈ H, a fact that is known as the “PauliPrinciple”.

While the symmetry or antisymmetry of states of multiple identical particlesis a separate axiom when such particles are described in this way as tensorproducts, we will see later on (chapter 34) that this phenomenon instead findsa natural explanation when particles are described in terms of quantum fields.

9.3 Indecomposable vectors and entanglement

If one is given a function f on a space X and a function g on a space Y , onecan form a product function fg on the product space X × Y by taking (forx ∈ X, y ∈ Y )

(fg)(x, y) = f(x)g(y)

However, most functions on X × Y are not decomposable in this manner. Sim-ilarly, for a tensor product of vector spaces, one has:

Definition (Decomposable and indecomposable vectors). A vector in V ⊗Wis called decomposable if it is of the form v ⊗ w for some v ∈ V,w ∈ W . If itcannot be put in this form it is called indecomposable.

Note that our basis vectors of V ⊗W are all decomposable since they areproducts of basis vectors of V and W . Linear combinations of these basis vectorshowever are in general indecomposable. If we think of an element of V ⊗Was a dimV by dimW matrix, with entries the coordinates with respect to ourbasis vectors for V ⊗W , then for decomposable vectors we get a special classof matrices, those of rank one.

In the physics context, the language used is:

Definition (Entanglement). An indecomposable state in the tensor productstate space HT = H1 ⊗H2 is called an entangled state.

The phenomenon of entanglement is responsible for some of the most surpris-ing and subtle aspects of quantum mechanical system. The Einstein-Podolsky-Rosen paradox concerns the behavior of an entangled state of two quantumsystems, when one moves them far apart. Then performing a measurement onone system can give one information about what will happen if one performsa measurement on the far removed system, introducing a sort of unexpectednon-locality.

109

Measurement theory itself involves crucially an entanglement between thestate of a system being measured, thought of as in a state space Hsystem, andthe state of the measurement apparatus, thought of as lying in a state spaceHapparatus. The laws of quantum mechanics presumably apply to the totalsystem Hsystem⊗Happaratus, with the counter-intuitive nature of measurementsappearing due to this decomposition of the world into two entangled parts: theone under study, and a much larger for which only an approximate descriptionin classical terms is possible. For much more about this, a recommended readingis Chapter 2 of [58].

9.4 Tensor products of representations

Given two representations of a group, one can define a new representation, thetensor product representation, by

Definition (Tensor product representation of a group). For (πV , V ) and (πW ,W )representations of a group G, one has a tensor product representation (πV⊗W , V⊗W ) defined by

(πV⊗W (g))(v ⊗ w) = πV (g)v ⊗ πW (g)w

One can easily check that πV⊗W is a homomorphism.To see what happens for the corresponding Lie algebra representation, one

computes (for X in the Lie algebra)

π′V⊗W (X)(v ⊗ w) =d

dtπV⊗W (etX)(v ⊗ w)t=0

=d

dt(πV (etX)v ⊗ πW (etX)w)t=0

=((d

dtπV (etX)v)⊗ πW (etX)w)t=0 + (πV (etX)v ⊗ (

d

dtπW (etX)w))t=0

=(π′V (X)v)⊗ w + v ⊗ (π′W (X)w)

which could also be written

π′V⊗W (X) = (π′V (X)⊗ 1W ) + (1V ⊗ π′W (X))

9.4.1 Tensor products of SU(2) representations

Given two representations (πV , V ) and (πW ,W ) of a group G, we can decom-pose each into irreducibles. To do the same for the tensor product of the tworepresentations, we need to know how to decompose the tensor product of twoirreducibles. This is a fundamental non-trivial problem for a group G, with theanswer for G = SU(2) as follows:

Theorem 9.1 (Clebsch-Gordan decomposition).The tensor product (πV n1⊗V n2 , V n1 ⊗ V n2) decomposes into irreducibles as

(πn1+n2, V n1+n2)⊕ (πn1+n2−2, V

n1+n2−2)⊕ · · · ⊕ (π|n1−n2|, V|n1−n2|)

110

Proof. One way to prove this result is to use highest-weight theory, raising andlowering operators, and the formula for the Casimir operator. We will not tryand show the details of how this works out, but in the next section give asimpler argument using characters. However, in outline (for more details, seefor instance section 5.2 of [55]), here’s how one could proceed:

One starts by noting that if vn1∈ Vn1

, vn2∈ Vn2

are highest weight vectorsfor the two representations, vn1⊗vn2 will be a highest weight vector in the tensorproduct representation (i.e. annihilated by π′n1+n2

(S+)), of weight n1 + n2.So (πn1+n2

, V n1+n2) will occur in the decomposition. Applying π′n1+n2(S−) to

vn1⊗vn2

one gets a basis of the rest of the vectors in (πn1+n2, V n1+n2). However,

at weight n1 +n2−2 one can find another kind of vector, a highest-weight vectororthogonal to the vectors in (πn1+n2 , V

n1+n2). Applying the lowering operatorto this gives (πn1+n2−2, V

n1+n2−2). As before, at weight n1 + n2 − 4 one findsanother, orthogonal highest weight vector, and gets another representation, withthis process only terminating at weight |n1 − n2|.

9.4.2 Characters of representations

A standard tool for dealing with representations that we have ignored so far isthat of associating to a representation an invariant called its character. Thiswill be a conjugation-invariant function on the group that only depends on theequivalence class of the representation. Given two representations constructedin very different ways, one can often check whether they are isomorphic just byseeing if their character functions match. The problem of identifying the possibleirreducible representations of a group can be attacked by analyzing the possiblecharacter functions of irreducible representations. We will not try and enterinto the general theory of characters here, but will just see what the charactersof irreducible representations are for the case of G = SU(2). These can be usedto give a simple argument for the Clebsch-Gordan decomposition of the tensorproduct of SU(2) representations. For this we don’t need general theoremsabout the relations of characters and representations, but can directly checkthat the irreducible representations of SU(2) correspond to distinct characterfunctions which are easily evaluated.

Definition (Character). The character of a representation (π, V ) of a group Gis the function on G given by

χV (g) = Tr(π(g))

Since the trace of a matrix is invariant under conjugation, χV in general willbe a complex valued, conjugation-invariant function on G. One can easily checkthat it will satisfy the relations

χV⊕W = χV + χW , χV⊗W = χV χW

For the case of G = SU(2), any element can be conjugated to be in thesubgroup U(1) of diagonal matrices. Knowing the weights of the irreducible

111

representations (πn, Vn) of SU(2), we know the characters to be the functions

χV n(

(eiθ 00 e−iθ

)) = einθ + ei(n−2)θ + · · ·+ e−i(n−2)θ + e−inθ (9.2)

As n gets large, this becomes an unwieldy expression, but one has

Theorem (Weyl character formula).

χV n(

(eiθ 00 e−iθ

)) =

ei(n+1)θ − e−i(n+1)θ

eiθ − e−iθ=

sin((n+ 1)θ)

sin(θ)

Proof. One just needs to use the identity

(einθ + ei(n−2)θ + · · ·+ e−i(n−2)θ + e−inθ)(eiθ − e−iθ) = ei(n+1)θ − e−i(n+1)θ

and equation 9.2 for the character.

To get a proof of 9.1, one can compute the character of the tensor producton the diagonal matrices using the Weyl character formula for the second factor(ordering things so that n2 > n1)

χV n1⊗V n2 =χV n1χV n2

=(ein1θ + ei(n1−2)θ + · · ·+ e−i(n1−2)θ + e−in1θ)ei(n2+1)θ − e−i(n2+1)θ

eiθ − e−iθ

=(ei(n1+n2+1)θ − e−i(n1+n2+1)θ) + · · ·+ (ei(n2−n1+1)θ − e−i(n2−n1+1)θ)

eiθ − e−iθ=χV n1+n2 + χV n1+n2−2 + · · ·+ χV n2−n1

So, when we decompose the tensor product of irreducibles into a direct sum ofirreducibles, the ones that must occur are exactly those of theorem 9.1.

9.4.3 Some examples

Some simple examples of how this works are:

• Tensor product of two spinors:

V 1 ⊗ V 1 = V 2 ⊕ V 0

This says that the four complex dimensional tensor product of two spinorrepresentations (which are each two complex dimensional) decomposesinto irreducibles as the sum of a three dimensional vector representationand a one dimensional trivial (scalar) representation.

Using the basis

(10

),

(01

)for V 1, the tensor product V 1⊗V 1 has a basis

(10

)⊗(

10

),

(10

)⊗(

01

),

(01

)⊗(

10

),

(01

)⊗(

01

)112

The vector

1√2

(

(10

)⊗(

01

)−(

01

)⊗(

10

)) ∈ V 1 ⊗ V 1

is clearly antisymmetric under permutation of the two factors of V 1⊗V 1.One can show that this vector is invariant under SU(2), by computingeither the action of SU(2) or of its Lie algebra su(2). So, this vectoris a basis for the component V 0 in the decomposition of V 1 ⊗ V 1 intoirreducibles.

The other component, V 2, is three dimensional, and has a basis(10

)⊗(

10

),

1√2

(

(10

)⊗(

01

)+

(01

)⊗(

10

)),

(01

)⊗(

01

)These three vectors span one-dimensional complex subspaces of weightsq = 2, 0,−2 under the U(1) ⊂ SU(2) subgroup(

eiθ 00 e−iθ

)They are symmetric under permutation of the two factors of V 1 ⊗ V 1.

We see that if we take two identical quantum systems with H = V 1 = C2

and make a composite system out of them, if they were bosons we wouldget a three dimensional state space V 2 = S2(V 1), transforming as a vector(spin one) under SU(2). If they were fermions, we would get a one-dimensional state space V 0 = Λ2(V 1) of spin zero (invariant under SU(2)).Note that in this second case we automatically get an entangled state, onethat cannot be written as a decomposable product.

• Tensor product of three or more spinors:

V 1⊗V 1⊗V 1 = (V 2⊕V 0)⊗V 1 = (V 2⊗V 1)⊕ (V 0⊗V 1) = V 3⊕V 1⊕V 1

This says that the tensor product of three spinor representations decom-poses as a four dimensional (“spin 3/2”) representation plus two copies ofthe spinor representation.

One can clearly generalize this and considerN -fold tensor products (V 1)⊗N

of the spinor representation. Taking N high enough one can get any ir-reducible representation of SU(2) that one wants this way, giving an al-ternative to our construction using homogeneous polynomials. Doing thishowever gives the irreducible as just one component of something larger,and one needs a method to project out the component one wants. Onecan do this using the action of the symmetric group SN on (V 1)⊗N andan understanding of the irreducible representations of SN . This relation-ship between irreducible representations of SU(2) and those of SN comingfrom looking at how both groups act on (V 1)⊗N is known as “Schur-Weyl

113

duality”, and generalizes to the case of SU(n), where one looks at N -foldtensor products of the defining representation of SU(n) matrices on Cn.For SU(n) this provides perhaps the most straight-forward constructionof all irreducible representations of the group.

9.5 Bilinear forms and tensor products

A different sort of application of tensor products that will turn out to be im-portant is to the description of bilinear forms, which generalize the dual spaceV ∗ of linear forms on V . We have

Definition (Bilinear forms). A bilinear form B on a vector space V over a fieldk (for us, k = R or C) is a map

B : (v, v′) ∈ V × V → B(v, v′) ∈ k

that is bilinear in both entries, i.e.

B(v + v′, v′′) = B(v, v′′) +B(v′, v′′), B(cv, v′) = cB(v, v′)

B(v, v′ + v′′) = B(v, v′) +B(v, v′′), B(v, cv′) = cB(v, v′)

where c ∈ k.If B(v′, v) = B(v, v′) the bilinear form is called symmetric, if B(v′, v) =

−B(v, v′) it is antisymmetric.

The relation to tensor products is

Theorem 9.2. The space of bilinear forms on V is isomorphic to V ∗ ⊗ V ∗.Proof. Given two linear forms α1 ∈ V ∗, α2 ∈ V ∗, one has a map

α1 ⊗ α2 ∈ V ∗ ⊗ V ∗ → B : B(v, v′) = α1(v)α2(v′)

Choosing a basis ej of V , the coordinate functions vj = e∗j provide a basis ofV ∗, so the vj ⊗ vk will be a basis of V ∗ ⊗ V ∗. The map above takes linearcombinations of these to bilinear forms, and is easily seen to be one-to-one andsurjective for such linear combinations.

Given a basis ej of V and dual basis vj of V ∗ (the coordinates), one canwrite the element of V ∗ ⊗ V ∗ corresponding to B as the sum∑

j,k

Bjkvj ⊗ vk

This expresses the bilinear form B in terms of a matrix B with entries Bjk,which can be computed as

Bjk = B(ej , ek)

In terms of the matrix B, the bilinear form is computed as

B(v, v′) =(v1 . . . vd

)B11 . . . B1d

......

...Bd1 . . . Bdd

v′1...v′d

= v ·Bv′

114

9.6 Symmetric and antisymmetric multilinear forms

The symmetric bilinear forms lie in S2(V ∗) ⊂ V ∗ ⊗ V ∗ and correspond tosymmetric matrices. Elements of V ∗ give linear functions on V , and one canget quadratic functions on V from elements B ∈ S2(V ∗) by taking

v ∈ V → B(v, v) = v ·Bv

Equivalently, in terms of tensor products, one gets quadratic functions as theproduct of linear functions by taking

(α, β) ∈ V ∗ × V ∗ → 1

2(α1 ⊗ α2 + α2 ⊗ α1) ∈ S2(V ∗)

and then evaluating at v ∈ V to get the number

1

2(α1(v)α2(v) + α2(v)α1(v))

This multiplication can be extended to a product on the space

S∗(V ∗) = ⊕NSN (V ∗)

(called the space of symmetric multilinear forms) by defining

(α1 ⊗ · · · ⊗ αj)(αj+1 ⊗ · · · ⊗ αN ) =1

N !

∑σ∈SN

ασ(1) ⊗ · · · ⊗ ασ(N)

One can show that S∗(V ∗) with this product is isomorphic to the algebra ofpolynomials on V . Evaluation of a polynomial at v is given by multiplying theαj(v) for the terms αj in the tensor product. Reflecting this isomorphism forthe case S1(V ∗) = V ∗, the symbol vj can equivalently mean the coordinatefunction vj ∈ V ∗, or the number vj(v).

Antisymmetric bilinear forms lie in Λ2(V ∗) ⊂ V ∗ ⊗ V ∗ and correspond toantisymmetric matrices. One can define a multiplication (called the “wedgeproduct”) on V ∗ that takes values in Λ2(V ∗) by

(α1, α2) ∈ V ∗ × V ∗ → α1 ∧ α2 =1

2(α1 ⊗ α2 − α2 ⊗ α1) ∈ Λ2(V ∗)

This multiplication can be extended to a product on the space

Λ∗(V ∗) = ⊕NΛN (V ∗)

(called the space of antisymmetric multilinear forms) by defining

(α1 ⊗ · · · ⊗ αj)(αj+1 ⊗ · · · ⊗ αN ) =1

N !

∑σ∈SN

(−1)|σ|ασ(1) ⊗ · · · ⊗ ασ(N)

One can use this to get a product on the space of antisymmetric multilinearforms of different degrees, giving something in many ways analogous to thealgebra of polynomials (although without a notion of evaluation at a point v).This plays a role in the description of fermions and will be considered in moredetail in chapter 27.

115


For more about the tensor product and tensor product of representations, seesection 6 of [74], or appendix B of [65]. Almost every quantum mechanics text-book will contain an extensive discussion of the Clebsch-Gordan decompositionfor the tensor product of two irreducible SU(2) representations.

A complete discussion of bilinear forms, together with the algebra of sym-metric and antisymmetric multilinear forms, can be found in [25].

116

Chapter 10

Energy, Momentum andTranslation Groups

We’ll now turn to the problem that conventional quantum mechanics coursesgenerally begin with: that of the quantum system describing a free particlemoving in physical space R3. This is something quite different than the classicalmechanical decription of a free particle, which will be reviewed in chapter 12.A common way of motivating this is to begin with the 1924 suggestion by deBroglie that, just as photons may behave like particles or waves, the same shouldbe true for matter particles. Photons carry an energy given by E = ~ω, whereω is the angular frequency, and de Broglie’s proposal was that matter particlesbehave like a wave with spatial dependence

eik·x

where x is the spatial position, and the momentum of the particle is p = ~k.This proposal was realized in Schrodinger’s early 1926 discovery of a version

of quantum mechanics, in which the state space H is a space of complex-valuedfunctions on R3, called “wavefunctions”. The operator

P = −i~∇

will have eigenvalues ~k, the de Broglie momentum, so it can be identified asthe momentum operator.

In this chapter our discussion will emphasize the central role of the momen-tum operator. This operator will have the same relationship to spatial trans-lations as the Hamiltonian operator does to time translations. In both cases,the operators are given by the Lie algebra representation corresponding to aunitary representation on the quantum state space H of groups of translations(translation in the three space and one time directions respectively).

One way to motivate the quantum theory of a free particle is that, whateverit is, it should have the same sort of behavior as in the classical case underthe translational and rotational symmetries of space-time. In chapter 12 we

117

will see that in the Hamiltonian form of classical mechanics, the components ofthe momentum vector give a basis of the Lie algebra of the spatial translationgroup R3, the energy a basis of the Lie algebra of the time translation groupR. Invoking the classical relationship between energy and momentum

E =|p|2

2m

used in non-relativistic mechanics relates the Hamiltonian and momentum oper-ators, giving the conventional Schrodinger differential equation for the wavefunc-tion of a free particle. We will examine the solutions to this equation, beginningwith the case of periodic boundary conditions, where spatial translations in eachdirection are given by the compact group U(1) (whose representations we havealready studied in detail).

10.1 Energy, momentum and space-time trans-lations

We have seen that it is a basic axiom of quantum mechanics that the observ-able operator responsible for infinitesimal time translations is the Hamiltonianoperator H, a fact that is expressed as the Schrodinger equation

i~d

dt|ψ〉 = H|ψ〉

When H is time-independent, one can understand this equation as reflecting theexistence of a unitary representation (U(t),H) of the group R of time transla-tions on the state space H. For the case of H infinite-dimensional, this is knownas Stone’s theorem for one-parameter unitary groups, see for instance chapter10.2 of [31] for details.

When H is finite-dimensional, the fact that a differentiable unitary repre-sentation U(t) of R on H is of the form

U(t) = e−i~ tH

for H a self-adjoint matrix follows from the same sort of argument as in theorem2.1. Such a U(t) provides solutions of the Schrodinger equation by

|ψ(t)〉 = U(t)|ψ(0)〉

The Lie algebra of R is also R and we get a Lie algebra representation of Rby taking the time derivative of U(t), which gives us

~d

dtU(t)|t=0 = −iH

Since this Lie algebra representation comes from taking the derivative of a uni-tary representation, −iH will be skew-adjoint, so H will be self-adjoint. The

118

minus sign is a convention, for reasons that will be explained in the discussionof momentum to come later.

Note that if one wants to treat the additive group R as a matrix group,related to its Lie algebra R by exponentiation of matrices, one can describe thegroup as the group of matrices of the form(

1 a0 1

)since (

1 a0 1

)(1 b0 1

)=

(1 a+ b0 1

)Since

e

0 a0 0

=

(1 a0 1

)the Lie algebra is just matrices of the form(

0 a0 0

)We will mostly though write the group law in additive form. We are inter-

ested in the group R as a group of translations acting on a linear space, and thecorresponding infinite dimensional representation induced on functions on thespace. The simplest case is when R acts on itself by translation. Here a ∈ Racts on q ∈ R (where q is a coordinate on R) by

q → a · q = q + a

and the induced representation π on functions uses

π(g)f(q) = f(g−1 · q)

to getπ(a)f(q) = f(q − a)

In the Lie algebra version of this representation, we will have

π′(a) = −a ddq

since

π(a)f = eπ′(a)f = e−a

ddq f(q) = f(q)− adf

dq+a2

2!

d2f

dq2+ · · · = f(q − a)

which for functions with appropriate properties is just Taylor’s formula. Notethat here the same a labels points of the Lie algebra and of the group. We arenot treating the group R as a matrix group, since we want an additive group

119

law. So Lie algebra elements are not defined as for matrix groups (things oneexponentiates to get group elements). Instead, we think of the Lie algebra asthe tangent space to the group at the identity, and then simply identify R as thetangent space at 0 (the Lie algebra) and R as the additive group. Note howeverthat the representation obeys a multiplicative law, with the homomorphismproperty

π(a+ b) = π(a)π(b)

so there is an exponential in the relation between π and π′.Since we now want to describe quantum systems that depend not just on

time, but on space variables q = (q1, q2, q3), we will have an action by unitarytransformations of not just the group R of time translations, but also the groupR3 of spatial translations. We will define the corresponding Lie algebra rep-resentations using self-adjoint operators P1, P2, P3 that play the same role forspatial translations that the Hamiltonian plays for time translations:

Definition (Momentum operators). For a quantum system with state spaceH given by complex valued functions of position variables q1, q2, q3, momentumoperators P1, P2, P3 are defined by

P1 = −i~ ∂

∂q1, P2 = −i~ ∂

∂q2, P3 = −i~ ∂

∂q3

These are given the name “momentum operators” since we will see that theireigenvalues have an intepretation as the components of the momentum vectorfor the system, just as the eigenvalues of the Hamiltonian have an interpretationas the energy. Note that while in the case of the Hamiltonian the factor of ~ kepttrack of the relative normalization of energy and time units, here it plays thesame role for momentum and length units. It can be set to one if appropriatechoices of units of momentum and length are made.

The differentiation operator is skew-adjoint since, using integration by partsone has for ψ ∈ H∫ +∞

−∞ψ(

d

dqψ)dq =

∫ +∞

−∞(d

dq(ψψ)− (

d

dqψ)ψ)dq = −

∫ +∞

−∞(d

dqψ)ψdq

The Pj are thus self-adjoint operators, with real eigenvalues as expected for anobservable operator. Multiplying by −i to get the corresponding skew-adjointoperator of a unitary Lie algebra representation we find

−iPj = −~ ∂

∂qj

Up to the ~ factor that depends on units, these are exactly the Lie algebrarepresentation operators on basis elements for the action of R3 on functions onR3 induced from translation:

π(a1, a2, a3)f(q1, q2, q3) = f(q1 − a1, q2 − a2, q3 − a3)

120

π′(a1, a2, a3) = a1(−iP1) +a2(−iP2) +a3(−iP3) = −~(a1∂

∂q1+a2

∂

∂q2+a3

∂

∂q3)

Note that the convention for the sign choice here is the opposite from thecase of the Hamiltonian (−iP = −~ d

dq vs. −iH = ~ ddt ). This means that the

conventional sign choice we have been using for the Hamiltonian makes it minusthe generator of translations in the time direction. The reason for this comesfrom considerations of special relativity, where the inner product on space-timehas opposite signs for the space and time dimensions . We will review thissubject in chapter 38 but for now we just need the relationship special relativitygives between energy and momentum. Space and time are put together in“Minkowski space”, which is R4 with indefinite inner product

< (u0, u1, u2, u3), (v0, v1, v2, v3) >= −u0v0 + u1v1 + u2v2 + u3v3

Energy and momentum are the components of a Minkowski space vector (p0 =E, p1, p2, p3) with norm-squared given by minus the mass-squared:

< (E, p1, p2, p3), (E, p1, p2, p3) >= −E2 + |p|2 = −m2

This is the formula for a choice of space and time units such that the speed oflight is 1. Putting in factors of the speed of light c to get the units right onehas

E2 − |p|2c2 = m2c4

Two special cases of this are:

• For photons, m = 0, and one has the energy momentum relation E = |p|c

• For velocities v small compared to c (and thus momenta |p| small com-pared to mc), one has

E =√|p|2c2 +m2c4 = c

√|p|2 +m2c2 ≈ c|p|2

2mc+mc2 =

|p|2

2m+mc2

In the non-relativistic limit, we use this energy-momentum relation todescribe particles with velocities small compared to c, typically droppingthe momentum-independent constant term mc2.

In later chapters we will discuss quantum systems that describe photons,as well as other possible ways of constructing quantum systems for relativisticparticles. For now though, we will stick to the non-relativistic case. To describea quantum non-relativistic particle we choose a Hamiltonian operator H suchthat its eigenvalues (the energies) will be related to the momentum operator

eigenvalues (the momenta) by the classical energy-momentum relation E = |p|22m :

H =1

2m(P 2

1 + P 22 + P 2

3 ) =1

2m|P|2 =

−~2

2m(∂2

∂q21

+∂2

∂q22

+∂2

∂q23

)

121

The Schrodinger equation then becomes:

i~∂

∂tψ(q, t) =

−~2

2m(∂2

∂q21

+∂2

∂q22

+∂2

∂q23

)ψ(q, t) =−~2

2m∇2ψ(q, t)

This is an easily solved simple constant coefficient second-order partial differ-ential equation. One method of solution is to separate out the time-dependence,by first finding solutions ψE to the time-independent equation

HψE(q) =−~2

2m∇2ψE(q) = EψE(q)

with eigenvalue E for the Hamiltonian operator and then use the fact that

ψ(q, t) = ψE(q)e−i~ tE

will give solutions to the full-time dependent equation

i~∂

∂tψ(q, t) = Hψ(q, t)

The solutions ψE(q) to the time-independent equation are just complex expo-nentials proportional to

ei(k1q1+k2q2+k3q3) = eik·q

satisfying−~2

2m(−i)2|k|2 =

~2|k|2

2m= E

We have found that solutions to the Schrodinger equation are given by linearcombinations of states |k〉 labeled by a vector k, which are eigenstates of themomentum and Hamiltonian operators with

Pj |k〉 = ~kj |k〉, H|k〉 =~2

2m|k|2|k〉

These are states with well-defined momentum and energy

pj = ~kj , E =|p|2

2m

so they satisfy exactly the same energy-momentum relations as those for a clas-sical non-relativistic particle.

While the quantum mechanical state space H contains states with the clas-sical energy-momentum relation, it also contains much, much more since itincludes linear combinations of such states. At t = 0 one has

|ψ〉 =∑k

ckeik·q

122

where ck are complex numbers, and the general time-dependent state will be

|ψ(t)〉 =∑k

ckeik·qe−it~

|k|22m

or, equivalently in terms of momenta p = ~k

|ψ(t)〉 =∑p

cpei~p·qe−

i~|p|22m t

10.2 Periodic boundary conditions and the groupU(1)

We have not yet discussed the inner product on our space of states when theyare given as wavefunctions on R3, and there is a significant problem with doingthis. To get unitary representations of translations, we need to use a translationinvariant, Hermitian inner product on wavefunctions, and this will have to beof the form

〈ψ1, ψ2〉 = C

∫R3

ψ1(q)ψ2(q)d3q

for some constant C. But if we try and compute the norm-squared of one ofour basis states |k〉 we find

〈k|k〉 = C

∫R3

(e−ik·q)(eik·q)d3q = C

∫R3

1 d3q =∞

As a result there is no value of C which will give these states a unit norm.In the finite dimensional case, a linear algebra theorem assures us that given a

self-adjoint operator, we can find an orthonormal basis of its eigenvectors. In thisinfinite dimensional case this is no longer true, and a much more sophisticatedformalism (the “spectral theorem for self-adjoint operators”) is needed to replacethe linear algebra theorem. This is a standard topic in treatments of quantummechanics aimed at mathematicians emphasizing analysis, but we will not tryand enter into this here. One place to find such a discussion is section 2.1 of[71].

One way to deal with the normalization problem is to replace the non-compact space by one of finite volume. We’ll consider first the simplified case ofa single spatial dimension, since once one sees how this works for one dimension,treating the others the same way is straight-forward. In this one dimensionalcase, one replaces R by the circle S1. This is equivalent to the physicist’s methodof imposing “periodic boundary conditions”, meaning to define the theory onan interval, and then identify the ends of the interval. One can then think ofthe position variable q as an angle φ and define the inner product as

〈ψ1, ψ2〉 =1

2π

∫ 2π

0

ψ1(φ)ψ2(φ)dφ

123

The state space is thenH = L2(S1)

the space of complex-valued square-integrable functions on the circle.Instead of the translation group R, we have the standard action of the

group SO(2) on the circle. Elements g(θ) of the group are rotations of the circlecounterclockwise by an angle θ, or if we parametrize the circle by an angle φ,just shifts

φ→ φ+ θ

Recall that in general we can construct a representation on functions from agroup action on a space by

π(g)f(x) = f(g−1 · x)

so we see that this rotation action on the circle gives a representation on H

π(g(θ))ψ(φ) = ψ(φ− θ)

If X is a basis of the Lie algebra so(2) (for instance taking the circle as the unit

circle in R2, rotations 2 by 2 matrices, X =

(0 −11 0

), g(θ) = eθX) then the

Lie algebra representation is given by taking the derivative

π′(X)f(φ) =d

dθf(φ− θ)|θ=0 = −f ′(φ)

so we have

π′(X) = − d

dφ

This operator is defined on a dense subspace of H = L2(S1) and is skew-adjoint,since (using integration by parts)

〈ψ1, ψ′2〉 =

1

2π

∫ 2π

0

ψ1d

dφψ2dφ

=1

2π

∫ 2π

0

(d

dφ(ψ1ψ2)− (

d

dφψ1)ψ2)dφ

=− 〈ψ′1, ψ2〉

The eigenfunctions of π′(X) are just the einφ, for n ∈ Z, which we will alsowrite as state vectors |n〉. These are orthonormal

〈n|m〉 = δnm

and provide a basis for the space L2(S1), a basis that corresponds to the de-composition into irreducibles of

L2(S1)

124

as a representation of SO(2) described above. One has

(π, L2(S1)) = ⊕n∈Z(πn,C)

where πn are the irreducible one-dimensional representations given by

πn(g(θ)) = einθ

The theory of Fourier series for functions on S1 says that one can expand anyfunction ψ ∈ L2(S1) in terms of this basis, i.e.

|ψ〉 = ψ(φ) =

+∞∑n=−∞

cneinφ =

+∞∑n=−∞

cn|n〉

where cn ∈ C. The condition that ψ ∈ L2(S1) corresponds to the condition

+∞∑n=−∞

|cn|2 <∞

on the coefficients cn. Using orthonormality of the |n〉 we find

cn = 〈n|ψ〉 =1

2π

∫ 2π

0

e−inφψ(φ)dφ

The Lie algebra of the group S1 is the same as that of the group (R,+),and the π′(X) we have found for the S1 action on functions is related to themomentum operator in the same way as in the R case. So, we can use the samemomentum operator

P = −i~ d

dφ

which satisfiesP |n〉 = ~n|n〉

By changing space to the compact S1 we now have momenta that instead oftaking on any real value, can only be integral numbers times ~. Solving theSchrodinger equation

i~∂

∂tψ(φ, t) =

P 2

2mψ(φ, t) =

−~2

2m

∂2

∂φ2ψ(φ, t)

as before, we find

EψE(φ) =−~2

2m

d2

dφ2ψE(φ)

an eigenvector equation, which has solutions |n〉, with

E =~2n2

2m

125

Writing a solution to the Schrodinger equation as

ψ(φ, t) =

+∞∑n=−∞

cneinφe−i

~n2

2m t

the cn will be determined from the initial condition of knowing the wavefunctionat time t = 0, according to the Fourier coefficient formula

cn =1

2π

∫ 2π

0

e−inφψ(φ, 0)dφ

To get something more realistic, we need to take our circle to have an ar-bitrary circumference L, and we can study our original problem by consideringthe limit L → ∞. To do this, we just need to change variables from φ to φL,where

φL =L

2πφ

The momentum operator will now be

P = −i~ d

dφL

and its eigenvalues will be quantized in units of 2π~L . The energy eigenvalues

will be

E =2π2~2n2

mL2

10.3 The group R and the Fourier transform

In the previous section, we imposed periodic boundary conditions, replacing thegroup R of translations by a compact group S1, and then used the fact thatunitary representations of this group are labeled by integers. This made theanalysis rather easy, with H = L2(S1) and the self-adjoint operator P = −i~ ∂

∂φbehaving much the same as in the finite-dimensional case: the eigenvectors ofP give a countable orthonormal basis of H. If one wants to, one can think of Pas an infinite-dimensional matrix.

Unfortunately, in order to understand many aspects of quantum mechanics,we can’t get away with this trick, but need to work with R itself. One reasonfor this is that the unitary representations of R are labeled by the same group,R, and we will find it very important to exploit this and treat positions andmomenta on the same footing (see the discussion of the Heisenberg group inchapter 11). What plays the role here of |n〉 = einφ, n ∈ Z will be the |k〉 = eikq,k ∈ R. These are functions on R that are irreducible representations under thetranslation action (π(a) acts on functions of q by taking q → q − a)

π(a)eikq = eik(q−a) = e−ikaeikq

We can try and mimic the Fourier series decomposition, with the coefficientscn that depend on the labels of the irreducibles replaced by a function f(k)depending on the label k of the irreducible representation of R.

126

Definition (Fourier transform). The Fourier transform of a function ψ is givenby

Fψ = ψ(k) ≡ 1√2π

∫ ∞−∞

e−ikqψ(q)dq

The definition makes sense for ψ ∈ L1(R), Lebesgue integrable functions onR. For the following, it is convenient to instead restrict to the Schwartz spaceS(R) of functions ψ such that the function and its derivatives fall off faster thanany power at infinity (which is a dense subspace of L2(R)). For more detailsabout the analysis and proofs of the theorems quoted here, one can refer to astandard textbook such as [69].

Given the Fourier transform of ψ, one can recover ψ itself:

Theorem (Fourier Inversion). For ψ ∈ S(R) the Fourier transform of a func-tion ψ ∈ S(R), one has

ψ(q) = F ψ =1√2π

∫ +∞

−∞eikqψ(k)dk

Note that F is the same linear operator as F , with a change in sign of theargument of the function it is applied to. Note also that we are choosing oneof various popular ways of normalizing the definition of the Fourier transform.In others, the factor of 2π may appear instead in the exponent of the complexexponential, or just in one of F or F and not the other.

The operators F and F are thus inverses of each other on S(R). One has

Theorem (Plancherel). F and F extend to unitary isomorphisms of L2(R)with itself. In other words∫ ∞

−∞|ψ(q)|2dq =

∫ ∞−∞|ψ(k)|2dk

Note that we will be using the same inner product on functions on R

〈ψ1, ψ2〉 =

∫ ∞−∞

ψ1(q)ψ2(q)dq

both for functions of q and their Fourier transforms, functions of k.An important example is the case of Gaussian functions where

Fe−αq2

2 =1√2π

∫ +∞

−∞e−ikqe−α

q2

2 dq

=1√2π

∫ +∞

−∞e−

α2 ((q+i 2k

α )2−( ikα )2)dq

=1√2πe−

k2

2α

∫ +∞

−∞e−

α2 q′2dq′

=1√αe−

k2

2α

127

A crucial property of the unitary operator F on H is that it diagonalizesthe differentiation operator and thus the momentum operator P . Under Fouriertransform, differential operators become just multiplication by a polynomial,giving a powerful technique for solving differential equations. Computing theFourier transform of the differentiation operator using integration by parts, wefind

dψ

dq=

1√2π

∫ +∞

−∞e−ikq

dψ

dqdq

=1√2π

∫ +∞

−∞(d

dq(e−ikqψ)− (

d

dqe−ikq)ψ)dq

=ik1√2π

∫ +∞

−∞e−ikqψdq

=ikψ(k)

So under Fourier transform, differentiation by q becomes multiplication by ik.This is the infinitesimal version of the fact that translation becomes multiplica-tion by a phase under Fourier transform. If ψa(q) = ψ(q + a), one has

ψa(k) =1√2π

∫ +∞

−∞e−ikqψ(q + a)dq

=1√2π

∫ +∞

−∞e−ik(q′−a)ψ(q′)dq′

=eikaψ(k)

Since p = ~k, we can easily change variables and work with p instead of k,and often will do this from now on. As with the factors of 2π, there’s a choiceof where to put the factors of ~ in the normalization of the Fourier transform.We’ll make the following choices, to preserve symmetry between the formulasfor Fourier transform and inverse Fourier transform:

ψ(p) =1√2π~

∫ +∞

−∞e−i

pq~ dq

ψ(q) =1√2π~

∫ +∞

−∞eipq~ dp

Note that in this case we have lost an important property that we had forfinite dimensional H and had managed to preserve by using S1 rather thanR as our space. If we take H = L2(R), the eigenvectors for the operator P(the functions eikq) are not square-integrable, so not in H. The operator Pis an unbounded operator and we no longer have a theorem saying that itseigenvectors give an orthornormal basis of H. As mentioned earlier, one wayto deal with this uses a general spectral theorem for self-adjoint operators on aHilbert space, for more details see Chapter 2 of [71].

128

10.3.1 Delta functions

One would like to think of the eigenvectors of the operator P as in some sensecontinuing to provide an orthonormal basis for H. One problem is that theseeigenvectors are not square-integrable, so one needs to expand one’s notion ofstate space H beyond a space like L2(R). Another problem is that Fouriertransforms of such eigenvectors (which will be eigenvectors of the position op-erator) gives something that is not a function but a distribution. The propergeneral formalism for handling state spacesH which include eigenvectors of bothposition and momentum operators seems to be that of “rigged Hilbert spaces”which this author confesses to never have mastered (the standard reference is[22]). As a result we won’t here give a rigorous discussion, but will use non-normalizable functions and distributions in the non-rigorous form in which theyare used in physics. The physics formalism is set up to work as if H was finitedimensional and allows easy manipulations which don’t obviously make sense.Our interest though is not in the general theory, but in very specific quantumsystems, where everything is determined by their properties as unitary grouprepresentations. For such systems, the general theory of rigged Hilbert spacesis not needed, since for the statements we are interested in various ways canbe found to make them precise (although we will generally not enter into thecomplexities needed to do so).

Given any function g(q) on R, one can try and define an element of the dualspace of the space of functions on R by integration, i.e by the linear operator

f →∫ +∞

−∞g(q)f(q)dq

(we won’t try and specify which condition on functions f or g is chosen to makesense of this). There are however some other very obvious linear functionals onsuch a function space, for instance the one given by evaluating the function atq = c:

f → f(c)

Such linear functionals correspond to generalized functions, objects which whenfed into the formula for integration over R give the desired linear functional.The most well-known of these is the one that gives this evaluation at q = c, itis known as the “delta function” and written as δ(q− c). It is the object which,if it were a function, would satisfy∫ +∞

−∞δ(q − c)f(q)dq = f(c)

To make sense of such an object, one can take it to be a limit of actual functions.For the δ-function, consider the limit as ε→ 0 of

gε =1√2πε

e−(q−c)2

2ε

129

which satisfy ∫ +∞

−∞gε(q)dq = 1

for all ε > 0 (one way to see this is to use the formula given earlier for theFourier transform of a Gaussian).

Heuristically (ignoring obvious problems of interchange of integrals thatdon’t make sense), one can write the Fourier inversion formula as follows

ψ(x) =1√2π

∫ +∞

−∞eikqψ(k)dk

=1√2π

∫ +∞

−∞eikq(

1√2π

∫ +∞

−∞e−ikq

′ψ(q′)dq′)dk

=1

2π

∫ +∞

−∞(

∫ +∞

−∞eik(q−q′)ψ(q′)dk)dq′

=

∫ +∞

−∞δ(q′ − q)ψ(q′)dq′

Taking the delta function to be an even function (so δ(x′ − x) = δ(x− x′)),one can interpret the above calculation as justifying the formula

δ(q − q′) =1

2π

∫ +∞

−∞eik(q−q′)dk

One then goes on to consider the eigenvectors

|k〉 =1√2πeikq

of the momentum operator as satisfying a replacement for the finite-dimensionalorthonormality relation, with the δ-function replacing the δnm:

〈k′|k〉 =

∫ +∞

−∞(

1√2πeik′q)(

1√2πeikq)dq =

1

2π

∫ +∞

−∞ei(k−k

′)qdq = δ(k − k′)

As mentioned before, we will usually work with the variable p = ~k, in whichcase we have

|p〉 =1√2π~

eipq~

and

δ(p− p′) =1

2π~

∫ +∞

−∞ei

(p−p′)q~ dq

For a more mathematically legitimate version of this calculation, one placeto look is Lecture 6 in the notes on physics by Dolgachev [13].

130


Every book about quantum mechanics covers this example of the free quantumparticle somewhere very early on, in detail. Our discussion here is unusualjust in emphasizing the role of the spatial translation groups and its unitaryrepresentations. Discussions of quantum mechanics for mathematicians (suchas [71]) typically emphasize the development of the functional analysis neededfor a proper description of the Hilbert space H and of the properties of generalself-adjoint operators on this state space. In this class we’re restricting attentionto a quite limited set of such operators coming from Lie algebra representations,so will avoid the general theory.

131

Chapter 11

The Heisenberg group andthe SchrodingerRepresentation

In our discussion of the free particle, we used just the actions of the groups R3 ofspatial translations and the group R of time translations, finding correspondingobservables, the self-adjoint momentum (P ) and Hamiltonian (H) operators.We’ve seen though that the Fourier transform involves a perfectly symmetricaltreatment of position and momentum variables. This allows us to introduce aposition operator Q acting on our state space H. We will analyze in detail inthis chapter the implications of extending the algebra of observable operators inthis way, most of the time restricting to the case of a single spatial dimension,since the physical case of three dimensions is an easy generalization.

The P and Q operators generate an algebra usually called the Heisenbergalgebra, since Werner Heisenberg and collaborators used it in the earliest workon a full quantum-mechanical formalism in 1925. It was quickly recognized byHermann Weyl that this algebra comes from a Lie algebra representation, witha corresponding group (called the Heisenberg group by mathematicians, theWeyl group by physicists). The state space of a quantum particle, either free ormoving in a potential, will be a unitary representation of this group, with thegroup of spatial translations a subgroup. Note that this particular use of a groupand its representation theory in quantum mechanics is both at the core of thestandard axioms and much more general than the usual characterization of thesignificance of groups as “symmetry groups”. The Heisenberg group does not inany sense correspond to a group of invariances of the physical situation (thereare no states invariant under the group), and its action does not commute withany non-zero Hamiltonian operator. Instead it plays a much deeper role, withits unique unitary representation determining much of the structure of quantummechanics.

Note: beginning with this chapter, we will always assume units for position

133

and momentum chosen so that ~ = 1 and no longer keep track of how thisdimensional constant appears in equations.

11.1 The position operator and the HeisenbergLie algebra

In the description of the state space H as functions of a position variable q, themomentum operator is

P = −i ddq

The Fourier transform F provides a unitary transformation to a description ofH as functions of a momentum variable p in which the momentum operator Pis just multiplication by p. Exchanging the role of p and q, one gets a positionoperator Q that acts as

Q = id

dp

when states are functions of p (the sign difference comes from the sign change

in F vs. F), or as multiplication by q when states are functions of q.

11.1.1 Position space representation

In the position space representation, taking as position variable q′, one hasnormalized eigenfunctions describing a free particle of momentum p

|p〉 =1√2πeipq

′

which satisfy

P |p〉 = −i ddq′

(1√2πeipq

′) = p(

1√2πeipq

′) = p|p〉

The operator Q in this representation is just the multiplication operator

Qψ(q′) = q′ψ(q′)

that multiplies a function of the position variable q′ by q′. The eigenvectors |q〉of this operator will be the δ-functions δ(q′ − q) since

Q|q〉 = q′δ(q′ − q) = qδ(q′ − q)

A standard convention in physics is to think of a state written in the notation|ψ〉 as being representation independent. The wavefunction in the position spacerepresentation can then be found by taking the coefficient of |ψ〉 in the expansionof a state in Q eigenfunctions |q〉, so

〈q|ψ〉 =

∫ +∞

−∞δ(q − q′)ψ(q′)dq′ = ψ(q)

134

and in particular

〈q|p〉 =1√2πeipq

11.1.2 Momentum space representation

In the momentum space description of H as functions of p′, the state is theFourier transform of the state in the position space representation, so the state|p〉 will be the function (actually, the distribution) on momentum space

F(1√2πeipq

′) =

1

2π

∫ +∞

−∞e−ip

′q′eipq′dq′ =

1

2π

∫ +∞

−∞ei(p−p

′)q′dq′ = δ(p− p′)

These are eigenfunctions of the operator P , which is a multiplication operatorin this representation

P |p〉 = p′δ(p′ − p) = pδ(p′ − p)

The position eigenfunctions are also given by Fourier transform

|q〉 = F(δ(q − q′)) =1√2π

∫ +∞

−∞e−ip

′q′δ(q − q′)dq′ =1√2πe−ip

′q

The position operator is

Q = id

dp

and |q〉 is an eigenvector with eigenvalue q

Q|q〉 = id

dp′(

1√2π~

e−ip′q) = q(

1√2πe−ip

′q) = q|q〉

Another way to see that this is the correct operator is to use the unitary trans-formation F and its inverse F that relate the position and momentum spacerepresentations. Going from position space to momentum space one has

Q→ FQF

and one can check that this transformed Q operator will act as i ddp′ on functions

of p′.

One can express momentum space wavefunctions as coefficients of the ex-pansion of a state ψ in terms of momentum eigenvectors

〈p|ψ〉 =

∫ +∞

−∞(

1√2πe−ipq

′)ψ(q′)dq′ = F(ψ(q)) = ψ(p)

135

11.1.3 Physical interpretation

With now both momentum and position operators on H, we have the standardset-up for describing a non-relativistic quantum particle that is discussed exten-sively early on in any quantum mechanics textbook, and one of these should beconsulted for more details and for explanations of the physical interpretationof this quantum system. The classically observable quantity corresponding tothe operator P is the momentum, and eigenvectors of P are the states thathave well-defined values for this (the eigenvalue). The momentum eigenvaluesand energy eigenvalues will have the correct non-relativistic energy momentumrelationship. Note that for the free particle P commutes with the Hamiltonian

H = P 2

2m so there is a conservation law: states with a well-defined momentumat one time always have the same momentum. This corresponds to an obviousphysical symmetry, the symmetry under spatial translations.

The operator Q on the other hand does not correspond to a physical sym-metry, since it does not commute with the Hamiltonian. We will see that itdoes generate a group action, and from the momentum space picture we cansee that this is a shift in the momentum, but such shifts are not symmetries ofthe physics and there is no conservation law for Q. The states in which Q hasa well-defined numerical value are the ones such that the position wavefunctionis a delta-function. If one prepares such a state at a given time, it will notremain a delta-function, but quickly evolve into a wavefunction that spreadsout in space.

Since the eigenfunctions of P and Q are non-normalizable, one needs aslightly different formulation of the measurement theory principle used for finitedimensional H. In this case, the probability of observing a position of a particlewith wavefunction ψ(q) in the interval [q1, q2] will be∫ q2

q1ψ(q)ψ(q)dq∫ +∞

−∞ ψ(q)ψ(q)dq

This will make sense for states |ψ〉 ∈ L2(R), which we will normalize to havenorm-squared one when discussing their physical interpretation. Then the sta-tistical expectation value for the measured position variable will be

〈ψ|Q|ψ〉

which can be computed in either the position or momentum space representa-tion.

Similarly, the probability of observing a momentum of a particle with momentum-space wavefunction ψ(q) in the interval [p1, p2] will be∫ p2

p1ψ(p)ψ(p)dp∫ +∞

−∞ ψ(p)ψ(p)dp

and for normalized states the statistical expectation value of the measured mo-mentum is

〈ψ|P |ψ〉

136

Note that states with a well-defined position (the delta-function states inthe position-space representation) are equally likely to have any momentumwhatsoever. Physically this is why such states quickly spread out. States witha well-defined momentum are equally likely to have any possible position. Theproperties of the Fourier transform imply the so-called “Heisenberg uncertaintyprinciple” that gives a lower bound on the product of a measure of uncertaintyin position times the same measure of uncertainty in momentum. Examplesof this that take on the lower bound are the Gaussian shaped functions whoseFourier transforms were computed earlier.

For much more about these questions, again most quantum mechanics text-books will contain an extensive discussion.

11.2 The Heisenberg Lie algebra

In either the position or momentum space representation the operators P andQ satisfy the relation

[Q,P ] = i1

Soon after this commutation relation appeared in early work on quantum me-chanics, Weyl realized that it can be interpreted as the relation between oper-ators one would get from a representation of a three-dimensional Lie algebra,now called the Heisenberg Lie algebra.

Definition (Heisenberg Lie algebra, d = 1). The Heisenberg Lie algebra h3 isthe vector space R3 with the Lie bracket defined by its values on a basis (X,Y, Z)by

[X,Y ] = Z, [X,Z] = [Y,Z] = 0

Writing a general element of h3 in terms of this basis as xX + yY + zZ,and grouping the x, y coordinates together (we will see that it is useful to thinkof the vector space h3 as R2 ⊕ R), the Lie bracket is given in terms of thecoordinates by

[(

(xy

), z), (

(x′

y′

), z′)] = (

(00

), xy′ − yx′)

Note that this is a non-trivial Lie algebra, but only minimally so. All Liebrackets of Z with anything else are zero. All Lie brackets of Lie brackets arealso zero (as a result, this is an example of what is known as a “nilpotent” Liealgebra).

The Heisenberg Lie algebra is isomorphic to the Lie algebra of 3 by 3 strictlyupper triangular real matrices, with Lie bracket the matrix commutator, by thefollowing isomorphism:

X ↔

0 1 00 0 00 0 0

, Y ↔

0 0 00 0 10 0 0

, Z ↔

0 0 10 0 00 0 0

137

xX + yY + zZ ↔

0 x z0 0 y0 0 0

and one has

[

0 x z0 0 y0 0 0

,

0 x′ z′

0 0 y′

0 0 0

] =

0 0 xy′ − x′y0 0 00 0 0

The generalization of this to higher dimensions is

Definition (Heisenberg Lie algebra). The Heisenberg Lie algebra h2d+1 is thevector space R2d+1 = R2d ⊕R with the Lie bracket defined by its values on abasis Xj , Yj , Z, (j = 1, . . . d) by

[Xj , Yk] = δjkZ, [Xj , Z] = [Yj , Z] = 0

Writing a general element as∑dj=1 xjXj +

∑dk=1 ykYk + zZ, in terms of coor-

dinates the Lie bracket is

[(

(xy

), z), (

(x′

y′

), z)] = (

(00

),x · y′ − y · x′)

One can write this Lie algebra as a Lie algebra of matrices for any d. Forinstance, in the physical case of d = 3, elements of the Heisenberg Lie algebracan be written

0 x1 x2 x3 z0 0 0 0 y3

0 0 0 0 y2

0 0 0 0 y1

0 0 0 0 0

11.3 The Heisenberg group

One can easily see that exponentiating matrices in h3 gives

exp

0 x z0 0 y0 0 0

=

1 x z + 12xy

0 1 y0 0 1

so the group with Lie algebra h3 will be the group of upper triangular 3 by 3 realmatrices with ones on the diagonal, and this group will be the Heisenberg groupH3. For our purposes though, it is better to work in exponential coordinates(i.e. labeling a group element with the Lie algebra element that exponentiatesto it).

Matrix exponentials in general satisfy the Baker-Campbell-Hausdorff for-mula, which says

eAeB = eA+B+ 12 [A,B]+ 1

12 [A,[A,B]]− 112 [B,[A,B]]+···

138

where the higher terms can all be expressed as repeated commutators. Thisprovides one way of showing that the Lie group structure is determined (forgroup elements expressible as exponentials) by knowing the Lie bracket. Forthe full formula and a detailed proof, see chapter 3 of [29]. One can easilycheck the first few terms in this formula by expanding the exponentials, but thedifficulty of the proof is that it is not at all obvious why all the terms can beorganized in terms of commutators.

For the case of the Heisenberg Lie algebra, since all multiple commutatorsvanish, the Baker-Campbell-Hausdorff formula implies for exponentials of ele-ments of h3

eAeB = eA+B+ 12 [A,B]

(a proof of this special case of Baker-Campbell-Hausdorff is in section 3.1 of [29]).We can use this to explicitly write the group law in exponential coordinates:

Definition (Heisenberg group, d = 1). The Heisenberg group H3 is the spaceR3 with the group law

(

(xy

), z)(

(x′

y′

)z′) = (

(x+ x′

y + y′

), z + z′ +

1

2(xy′ − yx′)) (11.1)

Note that the Lie algebra basis elements X,Y, Z each generate subgroupsof H3 isomorphic to R. Elements of the first two of these subgroups generatethe full group, and elements of the third subgroup are “central”, meaning theycommute with all group elements. Also notice that the non-commutative natureof the Lie algebra or group depends purely on the factor xy′ − yx′.

The generalization of this to higher dimensions is:

Definition (Heisenberg group). The Heisenberg group H2d+1 is the space R2d+1

with the group law

(

(xy

), z)(

(x′

y′

), z′) = (

(x + x′

y + y′

), z + z′ +

1

2(x · y′ − y · x′))

where the vectors here all live in Rd.

Note that in these exponential coordinates the exponential map relating theHeisenberg Lie algebra h2d+1 and the Heisenberg Lie group H2d+1 is just theidentity map.

11.4 The Schrodinger representation

Since it can be defined in terms of 3 by 3 matrices, the Heisenberg group H3

has an obvious representation on C3, but this representation is not unitary andnot of physical interest. What is of great interest is the infinite dimensionalrepresentation on functions of q for which the Lie algebra version is given bythe Q, P and unit operators:

139

Definition (Schrodinger representation, Lie algebra version). The Schrodingerrepresentation of the Heisenberg Lie algebra h3 is the representation (Γ′S , L

2(R))satisfying

Γ′S(X)ψ(q) = −iQψ(q) = −iqψ(q), Γ′S(Y )ψ(q) = −iPψ(q) = − d

dqψ(q)

Γ′S(Z)ψ(q) = −iψ(q)

Factors of i have been chosen to make these operators skew-adjoint and therepresentation thus unitary. They can be exponentiated, giving in the exponen-tial coordinates on H3 of equation 11.1

ΓS((

(x0

), 0))ψ(q) = e−xiQψ(q) = e−ixqψ(q)

ΓS((

(0y

), 0))ψ(q) = e−yiPψ(q) = e−y

ddqψ(q) = ψ(q − y)

ΓS((

(00

), z))ψ(q) = e−izψ(q)

For general group elements of H3 one has

Definition (Schrodinger representation, Lie group version). The Schrodingerrepresentation of the Heisenberg Lie group H3 is the representation (ΓS , L

2(R))satisfying

ΓS((

(xy

), z))ψ(q) = e−izei

xy2 e−ixqψ(q − y)

To check that this defines a representation, one computes

ΓS((

(xy

), z))ΓS((

(x′

y′

), z′))ψ(q) = ΓS((

(xy

), z))e−iz

′eix′y′

2 e−ix′qψ(q − y′)

=e−i(z+z′)ei

xy+x′y′2 e−ixqe−ix

′(q−y)ψ(q − y − y′)

=e−i(z+z′+ 1

2 (xy′−yx′))ei(x+x′)(y+y′)

2 e−i(x+x′)qψ(q − (y + y′))

=ΓS((

(x+ x′

y + y′

), z + z′ +

1

2(xy′ − yx′)))

The group analog of the Heisenberg commutation relations (often called the“Weyl form” of the commutation relations) is the relation

e−ixQe−iyP = e−ixye−iyP e−ixQ

One can derive this by calculating (using the Baker-Campbell-Hausdorff for-mula)

e−ixQe−iyP = e−i(xQ+yP )+ 12 [−ixQ,−iyP ]) = e−i

xy2 e−i(xQ+yP )

140

as well as the same product in the opposite order, and then comparing theresults.

Note that, for the Schrodinger representation, we have

ΓS((

(00

), z + 2π)) = ΓS((

(00

), z))

so the representation operators are periodic with period 2π in the z-coordinate.Some authors choose to define the Heisenberg group H3 as not R2 ⊕ R, butR2⊕S1, building this periodicity automatically into the definition of the group,rather than the representation.

We have seen that the Fourier transform F takes the Schrodinger represen-tation to a unitarily equivalent representation of H3, in terms of functions of p(the momentum space representation). The operators change as

ΓS(g)→ F ΓS(g)F

when one makes the unitary transformation to the momentum space represen-tation.

In typical physics quantum mechanics textbooks, one often sees calculationsmade just using the Heisenberg commutation relations, without picking a spe-cific representation of the operators that satisfy these relations. This turns outto be justified by the remarkable fact that, for the Heisenberg group, once onepicks the constant with which Z acts, all irreducible representations are uni-tarily equivalent. By unitarity this constant is −ic, c ∈ R. We have chosenc = 1, but other values of c would correspond to different choices of units. In asense, the representation theory of the Heisenberg group is very simple: there’sjust one irreducible representation. This is very different than the theory foreven the simplest compact Lie groups (U(1) and SU(2)) which have an infinityof inequivalent irreducibles labeled by weight or by spin. Representations of aHeisenberg group will appear in different guises (we’ve seen two, will see an-other in the discussion of the harmonic oscillator, and there are yet others thatappear in the theory of theta-functions), but they are all unitarily equivalent.This statement is known as the Stone-von Neumann theorem.

So far we’ve been modestly cavalier about the rigorous analysis needed tomake precise statements about the Schrodinger representation. In order to provea theorem like the Stone-von Neumann theorem, which tries to say somethingabout all possible representations of a group, one needs to invoke a great dealof analysis. Much of this part of analysis was developed precisely to be able todeal with general quantum mechanical systems and prove theorems about them.The Heisenberg group, Lie algebra and its representations are treated in detailin many expositions of quantum mechanics for mathematicians. Some goodreferences for this material are [71], and [31]. In depth discussions devoted tothe mathematics of the Heisenberg group and its representations can be foundin [38], [20] and [73].

In these references can be found a proof of the (not difficult)

Theorem. The Schrodinger representation ΓS described above is irreducible.

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and the much more difficult

Theorem (Stone-von Neumann). Any irreducible representation π of the groupH3 on a Hilbert space, satisfying

π′(Z) = −i1

is unitarily equivalent to the Schrodinger representation (ΓS , L2(R))

Note that all of this can easily be generalized to the case of d spatial di-mensions, for d finite, with the Heisenberg group now H2d+1 and the Stone-vonNeumann theorem still true. In the case of an infinite number of degrees offreedom, which is the case of interest in quantum field theory, the Stone-vonNeumann theorem no longer holds and one has an infinity of inequivalent irre-ducible representations, leading to quite different phenomena. For more on thistopic see chapter 37.

It is also important to note that the Stone-von Neumann theorem is for-mulated for Heisenberg group representations, not for Heisenberg Lie algebrarepresentations. For infinite-dimensional representations in cases like this, thereare representations of the Lie algebra that are “non-integrable”: they aren’tthe derivatives of Lie group representations. For general representations ofthe Heisenberg Lie algebra, i.e. the Heisenberg commutator relations, there arecounter-examples to the analog of the Stone von-Neumann theorem. It is onlyfor integrable representations that the theorem holds and one has a unique sortof irreducible representation.


For a lot more detail about the mathematics of the Heisenberg group, its Liealgebra and the Schrodinger representation, see [7], [38], [20] and [73]. An ex-cellent historical overview of the Stone-von Neumann theorem [57] by JonathanRosenberg is well worth reading.

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Chapter 12

The Poisson Bracket andSymplectic Geometry

We have seen that the quantum theory of a free particle corresponds to the con-struction of a representation of the Heisenberg Lie algebra in terms of operatorsQ and P . One would like to use this to produce quantum systems with a similarrelation to more non-trivial classical mechanical systems than the free particle.During the earliest days of quantum mechanics it was recognized by Dirac thatthe commutation relations of the Q and P operators somehow correspondedto the Poisson bracket relations between the position and momentum coordi-nates on phase space in the Hamiltonian formalism for classical mechanics. Inthis chapter we’ll give an outline of the topic of Hamiltonian mechanics andthe Poisson bracket, including an introduction to the symplectic geometry thatcharacterizes phase space.

The Heisenberg Lie algebra h2d+1 is usually thought of as quintessentiallyquantum in nature, but it is already present in classical mechanics, as the Liealgebra of degree zero and one polynomials on phase space, with Lie bracketthe Poisson bracket. The full Lie algebra of all functions on phase space (withLie bracket the Poisson bracket) is infinite dimensional, so not the sort of finitedimensional Lie algebra given by matrices that we have studied so far (although,historically, it is this kind of infinite dimensional Lie algebra that motivated thediscovery of the theory of Lie groups and Lie algebras by Sophus Lie during the1870s). In chapter 14 we will see that degree two polynomials on phase spacealso provide an important finite-dimensional Lie algebra.

12.1 Classical mechanics and the Poisson bracket

In classical mechanics in the Hamiltonian formalism, the space M = R2d

that one gets by putting together positions and the corresponding momentais known as “phase space”. Points in phase space can be thought of as uniquelyparametrizing possible initial conditions for classical trajectories, so another in-

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terpretation of phase space is that it is the space that uniquely parametrizessolutions of the equations of motion of a given classical mechanical system. Thebasic axioms of Hamiltonian mechanics can be stated in a way that parallelsthe ones for quantum mechanics.

Axiom (States). The state of a classical mechanical system is given by a pointin the phase space M = R2d, with coordinates qj , pj, for j = 1, . . . , d.

Axiom (Observables). The observables of a classical mechanical system are thefunctions on phase space.

Axiom (Dyamics). There is a distinguished observable, the Hamiltonian func-tion h, and states evolve according to Hamilton’s equations

qj =∂h

∂pj

pj = − ∂h∂qj

Specializing to the case d = 1, for any observable function f , Hamilton’sequations imply

df

dt=∂f

∂q

dq

dt+∂f

∂p

dp

dt=∂f

∂q

∂h

∂p− ∂f

∂p

∂h

∂q

We can define

Definition (Poisson bracket). There is a bilinear operation on functions on thephase space M = R2 (with coordinates (q, p)) called the Poisson bracket, givenby

(f1, f2)→ f1, f2 =∂f1

∂q

∂f2

∂p− ∂f1

∂p

∂f2

∂q

An observable f then evolves in time according to

df

dt= f, h

This relation is equivalent to Hamilton’s equations since it implies them bytaking f = q and f = p

q = q, h =∂h

∂p

p = p, h = −∂h∂q

For a non-relativistic free particle, h = p2

2m and these equations become

q =p

m, p = 0

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which just says that the momentum is the mass times the velocity, and is con-served. For a particle subject to a potential V (q) one has

h =p2

2m+ V (q)

and the trajectories are the solutions to

q =p

m, p = −∂V

∂q

which adds Newton’s second law

F = −∂V∂q

= ma = mq

to the definition of momentum in terms of velocity.One can easily check that the Poisson bracket has the properties

• Antisymmetryf1, f2 = −f2, f1

• Jacobi identity

f1, f2, f3+ f3, f1, f2+ f2, f3, f1 = 0

These two properties, together with the bilinearity, show that the Poissonbracket fits the definition of a Lie bracket, making the space of functions onphase space into an infinite dimensional Lie algebra. This Lie algebra is respon-sible for much of the structure of the subject of Hamiltonian mechanics, and itwas historically the first sort of Lie algebra to be studied.

The conservation laws of classical mechanics are best understood using thisLie algebra. From the fundamental dynamical equation

df

dt= f, h

we see that

f, h = 0 =⇒ df

dt= 0

and in this case the function f is called a “conserved quantity”, since it doesnot change under time evolution. Note that if we have two functions f1 and f2

on phase space such that

f1, h = 0, f2, h = 0

then using the Jacobi identity we have

f1, f2, h = −h, f1, f2 − f2, h, f1 = 0

This shows that if f1 and f2 are conserved quantities, so is f1, f2, so functionsf such that f, h = 0 make up a Lie subalgebra. It is this Lie subalgebrathat corresponds to “symmetries” of the physics, commuting with the timetranslation determined by the dynamical law given by h.

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12.2 The Poisson bracket and the HeisenbergLie algebra

A third fundamental property of the Poisson bracket that can easily be checkedis the

• Leibniz rule

f1f2, f = f1, ff2 + f1f2, f, f, f1f2 = f, f1f2 + f1f, f2

This property says that taking Poisson bracket with a function f acts on aproduct of functions in a way that satisfies the Leibniz rule for what happenswhen you take the derivative of a product. Unlike antisymmetry and the Ja-cobi identity, which reflect the Lie algebra structure on functions, the Leibnizproperty describes the relation of the Lie algebra structure to multiplication offunctions. At least for polynomial functions, it allows one to inductively reducethe calculation of Poisson brackets to the special case of Poisson brackets of thecoordinate functions q and p, for instance:

q2, qp = qq2, p+ q2, qp = q2q, p+ qq, pq = 2q2q, p = 2q2

The Poisson bracket is thus determined by its values on linear functions. Wewill define

Definition. Ω(·, ·) is the restriction of the Poisson bracket to M∗, the linearfunctions on M . Taking as basis vectors of M∗ the coordinate functions q andp, Ω is given on basis vectors by

Ω(q, q) = Ω(p, p) = 0, Ω(q, p) = −Ω(p, q) = 1

A general element of M∗ will be a linear combination cqq + cpp for someconstants cq, cp. For general pairs of elements in M∗, Ω will be given by

Ω(cqq + cpp, c′qq + c′pp) = cqc

′p − cpc′q (12.1)

We will often write elements of M∗ as the column vector of their coefficientscq, cp, identifying

cqq + cpp↔(cqcp

)Then one has

Ω(

(cqcp

),

(c′qc′p

)) = cqc

′p − cpc′q

Taking together linear functions on M and the constant function, one getsa three dimensional space with basis elements q, p, 1, and this space is closedunder Poisson bracket. This space is thus a Lie algebra, and is isomorphic tothe Heisenberg Lie algebra h3 (see section 11.2), with the isomorphism given onbasis elements by

X ↔ q, Y ↔ p, Z ↔ 1

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This isomorphism preserves the Lie bracket relations since

[X,Y ] = Z ↔ q, p = 1

It is convenient to choose its own notation for the dual phase space, so wewill often write M∗ =M. The three dimensional space we have identified withthe Heisenberg Lie algebra is then

M⊕R

We will denote elements of this space either by functions cqq + cpp+ c, or as

(

(cqcp

), c)

In this second notation, the Lie bracket is

[(

(cqcp

), c), (

(c′qc′p

), c)] = (

(00

),Ω(

(cqcp

),

(c′qc′p

)))

Notice that the non-trivial part of the Lie bracket structure is determined by Ω.In higher dimensions, coordinate functions q1, · · · , qd, p1, · · · , pd on M pro-

vide a basis for the dual space M consisting of the linear coefficient functionsof vectors in M . Taking as an additional basis element the constant function1, we have a 2d + 1 dimensional space with basis q1, · · · , qd, p1, · · · , pd, 1. ThePoisson bracket relations

qj , qk = pj , pk = 0, qj , pk = δjk

turn this space into a Lie algebra, isomorphic to the Heisenberg Lie algebrah2d+1. On general functions, the Poisson bracket will be given by the obviousgeneralization of the d = 1 case

f1, f2 =

d∑j=1

(∂f1

∂qj

∂f2

∂pj− ∂f1

∂pj

∂f2

∂qj) (12.2)

Elements of h2d+1 are functions on M = R2d of the form

cq1q1 + · · ·+ cqdqd + cp1p1 + · · ·+ cpdpd + c = cq · q + cp · p + c

(using the notation cq = (cq1 , . . . , cqd), cp = (cp1 , . . . , cpd)). We will oftendenote these by

(

(cqcp

), c)

This Lie bracket on h2d+1 is given by

[(

(cqcp

), c), (

(c′qc′p

), c)] = (

(00

),Ω(

(cqcp

),

(c′qc′p

)))

which depends just on the antisymmetric bilinear form

Ω(

(cqcp

),

(c′qc′p

)) = cq · c′p − cp · c′q (12.3)

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Digression. We have been careful here to keep track of the difference betweenphase space M = R2d and its dual M = M∗, since it is M⊕R that is giventhe structure of a Lie algebra (in this case h2d+1) by the Poisson bracket, and itis this Lie algebra we want to use in chapter 15 when we define a quantizationof the classical system. Taking duals, we find an isomorphism

M ⊕R↔ h∗2d+1

It is a general phenomenon that one can define a version of the Poisson bracketon functions on g∗, the dual of any Lie algebra g. This is because the Leibnizproperty ensures that the Poisson bracket only depends on Ω, its restriction tolinear functions, and linear functions on g∗ are just elements of g. So one candefine a Poisson bracket on functions on g∗ by first defining

Ω(X,X ′) = [X,X ′] (12.4)

for X,X ′ ∈ g = (g∗)∗, and then extending this to all functions on g∗ by theLeibniz property.

12.3 Symplectic geometry

We saw in chapter 4 that given a basis ej of a vector space V , a dual basis e∗jof V ∗ is given by taking e∗j = vj , where vj are the coordinate functions. If oneinstead is initially given the coordinate functions vj , one can construct a dualbasis of V = (V ∗)∗ by taking as basis vectors the first order linear differentialoperators given by differentiation with respect to the vj , in other words bytaking

ej =∂

∂vj

Elements of V are then identified with linear combinations of these operators.In effect, one is identifying vectors v with the directional derivative along thevector

v↔ v ·∇

We also saw in chapter 4 that an inner product 〈·, ·〉 on V provides anisomorphism of V and V ∗ by

v ∈ V ↔ lv(·) = 〈v, ·〉 ∈ V ∗ (12.5)

Such an inner product is the fundamental structure in Euclidean geometry,giving a notion of length of a vector and angle between two vectors, as wella group, the orthogonal group of linear transformations preserving the innerproduct. It is a symmetric, non-degenerate bilinear form on V .

A phase space M does not usually come with a choice of inner product. Wehave seen that the Poisson bracket gives us not a symmetric bilinear form, butan antisymmetric bilinear form Ω, defined on the dual spaceM. We will definean analog of an inner product, with symmetry replaced by antisymmetry:

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Definition (Symplectic form). A symplectic form ω on a vector space V is abilinear map

ω : V × V → R

such that

• ω is antisymmetric: ω(v, v′) = −ω(v, v′)

• ω is nondegenerate: if v 6= 0, then ω(v, ·) ∈ V ∗ is non-zero.

A vector space V with a symplectic form ω is called a symplectic vectorspace. The analog of Euclidean geometry, replacing the inner product by asymplectic form, is called symplectic geometry. In this sort of geometry, thereis no notion of length (since antisymmetry implies ω(v, v) = 0). There is ananalog of the orthogonal group, called the symplectic group, which consists oflinear transformations preserving ω, a group we will study in detail in chapter14.

Just as an inner product gives an identification of V and V ∗, a symplecticform can be used in a similar way, giving an identification of M and M. Usingthe symplectic form Ω onM, we can define an isomorphism by identifying basisvectors by

qj ∈M↔ Ω(·, qj) =− Ω(qj , ·) = − ∂

∂pj∈M

pj ∈M↔ Ω(·, pj) =− Ω(pj , ·) =∂

∂qj∈M

and in general

u ∈M↔ Ω(·, u) (12.6)

Note that unlike the inner product case, a choice of convention of minus signmust be made and is done here.

Recalling the discussion of bilinear forms from section 9.5, a bilinear form ona vector space V can be identified with an element of V ∗⊗V ∗. Taking V = M∗

we have V ∗ = (M∗)∗ = M , and the bilinear form Ω on M∗ is an element ofM ⊗M given by

Ω =

d∑j=1

(∂

∂qj⊗ ∂

∂pj− ∂

∂pj⊗ ∂

∂qj)

Under the identification 12.6 of M and M∗, Ω ∈M ⊗M corresponds to

ω =

d∑j=1

(qj ⊗ pj − pj ⊗ qj) ∈M∗ ⊗M∗ (12.7)

Another version of the identification of M and M is then given by

v ∈M → ω(v, ·) ∈M

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In the case of Euclidean geometry, one can show by Gram-Schmidt orthog-onalization that a basis ej can always be found that puts the inner product(which is a symmetric element of V ∗ ⊗ V ∗) in the standard form

n∑j=1

vj ⊗ vj

in terms of basis elements of V ∗, the coordinate functions vj . There is ananalogous theorem in symplectic geometry (for a proof, see for instance Propo-sition 1.1 of [7]), which says that a basis of a symplectic vector space V canalways be found so that the dual basis coordinate functions come in pairs qj , pj ,with the symplectic form ω the same one we have found based on the Poissonbracket, that given by equation 12.7. Note that one difference between Eu-clidean and symplectic geometry is that a symplectic vector space will alwaysbe even-dimensional.

Digression. For those familiar with differential manifolds, vector fields anddifferential forms, the notion of a symplectic vector space can be extended to:

Definition (Symplectic manifold). A symplectic manifold M is a manifold witha differential two-form ω(·, ·) (called a symplectic two-form) satisfying the con-ditions

• ω is non-degenerate, i.e. for a nowhere zero vector field X, ω(X, ·) is anowhere zero one-form.

• dω = 0, in which case ω is said to be closed.

The cotangent bundle T ∗N of a manifold N (i.e. the space of pairs of apoint on N together with a linear function on the tangent space at that point)provides one class of symplectic manifolds, generalizing the linear case N = Rd,and corresponding physically to a particle moving on N . A simple example thatis neither linear nor a cotangent bundle is the sphere M = S2, with ω the areatwo-form. The Darboux theorem says that, by an appropriate choice of localcoordinates, symplectic two-forms ω can always be put in the form

ω =

d∑j=1

dqj ∧ dpj

Unlike the linear case though, there will in general be no global choice of coor-dinates for which this true. Later on, our discussion of quantization will relycrucially on having a linear structure on phase space, so will not apply to generalsymplectic manifolds.

Note that there is no assumption here that M has a metric (i.e. it maynot be a Riemannian manifold). A symplectic two-form ω is a structure on amanifold analogous to a metric but with opposite symmetry properties. Whereasa metric is a symmetric non-degenerate bilinear form on the tangent space at

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each point, a symplectic form is an antisymmetric non-degenerate bilinear formon the tangent space.

Returning to vector spaces V , one can generalize the notion of a symplecticstructure by dropping the non-degeneracy condition on ω. The Leibniz propertycan still be used to extend this to a Poisson bracket on functions on V , whichis then called a Poisson structure on V . In particular, one can do this forV = g∗ for any Lie algebra g, using the choice of ω discussed at the end ofsection 12.2. So g∗ always has a Poisson structure, and on subspaces where ωis non-degenerate, a symplectic structure.

For instance, forV = h∗2d+1 = M ⊕R

one has a Poisson bracket on functions on V , but only on the subspace M isthis Poisson bracket non-degenerate on linear functions, making M a symplecticvector space. For another example one can take g = so(3), which is just R3,with antisymmetric bilinear form ω given by the vector cross-product. In thiscase it turns out that if one considers spheres of fixed radius in R3, ω providesa non-degenerate symplectic form on their tangent spaces and such spheres aresymplectic manifolds with symplectic two-form their area two-form.


Some good sources for discussions of symplectic geometry and the geometricalformulation of Hamiltonian mechanics are [2], [7] and [10].

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Chapter 13

Hamiltonian Vector Fieldsand the Moment Map

A basic feature of Hamiltonian mechanics is that, for any function f on phasespace M , one gets parametrized curves in phase space that solve Hamilton’sequations, and the tangent vectors of these parametrized curves provide a vectorfield on phase space. Such vector fields are called Hamiltonian vector fields, andthere is a distinguished choice of f , the Hamiltonian function h, which givesthe velocity vector fields for time evolution trajectories in phase space. Whena Lie group G acts on phase space, the infinitesimal action of the group alsoassociates to each element of g a vector field on phase space. When these areHamiltonian vector fields, we get (up to a constant) for each element L of g afunction µL that corresponds to that vector field. This map from g to functionson M is called the moment map, and it characterizes the action of G on phasespace. Quantization of such functions will give us the quantum observablescorresponding to the group action. For the case of the action of G = R3 onM = R6 by translations, the moment map gives the momentum, for the actionof G = SO(3) by rotations, the angular momentum.

13.1 Vector fields and the exponential map

One can think of a vector field on M = R2 as a choice of a two-dimensionalvector at each point in R2, so given by a vector-valued function

F(q, p) =

(Fq(q, p)Fp(q, p)

)Such a vector field determines a system of differential equations

dq

dt= Fq,

dp

dt= Fp

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Once we specify initial conditions

q(0) = q0, p(0) = p0

if Fq and Fp are differentiable functions these differential equations have a uniquesolution q(t), p(t), at least for some neighborhood of t = 0 (from the existenceand uniqueness theorem that can be found for instance in [36]). These solutionsq(t), p(t) describe trajectories in R2 with velocity vector F(q(t), p(t)) and suchtrajectories can be used to define the “flow” of the vector field: for each t this isthe map that takes the initial point (q(0), p(0)) ∈ R2 to the point (q(t), p(t)) ∈R2.

Another equivalent way to define vector fields on R2 is to use instead thedirectional derivative along the vector field, identifying

F(q, p)↔ F(q, p) ·∇ = Fq(q, p)∂

∂q+ Fp(q, p)

∂

∂p

The case of F a constant vector is just our previous identification of the vectorspace M with linear combinations of ∂

∂q and ∂∂p .

An advantage of defining vector fields in this way as first order linear differ-ential operators is that it shows that vector fields form a Lie algebra, where onetakes as Lie bracket the commutator of the differential operators. The commu-tator of two first-order differential operators is another first-order differentialoperator since higher order derivatives will cancel, using equality of mixed par-tial derivatives. In addition, such a commutator will satisfy the Jacobi identity.Not only do we get a Lie algebra, but also a representation of the Lie algebra,on functions on R2.

Given this Lie algebra of vector fields, one can ask what the correspondinggroup might be. This is not a finite dimensional matrix Lie algebra, so expo-nentiation of matrices will not give the group. One can however use the flow ofthe vector field X to define an analog of the exponential of a parameter t timesX:

Definition (Exponential map of a vector field). An exponential map for thevector field X on M is a map

exp(tX) : M →M

that depends on a parameter t ∈ R, is the identity map at t = 0, and satisfies

d

dtexp(tX)(m) = X(exp(tX)(m))

for m ∈M .

The flow of the vector field X is the map

ΦX : (t,m) ∈ R×M → ΦX(t,m) = exp(tX)(m)

154

If the vector field X is differentiable, exp(tX) will be a well-defined map forsome neighborhood of t = 0, and satisfy

exp(t1X)exp(t2X) = exp((t1 + t2)X)

thus providing a one-parameter group with derivative X at the identity.

Digression. For any manifold M , one has a Lie algebra of differentiable vectorfields with an associated Lie bracket. One also has an infinite dimensional Liegroup, the group of invertible maps from M to itself, such that the maps andtheir inverses are both differentiable. This group is called the diffeomorphismgroup of M and written Diff(M). Its Lie algebra is the Lie algebra of vectorfields.

The representation of the Lie algebra of vector fields on functions is thedifferential of the representation of Diff(M) on functions induced in the usualway from the action of Diff(M) on the space M .

13.2 Hamiltonian vector fields and canonical trans-formations

Our interest is not in general vector fields, but in vector fields corresponding toHamilton’s equations for some Hamiltonian function f , i.e. the case

Fq =∂f

∂p, Fp = −∂f

∂q

We call such vector fields Hamiltonian vector fields, defining:

Definition (Hamiltonian Vector Field). A vector field on M = R2 given by

∂f

∂p

∂

∂q− ∂f

∂q

∂

∂p= −f, ·

for some function f on M = R2 is called a Hamiltonian vector field and will bedenoted by Xf . In higher dimensions, Hamiltonian vector fields will be those ofthe form

Xf =

d∑j=1

(∂f

∂pj

∂

∂qj− ∂f

∂qj

∂

∂pj) = −f, · (13.1)

for some function f on M = R2d.

The simplest non-zero Hamiltonian vector fields are those for f a linearfunction. For cq, cp constants, if

f = cqq + cpp

then

Xf = cp∂

∂q− cq

∂

∂p

155

and the map

f → Xf

is just the isomorphism of M and M of equation 12.6.

For example, taking f = p, we have Xp = ∂∂q . The exponential map for this

vector field is

exp(tXp)(q0, p0) = (q0 + t, p0) (13.2)

Similarly, for f = q one has Xq = − ∂∂p and

exp(tXq)(q0, p0) = (q0, p0 − t) (13.3)

For quadratic functions f one gets vector fields Xf with components linear inthe coordinates. An important example, which describes a harmonic oscillatorand will be treated in much more detail in chapter 20, is the case

f =1

2(q2 + p2)

for which

Xf = p∂

∂q− q ∂

∂p

The trajectories satisfy

dq

dt= p,

dp

dt= −q

and are given by

q(t) = q(0) cos t+ p(0) sin t, p(t) = p(0) cos t− q(0) sin t

The exponential map is given by clockwise rotation through an angle t

exp(tXf )(q0, p0) = (q0 cos t+ p0 sin t, p0 cos t− q0 sin t)

The vector field Xf and the trajectories in the q − p plane look like this

156

The relation of vector fields to the Poisson bracket is given by

f1, f2 = Xf2(f1) = −Xf1(f2)

so one has in particular

q, f =∂f

∂p, p, f = −∂f

∂q

The definition we have given here of Xf (equation 13.1) carries with it achoice of how to deal with a confusing sign issue. Recall that vector fields on Mform a Lie algebra with Lie bracket the commutator of differential operators. Anatural question is that of how this Lie algebra is related to the Lie algebra offunctions on M (with Lie bracket the Poisson bracket).

The Jacobi identity implies

f, f1, f2 = f, f1, f2+ f2, f, f1 = f, f1, f2 − f, f2, f1

soXf1,f2 = Xf2

Xf1−Xf1

Xf2= −[Xf1

, Xf2] (13.4)

This shows that the map f → Xf of equation 13.1 that we defined betweenthese Lie algebras is not quite a Lie algebra homomorphism because of the -

157

sign in equation 13.4 (it is called a Lie algebra “antihomomorphism”). The mapthat is a Lie algebra homomorphism is

f → −Xf

To keep track of the minus sign here, one needs to keep straight the differencebetween

• The functions on phase space M are a Lie algebra, with a function f actingon the function space by the adjoint action

ad(f)(·) = f, ·

and

• The functions f provide vector fields Xf acting on functions on M , where

Xf (·) = ·, f = −f, ·

The first of these is what will be most relevant to us later when we quantizefunctions on M to get operators, preserving the Lie algebra structure. Thesecond is what one naturally gets from the geometrical action of a Lie group Gon the phase space M (see section 13.3). As a simple example, the function psatisfies

p, F (q) = −∂F∂q

so

p, · = −∂(·)∂q

is the infinitesimal action of translation in q on functions, whereas ∂∂q is the

vector field on M corresponding to infinitesimal translation in the position co-ordinate.

It is important to note that the Lie algebra homomorphism

f → Xf

is not an isomorphism, for two reasons:

• It is not injective (one-to-one), since functions f and f+C for any constantC correspond to the same Xf .

• It is not surjective since not all vector fields are Hamiltonian vector fields(i.e. of the form Xf for some f). One property that a vector field mustsatisfy in order to possibly be a Hamiltonian vector field is

Xg1, g2 = Xg1, g2+ g1, Xg2 (13.5)

for g1 and g2 on M . This is just the Jacobi identity for f, g1, g2, whenX = Xf .

158

Digression. For a general symplectic manifold M , the symplectic two-form ωgives one an analog of Hamilton’s equations. This is the following equality ofone-forms, relating a Hamiltonian function h and a vector field Xh determiningtime evolution of trajectories in M

iXhω = ω(Xh, ·) = dh

(here iX is interior product with the vector field X). The Poisson bracket inthis context can be defined as

f1, f2 = ω(Xf1, Xf2

)

Recall that a symplectic two-form is defined to be closed, satisfying the equa-tion dω = 0, which is then a condition on a three-form dω. Standard differentialform computations allow one to express dω(Xf1

, Xf2, Xf3

) in terms of Poissonbrackets of functions f1, f2, f3, and one finds that dω = 0 is the Jacobi identityfor the Poisson bracket.

The theory of “prequantization” (see [39], [31]) enlarges the phase space Mto a U(1) bundle with connection, where the curvature of the connection is thesymplectic form ω. Then the problem of lack of injectivity of the Lie algebrahomomorphism

f → −Xf

is resolved by instead using the map

f → −∇Xf + if (13.6)

where ∇X is the covariant derivative with respect to the connection. For detailsof this, see [39] or [31].

In our treatment of functions on phase space M , we have always been tak-ing such functions to be time-independent. One can abstractly interpret Mas the space of trajectories of a classical mechanical system, with coordinatesq, p having the interpretation of initial conditions q(0), p(0) of the trajectories.The exponential maps exp(tXh) give an action on the space of trajectories forHamiltonian function h, taking the trajectory with initial conditions given bym ∈M to the time-translated one with initial conditions given by exp(tXh)(m).One should really interpret the formula for Hamilton’s equations

df

dt= f, h

as meaningd

dtf(exp(tXf )(m))|t=0 = f(m), h(m)

for each m ∈M .Given a Hamiltonian vector field Xf , the maps

exp(tXf ) : M →M

159

are known to physicists as “canonical transformations”, and to mathematiciansas “symplectomorphisms”. We will not try and work out in any more detailhow the exponential map behaves in general. In chapter 14 we will see whathappens for f an order-two homogeneous polynomial in the qj , pk. In that casethe vector field Xf will take linear functions on M to linear functions, thusacting onM, in which case its behavior can be studied using the matrix for thelinear transformation with respect to the basis elements qj , pk.

Digression. The exponential map exp(tX) can be defined as above on a generalmanifold. For a symplectic manifold M , Hamiltonian vector fields Xf will havethe property that

exp(tXf )∗ω = ω

This is because

LXfω = (diXf + iXf d)ω = diXfω = dω(Xf , ·) = ddf = 0

13.3 Group actions on M and the moment map

Whenever one has an action of a Lie group G on a space M , an infinitesimalversion of this action is the map

L ∈ g→ XL

from g to vector fields on M that takes L to the vector field XL which acts onfunctions on M by

XLF (m) =d

dtF (etL ·m)|t=0

This map however is not a homomorphism (for the Lie bracket on vector fieldsthe commutator of derivatives), but an anti-homomorphism. To see why thisis, recall that when a group G acts on a space, we get a representation π onfunctions F on the space by

π(g)F (m) = F (g−1 ·m)

The derivative of this representation will be the Lie algebra representation

π′(L)F (m) =d

dtF (e−tL ·m)|t=0 = −XLF (m)

so we see that it is the map

L→ π′(L) = −XL

that will be a homomorphism.Given an action of a groupG onM , if for L ∈ g the vector fieldXL is a Hamil-

tonian vector field, we would like to find the function f such that XL = Xf .This will allow us to study infinitesimal group actions on M by using functions

160

on M and the Poisson bracket. Quantization will turn functions on M intooperators, turning the function f into an operator which will be the observablecorresponding to the infinitesimal group action by a Lie algebra element L.

Only for certain actions of G on M will the XL be Hamiltonian vector fields.A necessary condition is that XL satisfy equation 13.5

XLg1, g2 = XLg1, g2+ g1, XLg2

which is a property of Hamiltonian vector fields. Even when a function f existssuch that Xf = XL, it is only unique up to a constant, since f and f + C willgive the same vector field. We would like to choose these constants in such away that the map

L→ f

is a Lie algebra homomorphism from g to the Lie algebra of functions on M .When this is possible, the G-action is said to be a Hamiltonian G-action. Whenit is not possible, the G-symmetry of the classical phase space is said to havean “anomaly”.

We can define

Definition (Moment map). Given a Hamiltonian action of a Lie group G onM , a Lie algebra homomorphism

L→ µL

from g to functions on M is said to be a moment map if

XL = XµL

Equivalently, for functions F on M , µL satisfies

XµLF = −µL, F = XLF

Note what we are calling a moment map is sometimes called a “co-momentmap”, with the term “moment map” referring to a repackaged form of the sameinformation, the map

µ : M → g∗

where(µ(m))(L) = µL(m)

For G = H3, its action on phase space will be defined to be such thatthe moment map is just our identification of the Heisenberg Lie algebra withfunctions on phase space:

µL = L = cqq + cpp+ c ∈ h3

Here

XµL = −cq∂

∂p+ cp

∂

∂q

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The action of H3 on M for which this is the moment map will have XL = XµL

and one can check that this action is

(

(xy

), z) · (q0, p0) = (q0 + y, p0 − x) (13.7)

See equations 13.2 and 13.3, which show that this action corresponds to theexponential map for vector fields associated to Lie algebra basis elements q andp. For central elements of the Lie algebra (the constant functions), the vectorfield is zero, and the exponential map takes these to the identity map on M .

For d = 3 the translation group G = R3 is a subgroup of H7 of elements ofthe form

(

(0a

), 0)

since it acts on the phase space by translation in the position coordinates

(q0,p0)→ (q0 + a,p0)

Taking a to be the corresponding element in the Lie algebra of G = R3, thevector field on M is

Xa = a ·∇

and the moment map is given by

µa(q0,p0) = a · p0

(a function on M for each element a of the Lie algebra) or

µ(q0,p0)(a) = a · p0

(an element of the dual of the Lie algebra for each point m = (q0,p0) in phasespace).

For another example, consider the action of the group G = SO(3) of rota-tions on phase space M = R6, which gives a map from so(3) to vector fields onR6, taking for example

l1 ∈ so(3)→ Xl1 = −q3∂

∂q2+ q2

∂

∂q3− p3

∂

∂p2+ p2

∂

∂p3

(this is the vector field for an infinitesimal clockwise rotation in the q2− q3 andp2−p3 planes, in the opposite direction to the case of the vector field X 1

2 (q2+p2)

in the q − p plane of section 13.2). The moment map here gives the usualexpression for the 1-component of the angular momentum

µl1 = q2p3 − q3p2

since one can check from equation 13.1 that Xl1 = Xµl1. On basis elements of

so(3) one hasµlj (q0,p0) = (q0 × p0)j

162

Formulated as a map from M to so(3)∗, the moment map is

µ(q0,p0)(l) = (q0 × p0) · l

where l ∈ so(3).

Digression. For the case of M a general symplectic manifold, one can still de-fine the moment map, whenever one has a Lie group G acting on M , preservingthe symplectic form ω. The infinitesimal condition for such a G action is that

LXω = 0

where LX is the Lie derivative along the vector field X. Using the formula

LX = (d+ iX)2 = diX + iXd

for the Lie derivative acting on differential forms (iX is contraction with thevector field X), one has

(diX + iXd)ω = 0

and since dω = 0 we have

diXω = 0

When M is simply-connected, one-forms iXω whose differential is 0 (called“closed”) will be the differentials of a function (and called “exact”). So therewill be a function µ such that

iXω(·, ·) = ω(X, ·) = dµ(·)

although such a µ is only unique up to a constant.Given an element L ∈ g, the G action on M gives a vector field XL. When

we can choose the constants appropriately and find functions µL satisfying

iXLω(·, ·) = dµL(·)

such that the map

L→ µL

taking Lie algebra elements to functions on M (with Lie bracket the Poissonbracket) is a Lie algebra homomorphism, then this is called the moment map.One can equivalently work with

µ : M → g∗

by defining

(µ(m))(L) = µL(m)

An important class of symplectic manifolds M with an action of a Lie groupG, preserving the symplectic form, are the co-adjoint orbits Ol. These are the

163

manifolds one gets by acting on a chosen l ∈ g∗ by the co-adjoint action Ad∗,meaning the action of g on g∗ satisfying

(Ad∗(g) · l)(X) = l(Ad(g)X)

where X ∈ g, and Ad(g) is the usual adjoint action on g. For these cases, themoment map

µ : Ol → g∗

is just the inclusion map. Two simple examples are

• For g = h3, fixing a choice of c elements of g are linear functions onM = R2, so l ∈ g∗ is a point in M (evaluation of the function at thatpoint). The co-adjoint action is the action of H3 on M of equation 13.7.

• For g = so(3) the non-zero co-adjoint orbits are spheres, with radius thelength of l.


For a general discussion of vector fields on Rn, see [36]. See [2], [7] and [10] formore on Hamiltonian vector fields and the moment map. For more on the dualsof Lie algebras and co-adjoint orbits, see [8] and [38].

164

Chapter 14

Quadratic Polynomials andthe Symplectic Group

The Poisson bracket on functions on phase space M = R2d is determined by anantisymmetric bilinear form Ω on the dual phase spaceM = M∗. Just as thereis a group of linear transformations (the orthogonal group) leaving invariant aninner product, which is a symmetric bilinear form, here there is a group leavinginvariant Ω, the symplectic group Sp(2d,R). The Lie algebra sp(2d,R) of thisgroup can be identified with the Lie algebra of order-two polynomials onM , withLie bracket the Poisson bracket. Elements of Sp(2d,R) act on M, preservingΩ, and so provide a map of the Heisenberg Lie algebra h2d+1 =M⊕R to itself,preserving the Lie bracket (which is defined using Ω). The symplectic groupthus acts by automorphisms on h2d+1. This action has an infinitesimal version,reflected in the non-trivial Poisson brackets between order two and order onepolynomials on M .

The identification of elements L of the Lie algebra sp(2d,R) with order-twopolynomials µL on M is just the moment map for the action of the symplecticgroup Sp(2d,R) on phase space. The corresponding vector fields will have linearcoefficient functions. Quantization of these polynomial functions will providequantum observables corresponding to any Lie subgroup G ⊂ Sp(2d,R) (anyLie group G that acts on M preserving the symplectic form). Such group actionswill give rise to quantum observables, but they may or may not be “symmetries”,with the term “symmetry” usually meaning that one has µL, h = 0 for h theHamiltonian function.

14.1 The symplectic group

Recall that the orthogonal group can be defined as the group of linear transfor-mations preserving an inner product, which is a symmetric bilinear form. Wenow want to study the analog of the orthogonal group one gets by replacing

165

the inner product by the antisymmetric bilinear form Ω that determines thesymplectic geometry of phase space. We will define

Definition (Symplectic Group). The symplectic group Sp(2d,R) is the sub-group of linear transformations g of M∗ = R2d that satisfy

Ω(gv1, gv2) = Ω(v1, v2)

for v1, v2 ∈M∗

While this definition uses the dual phase space M∗ and Ω, it would have beenequivalent to have made the definition using M and ω, since these transforma-tions preserve the isomorphism between M and M∗ given by Ω (see equation12.6). For an action on M∗

u ∈M∗ → gu ∈M∗

the action on elements of M , which correspond to linear functions Ω(u, ·) onM∗ is given by

Ω(u, ·) ∈M → g · Ω(u, ·) = Ω(u, g−1(·)) = Ω(gu, ·) ∈M

Here the first equality uses the definition of the dual representation (see 4.2)and the second uses the invariance of Ω.

14.1.1 The symplectic group for d = 1

In order to study symplectic groups as groups of matrices, we’ll begin with thecase d = 1 and the group Sp(2,R). We can write Ω as

Ω(

(cqcp

),

(c′qc′p

)) = cqc

′p − cpc′q =

(cq cp

)( 0 1−1 0

)(c′qc′p

)(14.1)

Note that for the analogous case of the inner product, the same formula holdswith the elementary antisymmetric matrix replaced by the identity matrix.

A linear transformation g of M∗ will be given by(cqcp

)→(α βγ δ

)(cqcp

)The condition for Ω to be invariant under such a transformation is(

α βγ δ

)T (0 1−1 0

)(α βγ δ

)=

(0 1−1 0

)(14.2)

or (0 αδ − βγ

−αδ + βγ 0

)=

(0 1−1 0

)so

det

(α βγ δ

)= αδ − βγ = 1

166

This says that we can have any linear transformation with unit determinant.In other words, we find that Sp(2,R) = SL(2,R). We will see later that thisisomorphism with a special linear group only happens for d = 1.

For the analog of equation 14.2 in the inner product case, replace the el-ementary antisymmetric matrices by unit matrices, giving the condition thatdefines orthogonal matrices, gT g = 1.

Now turning to the Lie algebra, for group elements g ∈ GL(2,R) near theidentity, one can write g in the form g = etL where L is in the Lie algebragl(2,R). The condition that g acts on M preserving Ω implies that (differenti-ating 14.2)

d

dt((etL)T

(0 1−1 0

)etL) = (etL)T (LT

(0 1−1 0

)+

(0 1−1 0

)L)etL = 0

Setting t = 0, the condtion on L is

LT(

0 1−1 0

)+

(0 1−1 0

)L = 0 (14.3)

This requires that L must be of the form

L =

(a bc −a

)which is what one expects: L is in the Lie algebra sl(2,R) of 2 by 2 real matriceswith zero trace. The analog in the inner product case is just the conditiondefining elements of the Lie algebra of the orthogonal group, LT + L = 0.

The homogeneous degree two polynomials in p and q form a three-dimensionalsub-Lie algebra of the Lie algebra of functions on phase space, since the non-zero

Poisson bracket relations between them on a basis q2

2 ,p2

2 , qp are

q2

2,p2

2 = qp qp, p2 = 2p2 qp, q2 = −2q2

This Lie algebra is isomorphic to sl(2,R), with an explicit isomorphism givenby identifying basis elements as follows:

q2

2↔ E =

(0 10 0

)− p2

2↔ F =

(0 01 0

)− qp↔ G =

(1 00 −1

)(14.4)

The commutation relations amongst these matrices are

[E,F ] = G [G,E] = 2E [G,F ] = −2F

which are the same as the Poisson bracket relations between the correspondingquadratic polynomials.

We thus see that we have an isomorphism between the Lie algebra of degreetwo homogeneous polynomials with the Poisson bracket and the Lie algebra

167

of 2 by 2 trace-zero real matrices with the commutator as Lie bracket. Theisomorphism on general elements of these Lie algebras is

µL = −aqp+bq2

2− cp2

2=

1

2

(q p

)( b −a−a −c

)(qp

)↔ L =

(a bc −a

)(14.5)

We have seen that this is a Lie algebra isomorphism on basis elements, but onecan explicitly check that.

µL, µL′ = µ[L,L′]

The use of the notation µL for these quadratic functions reflects the factthat

L ∈ sl(2,R)→ µL

is a moment map. This is for the SL(2,R) action on phase space M = R2

corresponding to the above SL(2,R) action on M (under the identificationbetween M and M given by Ω). One can check the condition XL = XµL onvector fields on M , but we will not do this here, since for our purposes it is theaction on M that is important.

Two important subgroups of SL(2,R) are

• The subgroup of elements one gets by exponentiating G, which is isomor-phic to the multiplicative group of positive real numbers

etG =

(et 00 e−t

)Here one can explicitly see that this group has elements going off to infinity.

• Exponentiating the Lie algebra element E−F gives rotations of the plane

eθ(E−F ) =

(cos θ sin θ− sin θ cos θ

)Note that the Lie algebra element being exponentiated here is

E − F ↔ 1

2(p2 + q2)

which we will later re-encounter as the Hamiltonian function for the har-monic oscillator.

The group SL(2,R) is non-compact and thus its representation theory isquite unlike the case of SU(2). In particular, all of its non-trivial irreducibleunitary representations are infinite-dimensional, forming an important topic inmathematics, but one that is beyond our scope. We will be studying just onesuch irreducible representation, and it is a representation only of a double-coverof SL(2,R), not SL(2,R) itself.

168

14.1.2 The symplectic group for arbitary d

For general d, the symplectic group Sp(2d,R) is the group of linear transfor-mations g of M that leave Ω (see 12.3) invariant, i.e. satisfy

Ω(g

(cqcp

), g

(c′qc′p

)) = Ω(

(cqcp

),

(c′qc′p

))

where cq, cp are d-dimensional vectors. By essentially the same calculation as inthe d = 1 case, we find the d-dimensional generalization of equation 14.2. Thissays that Sp(2d,R) is the group of real 2d by 2d matrices g satisfying

gT(

0 1−1 0

)g =

(0 1−1 0

)where 0 is the d by d zero matrix, 1 the d by d unit matrix.

Again by a similar argument to the d = 1 case where the Lie algebra sp(2,R)was determined by the condition 14.3, sp(2d,R) is the Lie algebra of 2d by 2dmatrices L satisfying

LT(

0 1−1 0

)+

(0 1−1 0

)L = 0

Such matrices will be those with the block-diagonal form

L =

(A BC −AT

)(14.6)

where A,B,C are d by d real matrices, with B and C symmetric, i.e.

B = BT , C = CT

The generalization of 14.5 is

Theorem 14.1. The Lie algebra sp(2d,R) is isomorphic to the Lie algebra oforder two homogeneous polynomials on M = R2d by the isomorphism (using avector notation for the coefficient functions q1, · · · , qd, p1, · · · , pd)

L↔ µL

where

µL =1

2

(q p

)L

(0 −11 0

)(qp

)=

1

2

(q p

)( B −A−AT −C

)(qp

)=

1

2(q ·Bq− 2q ·Ap− p · Cp) (14.7)

169

We will postpone the proof of this theorem until section 14.2, since it is easierto first study Poisson brackets between order two and order one polynomials,then use this to prove the theorem about Poisson brackets between order twopolynomials.

The Lie algebra sp(2d,R) has a subalgebra gl(d,R) consisting of matricesof the form

L =

(A 00 −AT

)or, in terms of polynomials, polynomials

−q ·Ap = −(ATp) · q

Here A is any real d by d matrix. This shows that one way to get symplectictransformations is to take any linear transformation of the position coordinates,together with the dual linear transformation (see definition 4.2) on momentumcoordinates. In this way, any linear group of symmetries of the position spacebecomes a group of symmetries of phase-space. An example of this is the groupSO(d) of spatial rotations, with Lie algebra so(d) ⊂ gl(d,R), the antisymmetricd by d matrices, for which −AT = A. In the case d = 3, µL gives the standardexpression for the angular momentum as a function of the qj , pk coordinates onphase space. For example, taking L = l1, one has

µl1 = q2p3 − q3p2

the standard expression for angular momentum about the 1-axis.Another important subgroup comes from taking A = 0, B = 1, C = −1,

which gives

µL =1

2(|q|2 + |p|2)

which will be the Hamiltonian function for a d-dimensional harmonic oscillator.Exponentiating, one gets an SO(2) subgroup, one that acts on phase space in away that mixes position and momentum coordinates, so cannot be understoodjust in terms of configuration space.

14.2 The symplectic group and automorphismsof the Heisenberg group

Returning to the d = 1 case, we have found two three-dimensional Lie alge-bras (h3 and sl(2,R)) as subalgebras of the infinite dimensional Lie algebra offunctions on phase space:

• h3, the Lie algebra of linear polynomials on M , with basis 1, q, p.

• sl(2,R), the Lie algebra of order two homogeneous polynomials on M ,with basis q2, p2, qp.

170

Taking all quadratic polynomials, we get a six-dimensional Lie algebra withbasis elements 1, q, p, qp, q2, p2. This is not the direct product of h3 and sl(2,R)since there are nonzero Poisson brackets

qp, q =− q, qp, p = p

p2

2, q =− p, q

2

2, p = q

(14.8)

These relations show that operating on a basis of linear functions on M bytaking the Poisson bracket with something in sl(2,R) (a quadratic function)provides a linear transformation on M∗. In this section we will see that thisis a reflection of the fact that SL(2,R) acts on the Heisenberg group H3 byautomorphisms.

Recall the definition 11.1 of the Heisenberg group H3 as elements

(

(xy

), z) ∈M⊕R

with the group law

(

(xy

), z)(

(x′

y′

), z′) = (

(x+ x′

y + y′

), z + z′ +

1

2Ω(

(xy

),

(x′

y′

)))

Elements g ∈ SL(2,R) act on H3 by

(

(xy

), z)→ φg((

(xy

), z)) = (g

(xy

), z) (14.9)

This is an example of

Definition (Group automorphisms). If an action of elements g of a group Gon a group H

h ∈ H → φg(h) ∈ Hsatisfies

φg(h1)φg(h2) = φg(h1h2)

for all g ∈ G and h1, h2 ∈ H, the group G is said to act on H by automorphisms.Each map φg is an automorphism of H. Note that since φg is an action of G,we have φg1g2

= φg1φg2

.

Here G = SL(2,R), H = H3 and φg given above is an action by automorphismssince

φg(

(xy

), z)φg(

(x′

y′

), z′) =(g

(xy

), z)(g

(x′

y′

), z′)

=(g

(x+ x′

y + y′

), z + z′ +

1

2Ω(g

(xy

), g

(x′

y′

)))

=(g

(x+ x′

y + y′

), z + z′ +

1

2Ω(

(xy

),

(x′

y′

)))

=φg((

(xy

), z)(

(x′

y′

), z′)) (14.10)

171

One can consider the Lie algebra h3 instead of the group H3, and therewill again be an action of SL(2,R) by automorphisms. Denoting elementscqq + cpp+ c of the Lie algebra h3 =M⊕R (see section 12.2) by

(

(cqcp

), c)

the Lie bracket is

[(

(cqcp

), c), (

(c′qc′p

), c′)] = (

(00

), cqc

′p − cpc′q) = (

(00

),Ω(

(cqcp

),

(c′qc′p

)))

This Lie bracket just depends on Ω, so acting on M by g ∈ SL(2,R) will givea map of h3 to itself preserving the Lie bracket. More explicitly, an SL(2,R)group element acts on the Lie algebra h3 by

X = (

(cqcp

), c) ∈ h3 → φg(X) = (g

(cqcp

), c)

These φg are just the same maps φg that give automorphisms of the groupstructure of H3 since the exponential map relating h3 and H3 in these coordi-nates is just the identification

(

(cqcp

), c)↔ (

(xy

), z)

The φg provide an example of

Definition (Lie algebra automorphisms). If an action of elements g of a groupG on a Lie algebra h

X ∈ h→ φg(X) ∈ h

satisfies[φg(X), φg(Y )] = φg([X,Y ])

for all g ∈ G and X,Y ∈ h, the group is said to act on h by automorphisms.The action of φg on h is an automorphism of h and we have the relation φg1g2

=φg1φg2 .

Two examples are

• In the case discussed above, SL(2,R) acts on h3 by automorphisms. Withrespect to the decomposition

h3 =M⊕R

SL(2,R) acts just on M, by linear transformations preserving Ω.

• If h is the Lie algebra of a Lie group H, the adjoint representation (Ad, h)gives an action

X ∈ h→ Ad(h)(X) = hXh−1

172

of H on h by automorphisms φh = Ad(h). For the case of the Heisenberggroup, one can check that the adjoint representation of H3 on h3 leavesinvariant the M component, only acting on the R component. Note thatthis is opposite behavior to the co-adjoint action of H3 on h∗3, which actson M by translations (as in equation 13.7).

The SL(2,R) action on h3 by Lie algebra automorphisms has an infinitesimalversion (i.e. for group elements infinitesimally close to the identity), an action ofthe Lie algebra of SL(2,R) on h3. This is defined for L ∈ sl(2,R) and X ∈ h3

by

L ·X =d

dt(etL ·X)|t=0 (14.11)

Computing this, one finds

L · ((cqcp

), c) = (L

(cqcp

), 0) (14.12)

so L acts on h3 =M⊕R just by matrix multiplication on vectors in M.More generally, one has

Definition 14.1 (Lie algebra derivations). If an action of a Lie algebra g on aLie algebra h

X ∈ h→ Z ·X ∈ h

satisfies[Z ·X,Y ] + [X,Z · Y ] = Z · [X,Y ]

for all Z ∈ g and X,Y ∈ h, the Lie algebra g is said to act on h by derivations.The action of an element Z on h is a derivation of h.

Given an action of a Lie group G on a Lie algebra h by automorphisms,taking the derivative as in 14.11 gives an action of g on h by derivations since

Z·[X,Y ] =d

dt(φetL([X,Y ]))|t=0 =

d

dt([φetLX,φetLY ])|t=0 = [Z·X,Y ]+[X,Z·Y ]

We will often refer to this action of g on h as the infinitesimal version of theaction of G on h.Two examples are

• The case above, where sl(2,R) acts on h3 by derivations.

• The adjoint representation of a Lie algebra h on itself gives an action of hon itself by derivations, with

X ∈ h→ Z ·X = ad(Z)(X) = [Z,X]

The Poisson brackets between degree two and degree one polynomials dis-cussed at the beginning of this section give explicitly the action of sl(2,R) onh3 by derivations. For a general L ∈ sl(2,R) and cqq + cpp+ c ∈ h3 we have

µL, cqq + cpp+ c = c′qq + c′pp,

(c′qc′p

)=

(acq + bcpccq − acp

)= L

(cqcp

)(14.13)

173

(here µL is given by 14.5). We see that this is just the action of sl(2,R) byderivations on h3 of equation 14.12, the infinitesimal version of the action ofSL(2,R) = Sp(2,R) on h3 by automorphisms.

Note that in the larger Lie algebra of all polynomials on M of order two orless, the action of sl(2,R) on h3 by derivations is part of the adjoint action ofthe Lie algebra on itself, since it is given by the Poisson bracket (which is theLie bracket), between order two and order one polynomials.

The generalization to the case of arbitrary d is

Theorem. The sp(2d,R) action on h2d+1 =M⊕R by derivations is

L · (cq · q + cp · p + c) = µL, cq · q + cp · p + c = c′q · q + c′p · p (14.14)

where (c′qc′p

)= L

(cqcp

)or, equivalently (see section 4.1), on coordinate function basis vectors of M onehas

µL,(

qp

) = LT

(qp

)Proof. One can first prove 14.14 for the cases when only one of A,B,C is non-zero, then the general case follows by linearity. For instance, taking the specialcase

L =

(0 B0 0

), µL =

1

2q ·Bq

one can show that the action on coordinate functions (the basis vectors of M)is

1

2q ·Bq,

(qp

) = LT

(qp

)=

(0Bq

)by computing

1

2

∑j,k

qjBjkqk, pl =1

2

∑j,k

(qjBjkqk, pl+ qjBjk, plqk)

=1

2(∑j

qjBjl +∑k

Blkqk)

=∑j

Bljqj (since B = BT )

Repeating for A and C one finds that for general L one has

µL,(

qp

) = LT

(qp

)Since an element in M can be written as(

cq cp)LT(

qp

)= (L

(cqcp

))T(

qp

)174

we have (c′qc′p

)= L

(cqcp

)

We can now prove theorem 14.1 as follows:

Proof.L→ µL

is clearly a vector space isomorphism of a space of matrices and one of quadraticpolynomials. To show that it is a Lie algebra isomorphism, one can use theJacobi identity for the Poisson bracket to show

µL, µL′ , cq ·q+cp ·p−µL′ , µL, cq ·q+cp ·p = µL, µL′, cq ·q+cp ·p

The left-hand side of this equation is c′′q · q + c′′p · p, where(c′′qc′′p

)= (LL′ − L′L)

(cqcp

)As a result, the right-hand side is the linear map given by

µL, µL′ = µ[L,L′]


For more on symplectic groups and the isomorphism between sp(2d,R) andhomogeneous degree two polynomials, see chapter 14 of [27] or chapter 4 of [20].Chapter 15 of [27] and chapter 1 of [20] discuss the action of the symplecticgroup on the Heisenberg group and Lie algebra by automorphisms.

175

Chapter 15

Quantization

Given any Hamiltonian classical mechanical system with phase space R2d, phys-ics textbooks have a standard recipe for producing a quantum system, by amethod known as “canonical quantization”. We will see that for linear functionson phase space, this is just the construction we have already seen of a unitaryrepresentation Γ′S of the Heisenberg Lie algebra, the Schrodinger representation,and the Stone-von Neumann theorem assures us that this is the unique suchconstruction, up to unitary equivalence. We will also see that this recipe canonly ever be partially successful, with the Schrodinger representation extendingto give us a representation of a sub-algebra of the algebra of all functions onphase space (the polynomials of degree two and below), and a no-go theoremshowing that this cannot be extended to a representation of the full infinitedimensional Lie algebra. Recipes for quantizing higher-order polynomials willalways suffer from a lack of uniqueness, a phenomenon known to physicists asthe existence of “operator ordering ambiguities”.

In later chapters we will see that this quantization prescription does giveunique quantum systems corresponding to some Hamiltonian systems (in par-ticular the harmonic oscillator and the hydrogen atom), and does so in a mannerthat allows a description of the quantum system purely in terms of representa-tion theory.

15.1 Canonical quantization

Very early on in the history of quantum mechanics, when Dirac first saw theHeisenberg commutation relations, he noticed an analogy with the Poissonbracket. One has

q, p = 1 and − i

~[Q,P ] = 1

as well asdf

dt= f, h and

d

dtO(t) = − i

~[O, H]

177

where the last of these equations is the equation for the time dependence of aHeisenberg picture observable O(t) in quantum mechanics. Dirac’s suggestionwas that given any classical Hamiltonian system, one could “quantize” it byfinding a rule that associates to a function f on phase space a self-adjointoperator Of (in particular Oh = H) acting on a state space H such that

Of,g = − i~

[Of , Og]

This is completely equivalent to asking for a unitary representation (π′,H)of the infinite dimensional Lie algebra of functions on phase space (with thePoisson bracket as Lie bracket). To see this, note that one can choose unitsfor momentum p and position q such that ~ = 1. Then, as usual getting askew-adjoint Lie algebra representation operator by multiplying a self-adjointoperator by −i, setting

π′(f) = −iOfthe Lie algebra homomorphism property

π′(f, g) = [π′(f), π′(g)]

corresponds to−iOf,g = [−iOf ,−iOg] = −[Of , Og]

so one has Dirac’s suggested relation.Recall that the Heisenberg Lie algebra is isomorphic to the three-dimensional

sub-algebra of functions on phase space given by linear combinations of the con-stant function, the function q and the function p. The Schrodinger representa-tion ΓS provides a unitary representation not of the Lie algebra of all functionson phase space, but of these polynomials of degree at most one, as follows

O1 = 1, Oq = Q, Op = P

so

Γ′S(1) = −i1, Γ′S(q) = −iQ = −iq, Γ′S(p) = −iP = − d

dq

Moving on to quadratic polynomials, these can also be quantized, as follows

O p2

2

=P 2

2, O q2

2

=Q2

2

For the function pq one can no longer just replace p by P and q by Q since theoperators P and Q don’t commute, and PQ or QP is not self-adjoint. Whatdoes work, satisfying all the conditions to give a Lie algebra homomorphism is

Opq =1

2(PQ+QP )

This shows that the Schrodinger representation Γ′S that was defined as arepresentation of the Heisenberg Lie algebra h3 extends to a unitary Lie algebra

178

representation of a larger Lie algebra, that of all quadratic polynomials on phasespace, a representation that we will continue to denote by Γ′S and refer to as theSchrodinger representation. On a basis of homogeneous order two polynomialswe have

Γ′S(p2

2) = −iP

2

2=i

2

d2

dq2

Γ′S(q2

2) = −iQ

2

2= − i

2q2

Γ′S(pq) =−i2

(PQ+QP )

Restricting Γ′S to just linear combinations of these homogeneous order two poly-nomials (which give the Lie algebra sl(2,R), recall equation 14.4) we get a Liealgebra representation of sl(2,R) called the metaplectic representation.

Restricted to the Heisenberg Lie algebra, the Schrodinger representation Γ′Sexponentiates to give a representation ΓS of the corresponding Heisenberg Liegroup (see 11.4). As an sl(2,R) representation however, one can show that Γ′Shas the same sort of problem as the spinor representation of su(2) = so(3), whichwas not a representation of SO(3), but only of its double cover SU(2) = Spin(3).To get a group representation, one must go to a double cover of the groupSL(2,R), which will be called the metaplectic group and denoted Mp(2,R).

The source of the problem is the subgroup of SL(2,R) generated by expo-nentiating the Lie algebra element

1

2(p2 + q2)↔ E − F =

(0 1−1 0

)When we study the Schrodinger representation using its action on the quantumharmonic oscillator state space H in chapter 20 we will see that the Hamiltonianis the operator

1

2(P 2 +Q2)

and this has half-integer eigenvalues. As a result, trying to exponentiate Γ′Sgives a representation of SL(2,R) only up to a sign, and one needs to go to thedouble cover Mp(2,R) to get a true representation.

One should keep in mind though that, since SL(2,R) acts non-trivially byautomorphisms on H3, elements of these two groups do not commute. TheSchrodinger representation is a representation not of the product group, but ofsomething called a “semi-direct product” which will be discussed in more detailin chapter 16.

15.2 The Groenewold-van Hove no-go theorem

If one wants to quantize polynomial functions on phase space of degree greaterthan two, it quickly becomes clear that the problem of “operator ordering am-biguities” is a significant one. Different prescriptions involving different ways

179

of ordering the P and Q operators lead to different Of for the same functionf , with physically different observables (although the differences involve thecommutator of P and Q, so higher-order terms in ~).

When physicists first tried to find a consistent prescription for producing anoperator Of corresponding to a polynomial function on phase space of degreegreater than two, they found that there was no possible way to do this consistentwith the relation

Of,g = − i~

[Of , Og]

for polynomials of degree greater than two. Whatever method one devises forquantizing higher degree polynomials, it can only satisfy that relation to lowestorder in ~, and there will be higher order corrections, which depend upon one’schoice of quantization scheme. Equivalently, it is only for the six-dimensional Liealgebra of polynomials of degree up to two that the Schrodinger representationgives one a Lie algebra representation, and this cannot be consistently extendedto a representation of a larger subalgebra of the functions on phase space. Thisproblem is made precise by the following no-go theorem

Theorem (Groenewold-van Hove). There is no map f → Of from polynomialson R2 to self-adjoint operators on L2(R) satisfying

Of,g = − i~

[Of , Og]

andOp = P, Oq = Q

for any Lie subalgebra of the functions on R2 larger than the subalgebra ofpolynomials of degree less than or equal to two.

Proof. For a detailed proof, see section 5.4 of [7], section 4.4 of [20], or chapter16 of [27]. In outline, the proof begins by showing that taking Poisson brack-ets of polynomials of degree three leads to higher order polynomials, and thatfurthermore for degree three and above there will be no finite-dimensional sub-algebras of polynomials of bounded degree. The assumptions of the theoremforce certain specific operator ordering choices in degree three. These are thenused to get a contradiction in degree four, using the fact that the same degreefour polynomial has two different expressions as a Poisson bracket:

q2p2 =1

3q2p, p2q =

1

9q3, p3

15.3 Canonical quantization in d dimensions

One can easily generalize the above to the case of d dimensions, with theSchrodinger representation ΓS now giving a unitary representation of the Heisen-berg group H2d+1, with the corresponding Lie algebra representation given by

Γ′S(qj) = −iQj , Γ′S(pj) = −iPj

180

which satisfy the Heisenberg relations

[Qj , Pk] = iδjk

Generalizing to quadratic polynomials in the phase space coordinate func-tions, we have

Γ′S(qjqk) = −iQjQk, Γ′S(pjpk) = −iPjPk, Γ′S(qjpk) = − i2

(QjPk + PkQj)

(15.1)One can exponentiate these operators to get a representation on the same Hof Mp(2d,R), a double cover of the symplectic group Sp(2d,R). This phe-nomenon will be examined carefully in later chapters, starting with chapter 18and the calculation in section 18.2.1, followed by discussion in later chaptersusing a different (but unitarily equivalent) representation that appears in thequantization of the harmonic oscillator. The Groenewold-van Hove theorem im-plies that we cannot find a unitary representation of a larger group of canonicaltransformations extending this one on the Heisenberg and metaplectic groups.

15.4 Quantization and symmetries

The Schrodinger representation is thus a representation of the groups H2d+1

and Mp(2d,R), with the Lie algebra representation providing observables cor-responding to elements of the Lie algebras h2d+1 (linear combinations of Qjand Pk) and sp(2d,R) (linear combinations of order-two combinations of Qjand Pk). The observables that commute with the Hamiltonian operator H willmake up a Lie algebra of symmetries of the quantum system, and will take en-ergy eigenstates to energy eigenstates of the same energy. Some examples forthe physical case of d = 3 are:

• The group R3 of translations in coordinate space is a subgroup of theHeisenberg group and has a Lie algebra representation as linear combina-tions of the operators −iPj . If the Hamiltonian is position-independent,for instance the free particle case of

H =1

2m(P 2

1 + P 22 + P 2

3 )

then the momentum operators correspond to symmetries. Note that theposition operators Qj do not commute with this Hamiltonian, and so donot correspond to a symmetry of the dynamics.

• The group SO(3) of spatial rotations is a subgroup of Sp(6,R) and theoperators

−i(Q2P3 −Q3P2), −i(Q3P1 −Q1P3, −i(Q1P2 −Q2P1)

are a basis for a Lie algebra representation of so(3). These are the sameoperators that were studied in chapter 8 under the name ρ′(lj). They will

181

be symmetries of rotationally invariant Hamiltonians, for instance the freeparticle as above, or the particle in a potential

H =1

2m(P 2

1 + P 22 + P 2

3 ) + V (Q1, Q2, Q3)

when the potential only depends on the combination Q21 +Q2

2 +Q23.

15.5 More general notions of quantization

The definition given here of quantization using the Schrodinger representationof h2d+1 only allows the construction of a quantum system based on a classicalphase space for the linear case of M = R2d. For other sorts of classical systemsone needs other methods to get a corresponding quantum system. One possibleapproach is the path integral method, which starts with a choice of configurationspace and Lagrangian, and will be discussed in chapter 32.

Digression. The name “geometric quantization” refers to attempt to generalizequantization to the case of any symplectic manifold M , starting with the ideaof prequantization (see equation 13.6). This gives a representation of the Liealgebra of functions on M on a space of sections of a line bundle with connection∇, with ∇ a connection with curvature ω, where ω is the symplectic form onM . One then has to deal with two problems

• The space of all functions on M is far too big, allowing states localized inboth position and coordinate variables in the case M = R2d. One needssome way to cut down this space to something like a space of functionsdepending on only half the variables (e.g. just the positions, or just themomenta). This requires finding an appropriate choice of a so-called “po-larization” that will accomplish this.

• To get an inner product on the space of states, one needs to introducea twist by a “square root” of a certain line bundle, something called the“metaplectic correction”.

For more details, see for instance [31] or [82].Geometric quantization focuses on finding an appropriate state space. An-

other general method, the method of “deformation quantization” focuses insteadon the algebra of operators, with a quantization given by finding an appropriatenon-commutative algebra that is in some sense a deformation of a commuta-tive algebra of functions. To first order the deformation in the product law isdetermined by the Poisson bracket.

Starting with any Lie algebra g, one can in principle use 12.4 to get a Pois-son bracket on functions on the dual space g∗, and then take the quantizationof this to be the algebra of operators known as the universal enveloping alge-bra U(g). This will in general have many different irreducible representationsand corresponding possible quantum state spaces. The co-adjoint orbit philoso-phy posits an approximate matching between orbits in g∗ under the dual of the

182

adjoint representation (which are symplectic manifolds) and irreducible repre-sentations. Geometric quantization provides one possible method for trying toassociate representations to orbits. For more details, see [38].

None of the general methods of quantization is fully satisfactory, with eachrunning into problems in certain cases, or not providing a construction with allthe properties that one would want.


Just about all quantum mechanics textbooks contain some version of the discus-sion here of canonical quantization starting with classical mechanical systemsin Hamiltonian form. For discussions of quantization from the point of view ofrepresentation theory, see [7] and chapters 14-16 of [27]. For a detailed discus-sion of the Heisenberg group and Lie algebra, together with their representationtheory, also see chapter 2 of [38].

183

Chapter 16

Semi-direct Products

The theory of a free particle depends crucially on the group of symmetriesof three-dimensional space, a group which includes a subgroup R3 of spatialtranslations, and a subgroup SO(3) of rotations. The second subgroup acts non-trivially on the first, since the direction of a translation is rotated by an elementof SO(3). In later chapters dealing with special relativity, these symmetrygroups get enlarged to include a fourth dimension, time, and the theory of afree particle will again be determined by these symmetry groups. In chapters 13and 14 we saw that there are two groups acting on phase space: the Heisenberggroup H2d+1 and the symplectic group Sp(2d,R). In this situation also, thesecond group acts non-trivially on the first by automorphisms (see 14.10).

This situation of two groups, with one acting on the other by automor-phisms, allows one to construct a new sort of product of the two groups, calledthe semi-direct product, and this will be the topic for this chapter. We’ll alsobegin the study of representations of such groups, outlining what happens whenthe first group is commutative. Chapter 18 will describe how the Schrodingerrepresentation of H2d+1 extends to become a representation (up to a sign am-biguity) of the semi-direct product of H2d+1 and Sp(2d,R). In chapter 17 we’llconsider the cases of the semi-direct product of translations and rotations intwo and three dimensions, and there see how the irreducible representations areprovided by the quantum state space of a free particle.

16.1 An example: the Euclidean group

Given two groups G′ and G′′, one can form the product group by taking pairsof elements (g′, g′′) ∈ G′ ×G′′. However, when the two groups act on the samespace, but elements of G′ and G′′ don’t commute, a different sort of productgroup is needed. As an example, consider the case of pairs (a2, R2) of elementsa2 ∈ R3 and R2 ∈ SO(3), acting on R3 by translation and rotation

v→ (a2, R2) · v = a2 +R2v

185

If we then act on the result with (a1, R1) we get

(a1, R1) · ((a2, R2) · v) = (a1, R1) · (a2 +R2v) = a1 +R1a2 +R1R2v

Note that this is not what we would get if we took the product group law onR3 × SO(3), since then the action of (a1, R1)(a2, R2) on R3 would be

v→ a1 + a2 +R1R2v

To get the correct group action on R3, we need to take R3 × SO(3) not withthe product group law, but instead with the group law

(a1, R1)(a2, R2) = (a1 +R1a2, R1R2)

This group law differs from the standard product law, by a term R1a2, whichis the result of R1 ∈ SO(3) acting non-trivially on a2 ∈ R3. We will denote theset R3 × SO(3) with this group law by

R3 o SO(3)

This is the group of transformations of R3 preserving the standard inner prod-uct.

The same construction works in arbitrary dimensions, where one has

Definition (Euclidean group). The Euclidean group E(d) (sometimes writtenISO(d) for “inhomogeneous” rotation group) in dimension d is the product ofthe translation and rotation groups of Rd as a set, with multiplication law

(a1, R1)(a2, R2) = (a1 +R1a2, R1R2)

(where aj ∈ Rd, Rj ∈ SO(d)) and can be denoted by

Rd o SO(d)

E(d) can also be written as a matrix group, taking it to be the subgroupof GL(d + 1,R) of matrices of the form (R is a d by d orthogonal matrix, a ad-dimensional column vector) (

R a0 1

)One gets the multiplication law for E(d) from matrix multiplication since(

R1 a1

0 1

)(R2 a2

0 1

)=

(R1R2 a1 +R1a2

0 1

)

16.2 Semi-direct product groups

The Euclidean group example of the previous section can be generalized to thefollowing

186

Definition (Semi-direct product group). Given a group K, a group N , and anaction φ of K on N by automorphisms

φk : n ∈ N → φk(n) ∈ N

the semi-direct product N oK is the set of pairs (n, k) ∈ N ×K with group law

(n1, k1)(n2, k2) = (n1φk1(n2), k1k2)

One can easily check that this satisfies the group axioms. The inverse is

(n, k)−1 = (φk−1(n−1), k−1)

Checking associativity, one finds

((n1, k1)(n2, k2))(n3, k3) =(n1φk1(n2), k1k2)(n3, k3)

=(n1φk1(n2)φk1k2

(n3), k1k2k3)

=(n1φk1(n2)φk1

(φk2(n3)), k1k2k3)

=(n1φk1(n2φk2

(n3)), k1k2, k3)

=(n1, k1)(n2φk2(n3), k2k3)

=(n1, k1)((n2, k2)(n3, k3))

The notation N o K for this construction has the weakness of not explicitlyindicating the automorphism φ which it depends on. There may be multiplepossible choices for φ, and these will always include the trivial choice φk = 1 forall k ∈ K, which will give the standard product of groups.

Digression. For those familiar with the notion of a normal subgroup, N is anormal subgroup of N oK. A standard notation for “N is a normal subgroupof G” is N G. The symbol N oK is supposed to be a mixture of the × and symbols (note that some authors define it to point in the other direction).

The Euclidean group E(d) is an example with N = Rd,K = SO(d). Fora ∈ Rd, R ∈ SO(d) one has

φR(a) = Ra

In chapter 40 we will see another important example, the Poincare group whichgeneralizes E(3) to include a time dimension, treating space and time accordingto the principles of special relativity.

The most important example for quantum theory is

Definition (Jacobi group). The Jacobi group in d dimensions is the semi-directproduct group

GJ(d) = H2d+1 o Sp(2d,R)

If we write elements of the group as

((

(cqcp

), c), k)

187

where k ∈ Sp(2d,R), then the automorphism φk that defines the Jacobi groupis given by

φk((

(cqcp

), c)) = (k

(cqcp

), c) (16.1)

Note that the Euclidean group E(d) is a subgroup of the Jacobi group GJ(d),the subgroup of elements of the form

((

(0cp

), 0),

(R 00 R

))

where R ∈ SO(d). The

(

(0cp

), 0) ⊂ H2d+1

make up the group Rd of translations in the qj coordinates, and the

k =

(R 00 R

)⊂ Sp(2d,R)

are symplectic transformations since

Ω(k

(cqcp

), k

(c′qc′p

)) =Rcq ·Rc′p −Rcp ·Rc′q

=cq · c′p − cp · c′q

=Ω(

(cqcp

),

(c′qc′p

))

(R is orthogonal so preserves dot products).

16.3 Semi-direct product Lie algebras

We have seen that semi-direct product Lie groups can be constructed by takinga product N ×K of Lie groups as a set, and imposing a group multiplicationlaw that uses an action of K on N by automorphisms. In a similar manner, onecan construct semi-direct product Lie algebras n o k by taking the direct sumof n and k as vector spaces, and defining a Lie bracket that uses an action of non k by derivations (the infinitesimal version of automorphisms, see definition14.1).

Considering first the example E(d) = RdoSO(d), recall that elements E(d)can be written in the form (

R a0 1

)for R ∈ SO(d) and a ∈ Rd. The tangent space to this group at the identity willbe given by matrices of the form (

X a0 0

)188

where X is an antisymmetric d by d matrix and a ∈ Rd. Exponentiating suchmatrices will give elements of E(d).

The Lie bracket is then given by the matrix commutator, so

[

(X1 a1

0 0

),

(X2 a2

0 0

)] =

([X1, X2] X1a2 −X2a1

0 0

)(16.2)

So the Lie algebra of E(d) will be given by taking the sum of Rd (the Liealgebra of Rd) and so(d), with elements pairs (a, X) with a ∈ Rd and X anantisymmetric d by d matrix. The infinitesimal version of the rotation actionof SO(d) on Rd by automorphisms

φR(a) = Ra

isd

dtφetX (a)|t=0 =

d

dt(etXa)|t=0 = Xa

Just in terms of such pairs, the Lie bracket can be written

[(a1, X1), (a2, X2)] = (X1a2 −X2a1, [X1, X2])

We can define in general

Definition (Semi-direct product Lie algebra). Given Lie algebras k and n, andan action of elements Y ∈ k on n by derivations

X ∈ n→ Y ·X ∈ n

the semi-direct product nok is the set of pairs (X,Y ) ∈ n⊕k with the Lie bracket

[(X1, Y1), (X2, Y2)] = ([X1, X2] + Y1 ·X2 − Y2 ·X1, [Y1, Y2])

One can easily see that in the special case of the Lie algebra of E(d) thisagrees with the construction above.

In section 14.1.2 we studied the Lie algebra of all polynomials of degreeat most two in d-dimensional phase space coordinates qj , pj , with the Poissonbracket as Lie bracket. There we found two Lie subalgebras, the degree zeroand one polynomials (isomorphic to h2d+1), and the homogeneous degree twopolynomials (isomorphic to sp(2d,R)) with the second subalgebra acting on thefirst by derivations as in equation 14.14.

Recall from chapter 14 that elements of this Lie algebra can also be writtenas pairs

((

(cqcp

), c), L)

of elements in h2d+1 and sp(2d,R), with this pair corresponding to the polyno-mial

µL + cq · q + cp · p + c

189

In terms of such pairs, the Lie bracket is given by

[((

(cqcp

), c), L), ((

(c′qc′p

), c), L′)] = ((L

(c′qc′p

)−L′

(cqcp

),Ω(

(cqcp

),

(c′qc′p

))), [L,L′])

which satisfies the definition above.So one example of a semi-direct product Lie algebra is

h2d+1 o sp(2d,R)

and from the discussion in chapter 14.2 one can see that this is the Lie algebraof the semi-direct product group

GJ(d) = H2d+1 o Sp(2d,R)

.The Lie algebra of E(d) will be a sub-Lie algebra of this, consisting of ele-

ments of the form

((

(0cp

), 0),

(X 00 X

))

where X is an antisymmetric d by d matrix.

Digression. Just as E(d) can be identified with a group of d+1 by d+1 matrices,the Jacobi group GJ(d) is also a matrix group and one can in principle work withit and its Lie algebra using usual matrix methods. The construction is slightlycomplicated and represents elements of GJ(d) as matrices in Sp(2d+1,R) . Seesection 8.5 of [8] for details of the d = 1 case.


Semi-direct products are not commonly covered in detail in either physics ormathematics textbooks, with the exception of the case of the Poincare group ofspecial relativity, which will be discussed in chapter 40. Some textbooks thatdo cover the subject include section 3.8 of [65], chapter 6 of [28] and [8].

190

Chapter 17

The Quantum Free Particleas a Representation of theEuclidean Group

In this chapter we will explicitly construct unitary representations of the Eu-clidean groups E(2) and E(3) of spatial symmetries in two and three dimensions.These groups commute with the Hamiltonian of the free particle, and their irre-ducible representations will be given just by the quantum state space of a freeparticle (of fixed energy) in either two ot three spatial dimensions. The mo-mentum operators Pj will provide the infinitesimal action of translations on thestate space, while angular momentum operators Lk will provide the infinitesi-mal rotation action (there will be only one of these in two dimensions, three inthree dimensions).

The Hamiltonian of the free particle is proportional to the operator |P|2.This is a quadratic operator that commutes with the action of all the elementsof the Lie algebra of the Euclidean group, and so is a Casimir operator play-ing an analogous role to that of the SO(3) Casimir operator |L|2 of section8.4. Irreducible representations will be labeled by the eigenvalue of this oper-ator, which in this case will be proportional to the energy. In the Schrodingerrepresentation where the Pj are differentiation operators, this will be a second-order differential operator, and the eigenvalue equation will be a second-orderdifferential equation (the time-independent Schrodinger equation).

Using the Fourier transform, the space of solutions of the Schrodinger equa-tion of fixed energy becomes something much easier to analyze, the space offunctions on momentum space supported only on the subspace of momenta ofa fixed length. In the case of E(2) this is just a circle, whereas for E(3) it is asphere. In both cases, for each radius one gets an irreducible representation.

In the case of E(3) other classes of irreducible representations can be con-structed. This can be done by introducing multi-component wavefunctions,with a new action of the rotation group SO(3). A second Casimir operator is

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available in this case, and irreducible representations are eigenfunctions of thisoperator in the space of wavefunctions of fixed energy. The eigenvalue of thesecond Casimir operator turns out to be an integer, known to physicists as the“helicity”.

17.1 The quantum free particle and representa-tions of E(2)

We’ll begin with the case of two spatial dimensions, partly for simplicity, partlybecause physical systems that are translationally invariant in one direction canoften be treated as effectively two dimensional. A basis for the Lie algebra ofE(2) is given by the functions

l = q1p2 − q2p1, p1, p2

on the d = 2 phase space M = R4. The non-zero Lie bracket relations are givenby the Poisson brackets

l, p1 = p2, l, p2 = −p1

and there is an isomorphism of this Lie algebra with a matrix Lie algebra of 3by 3 matrices given by

l↔

0 −1 01 0 00 0 0

, p1 ↔

0 0 10 0 00 0 0

, p2 ↔

0 0 00 0 10 0 0

Writing this Lie algebra in terms of linear and quadratic functions on the

phase space shows that it can be realized as a sub-Lie algebra of the JacobiLie algebra gJ(2). Quantization via the Schrodinger representation Γ′S thenprovides a unitary representation of the Lie algebra of E(2) on the state spaceH of functions of the position variables q1, q2, in terms of operators

Γ′S(p1) = −iP1 = − ∂

∂q1, Γ′S(p2) = −iP1 = − ∂

∂q2(17.1)

and

Γ′S(l) = −iL = −i(Q1P2 −Q2P1) = −(q1∂

∂q2− q2

∂

∂q1) (17.2)

The Hamiltonian operator for the free particle is

H =1

2m(P 2

1 + P 22 ) = − 1

2m(∂2

∂q21

+∂2

∂q22

)

and solutions to the Schrodinger equation can be found by solving the eigenvalueequation

Hψ(q1, q2) = − 1

2m(∂2

∂q21

+∂2

∂q22

)ψ(q1, q2) = Eψ(q1, q2)

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The operators L,P1, P2 commute with H and so provide a representation of theLie algebra of E(2) on the space of wavefunctions of energy E.

This construction of irreducible representations of E(2) is similar in spiritto the construction of irreducible representations of SO(3) in section 8.4. Therethe Casimir operator L2 commuted with the SO(3) action, and gave a differ-ential operator on functions on the sphere whose eigenfunctions were spacesof dimension 2l + 1 with eigenvalue l(l + 1), for l non-negative and integral.For E(2) the quadratic function p2

1 + p22 Poisson commutes with l, p1, p2. After

quantization,

|P|2 = P 21 + P 2

2

is a second-order differential operator which commutes with L,P1, P2. This op-erator has infinite-dimensional eigenspaces that each carry an irreducible rep-resentation of E(2). They are characterized by a non-negative eigenvalue thathas physical interpretation as 2mE where m,E are the mass and energy of afree quantum particle moving in two spatial dimensions.

Recall from our discussion of the free particle in chapter 10 that

ψp(q) = eip·q = |p〉

is a solution of the time-independent Schrodinger equation with energy

E =|p|2

2m> 0

and such |p〉 give a sort of continuous basis of H, even though these are notsquare-integrable functions. The formalism for working with them uses distri-butions and the orthonormality relation

〈p|p′〉 = δ(p− p′)

An arbitrary ψ(q) ∈ H can be written as a continuous linear combination of

the |p〉, i.e. as an inverse Fourier transform of a function ψ(p) on momentumspace as

ψ(q) =1

2π

∫∫eip·qψ(p)d2p

In momentum space the time-independent Schrodinger equation becomes

(|p|2

2m− E)ψ(p) = 0

so we get a solution for any choice of ψ(p) that is non-zero only on the circle|p|2 = 2mE (we won’t try to characterize which class of such functions toconsider, which would determine which class of functions solving the Schrodingerequation we end up with after Fourier transform).

193

Going to polar coordinates p = (p cos θ, p sin θ), the space of solutions to

the time-independent Schrodinger equation at energy E is given by ψ(p) of theform

ψ(p) = ψE(θ)δ(p2 − 2mE)

To put this delta-function in a more useful form, note that for p ≈√

2mE onehas the linear approximation

p2 − 2mE ≈ 1

2√

2mE(p−

√2mE)

so one has the equality of distributions

δ(p2 − 2mE) =1

2√

2mEδ(p−

√2mE)

It is this space of functions ψE(θ) of functions on the circle of radius√

2mEthat will provide an infinite-dimensional representation of the group E(2), onethat turns out to be irreducible, although we will not show that here. The

194

position space wavefunction corresponding to ψE(θ) will be

ψ(q) =1

2π

∫∫eip·qψE(θ)δ(p2 − 2mE)pdpdθ

=1

2π

∫∫eip·qψE(θ)

1

2√

2mEδ(p−

√2mE)pdpdθ

=1

4π

∫ 2π

0

ei√

2mE(q1 cos θ+q2 sin θ)ψE(θ)dθ

Functions ψE(θ) with simple behavior in θ will correspond to wavefunctions

with more complicated behavior in position space. For instance, taking ψE(θ) =e−inθ one finds that the wavefunction along the q2 direction is given by

ψ(0, q) =1

4π

∫ 2π

0

ei√

2mE(q sin θ)e−inθdθ

=1

2Jn(√

2mEq)

where Jn is the n’th Bessel function.Equations 17.1 and 17.2 give the representation of the Lie algebra of E(2)

on wavefunctions ψ(q). The representation of this Lie algebra on the ψE(θ) is

just given by the Fourier transform, and we’ll denote this Γ′S . Using the formulafor the Fourier transform we find that

Γ′S(p1) = − ∂

∂q1= −ip1 = −i

√2mE cos θ

Γ′S(p2) = − ∂

∂q2= −ip2 = −i

√2mE sin θ

are multiplication operators and, taking the Fourier transform of 17.2 gives thedifferentiation operator

Γ′S(l) =− (p1∂

∂p2− p2

∂

∂p1)

=− ∂

∂θ

(use integration by parts to show qj = i ∂∂pj

and thus the first equality, then the

chain rule for functions f(p1(θ), p2(θ)) for the second).This construction of a representation of E(2) starting with the Schrodinger

representation gives the same result as starting with the action of E(2) onconfiguration space, and taking the induced action on functions on R2 (thewavefunctions). To see this, note that E(2) has elements (a, R(φ)) which canbe written as a product (a, R(φ)) = (a,1)(0, R(φ)) or, in terms of matricescosφ − sinφ a1

sinφ cosφ a2

0 0 1

=

1 0 a1

0 1 a2

0 0 1

cosφ − sinφ 0sinφ cosφ 0

0 0 1

195

The group has a unitary representation

(a, R(φ))→ u(a, R(φ))

on the position space wavefunctions ψ(q), given by the induced action on func-tions from the action of E(2) on position space R2

u(a, R(φ))ψ(q) =ψ((a, R(φ))−1 · q)

= = ψ((−R(−φ)a, R(−φ)) · q)

=ψ(R(−φ)(q− a))

This is just the Schrodinger representation ΓS of the Jacobi group GJ(2), re-stricted to the subgroup E(2) of transformations of phase space that are transla-tions in q and rotations in both q and p vectors, preserving their inner product(and thus the symplectic form). One can see this by considering the actionof translations as the exponential of the Lie algebra representation operatorsΓ′S(pj) = −iPj

u(a,1)ψ(q) = e−i(a1P1+a2P2)ψ(q) = ψ(q− a)

and the action of rotations as the exponential of the Γ′S(l) = −iL

u(0, R(φ))ψ(q) = e−iφLψ(q) = ψ(R(−φ)q)

One also has a Fourier-transformed version u of this representation, withtranslations now acting by multiplication operators on the ψE

u(a,1)ψE(θ) = e−i(a·p)ψE(θ) = e−i√

2mE(a1 cos θ+a2 sin θ)ψE(θ) (17.3)

and rotations acting by rotation in momentum space

u(0, R(φ))ψE(θ) = ψE(θ − φ) (17.4)

Although we won’t prove it here, the representations constructed this wayprovide essentially all the unitary irreducible representations of E(2), parame-trized by a real number E > 0. The only other ones are those on which thetranslations act trivially, corresponding to E = 0, with SO(2) acting as anirreducible representation. We have seen that such SO(2) representations areone-dimensional, and characterized by an integer, the weight, so get anotherclass of E(2) irreducible representations, labeled by an integer, but they arejust one-dimensional representations on C.

17.2 The case of E(3)

In the physical case of three spatial dimensions, the state space of the theory of aquantum free particle is again a Euclidean group representation, with the samerelationship to the Schrodinger representation as in two spatial dimensions. The

196

main difference is that the rotation group is now three dimensional and non-commutative, so instead of the single Lie algebra basis element l we have threeof them, satisfying Poisson bracket relations that are the Lie algebra relationsof so(3)

l1, l2 = l3, l2, l3 = l1, l3, l1 = l2

The pj give the other three basis elements of the Lie algebra of E(3). Theycommute amongst themselves and the action of rotations on vectors providesthe rest of the non-trivial Poisson bracket relations

l1, p2 = p3, l1, p3 = −p2

l2, p1 = −p3, l2, p3 = p1

l3, p1 = p2, l3, p2 = −p1

An isomorphism of this Lie algebra with a Lie algebra of matrices is givenby

l1 ↔

0 0 0 00 0 −1 00 1 0 00 0 0 0

l2 ↔

0 0 1 00 0 0 0−1 0 0 00 0 0 0

l3 ↔

0 −1 0 01 0 0 00 0 0 00 0 0 0

p1 ↔

0 0 0 10 0 0 00 0 0 00 0 0 0

p2 ↔

0 0 0 00 0 0 10 0 0 00 0 0 0

p3 ↔

0 0 0 00 0 0 00 0 0 10 0 0 0

The lj are quadratic functions in the qj , pj , given by the classical mechanical

expression for the angular momentum

l = q× p

or, in components

l1 = q2p3 − q3p2, l2 = q3p1 − q1p3, l3 = q1p2 − q2p1

The Euclidean group E(3) is a subgroup of the Jacobi group GJ(3) in thesame way as in two dimensions, and the Schrodinger representation ΓS providesa representation of E(3) with Lie algebra version

Γ′S(l1) = −iL1 = −i(Q2P3 −Q3P2) = −(q2∂

∂q3− q3

∂

∂q2)

Γ′S(l2) = −iL2 = −i(Q3P1 −Q1P3) = −(q3∂

∂q1− q1

∂

∂q3)

Γ′S(l3) = −iL3 = −i(Q1P2 −Q2P1) = −(q1∂

∂q2− q2

∂

∂q1)

197

Γ′S(pj) = −iPj = − ∂

∂qj

These are just the infinitesimal versions of the action of E(3) on functionsinduced from its action on position space R3. Given an element g = (a, R) ∈E(3) ⊂ GJ(3) we have a unitary transformation on wavefunctions

u(a, R)ψ(q) = ΓS(g)ψ(q) = ψ(g−1 · q) = ψ(R−1(q− a))

These group elements will be a product of a translation and a rotation, andthe unitary transformations u are exponentials of the Lie algebra actions above,with

u(a,1)ψ(q) = e−i(a1P1+a2P2+a3P3)ψ(q) = ψ(q− a)

for a translation by a, and

u(0, R(φ, ej))ψ(q) = e−iφLjψ(q) = ψ(R(−φ, ej)q)

for R(φ, ej) a rotation about the j-axis by angle φ.This representation of E(3) on wavefunctions is reducible, since in terms of

momentum eigenstates, rotations will only take eigenstates with one value of themomentum to those with another value of the same length-squared. We can getan irreducible representation by using the Casimir operator P 2

1 +P 22 +P 2

3 , whichcommutes with all elements in the Lie algebra of E(3). The Casimir operatorwill act on an irreducible representation as a scalar, and the representation willbe characterized by that scalar. The Casimir operator is just 2m times theHamiltonian

H =1

2m(P 2

1 + P 22 + P 2

3 )

and so the constant characterizing an irreducible will just be the energy 2mE.Our irreducible representation will be on the space of solutions of the time-independent Schrodinger equation

1

2m(P 2

1 + P 22 + P 2

3 )ψ(q) = − 1

2m(∂2

∂q21

+∂2

∂q22

+∂2

∂q23

)ψ(q) = Eψ(q)

Using the Fourier transform

ψ(q) =1

(2π)32

∫R3

eip·qψ(p)d3p

the time-independent Schrodinger equation becomes

(|p|2

2m− E)ψ(p) = 0

and we have distributional solutions

ψ(p) = ψE(p)δ(|p|2 − 2mE)

198

characterized by functions ψE(p) defined on the sphere |p|2 = 2mE.

Such complex-valued functions on the sphere of radius√

2mE provide aFourier-transformed version u of the irreducible representation of E(3). Herethe action of the group E(3) is by

u(a,1)ψE(p) = e−i(a·p)ψE(p)

for translations, by

u(0, R)ψE(p) = ψE(R−1p)

for rotations, and by

u(a, R)ψE(p) = u(a,1)u(0, R)ψE(p) = e−ia·R−1pψE(R−1p)

for a general element.

17.3 Other representations of E(3)

For the case of E(3), besides the representations parametrized by E > 0 con-structed above, as in the E(2) case there are finite-dimensional representationswhere the translation subgroup of E(3) acts trivially. Such irreducible represen-tations are just the spin-s representations (ρs,C

2s+1) of SO(3) for s = 0, 1, 2, . . ..

E(3) has some structure not seen in the E(2) case, which can be used toconstruct new classes of infinite-dimensional irreducible representations. Thiscan be seen from two different points of view:

• There is a second Casimir operator which one can show commutes withthe E(3) action, given by

L ·P = L1P1 + L2P2 + L3P3

• The group SO(3) acts on momentum vectors by rotation, with orbit ofthe group action the sphere of momentum vectors of fixed energy E > 0.This is the sphere on which the Fourier transform of the wavefunctionsin the representation is supported. Unlike the corresponding circle inthe E(2) case, here there is a non-trivial subgroup of the rotation groupSO(3) which leaves a given momentum vector invariant. This is just theSO(2) ⊂ SO(3) subgroup of rotations about the axis determined by themomentum vector, and it is different for different points in momentumspace.

199

For single-component wavefunctions, a straightforward computation showsthat the second Casimir operator L ·P acts as zero. By introducing wavefunc-tions with several components, together with an action of SO(3) that mixes thecomponents, it turns out that one can get new irreducible representations, witha non-zero value of the second Casimir corresponding to a non-trivial weight ofthe action of the SO(2) of rotations about the momentum vector.

One can construct such multiple-component wavefunctions as representa-tions of E(3) by taking the tensor product of our irreducible representation onwavefunctions of energy E (call this HE) and the finite dimensional irreduciblerepresentation C2s+1

HE ⊗C2s+1

The Lie algebra representation operators for the translation part of E(3) act asmomentum operators on HE and as 0 on C2s+1. For the SO(3) part of E(3),we get operators we can write as

Jj = Lj + Sj

where Lj acts on HE and Sj = ρ′(lj) acts on C2s+1.This tensor product representation will not be irreducible, but its irreducible

components can be found by taking the eigenspaces of the second Casimir op-

200

erator, which will now beJ ·P

We will not work out the details of this here (although details can be foundin chapter 31 for the case s = 1

2 , where SO(3) is replaced by Spin(3)). Whathappens is that the tensor product breaks up into irreducibles as

HE ⊗C2s+1 =

n=s⊕n=−s

HE,n

where n is an integer taking values from −s to s that is called the “helicity”.HE,n is the subspace of the tensor product on which the first Casimir |P|2 takesthe value 2mE, and the second Casimir J · P takes the value np, where p =√

2mE. The physical interpretation of the helicity is that it is the component ofangular momentum along the axis given by the momentum vector. The helicitycan also be thought of as the weight of the action of the SO(2) subgroup ofSO(3) corresponding to rotations about the axis of the momentum vector.

Choosing E > 0 and n ∈ Z, the representations on HE,n (which we haveconstructed using some s such that |s| ≥ n) give all possible irreducible repre-sentations of E(3). The representation spaces have a physical interpretation asthe state space for a free quantum particle of energy E which carries an “inter-nal” quantized angular momentum about its direction of motion, given by thehelicity.


The angular momentum operators are a standard topic in every quantum me-chanics textbook, see for example chapter 12 of [63]. The characterization hereof free-particle wavefunctions at fixed energy as giving irreducible representa-tions of the Euclidean group is not so conventional, but it is just an example ofa non-relativistic version of the conventional description of relativistic quantumparticles in terms of representations of the Poincare group (see chapter 40). Inthe Poincare group case the analog of the E(3) irreducible representations ofnon-zero helicity considered here will be irreducible representations labeled bya non-zero mass and an irreducible representation of SO(3) (the spin). In thatcase for massless particles one will again see representations labeled by a helicity(an irreducible representation of SO(2)), but there is no analog of such masslessparticles in the E(3) case.

For more details about representations of E(2) and E(3), see [73] or [76](which is based on [72]).

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Chapter 18

Representations ofSemi-direct Products

In this chapter we will examine some aspects of representations of semi-directproducts, in particular for the case of the Jacobi group and its Lie algebra, aswell as the case of N oK, for N commutative.

The Schrodinger representation provides a unitary representation of theHeisenberg group, one that carries extra structure arising from the fact thatthe symplectic group acts on the Heisenberg group by automorphisms. Eachelement of the symplectic group takes a given construction of the Schrodingerrepresentation to a unitarily equivalent one, providing an operator on the statespace called an “intertwining operator”. These intertwining operators will give(up to a phase factor), a representation of the symplectic group. Up to theproblem of the phase factor, the Schrodinger representation in this way extendsto a representation of the full Jacobi group. To explicitly find the phase factor,one can start with the Lie algebra representation, where the sp(2d,R) action isgiven by quantizing quadratic functions on phase space. It turns out that, for afinite dimensional phase space, this gives a representation up to sign, which canbe turned into a true representation by taking a double cover (called Mp(2d,R))of Sp(2d,R).

In later chapters, we will find that many actions of groups on quantum sys-tems can be understood as subgroups of this Mp(2d,R), with the correspondingobservables arising as the quadratic combinations of momentum and positionoperators determined by the moment map.

The Euclidean group E(d) is a subgroup of the Jacobi group and we saw inchapter 17 how some of its representations can be understood by restricting theSchrodinger representation to this subgroup. More generally, this is an exampleof a semi-direct product N oK with N commutative. In such cases irreduciblerepresentations can be characterized in terms of the action of K on irreduciblerepresentations of N , and the irreducible representations of certain subgroupsof K.

203

The reader should be warned that much of the material included in thischapter is not well-motivated by its applications to non-relativistic quantummechanics, where it is not obviously needed. The motivation is rather providedby the more complicated case of relativistic quantum field theory, but it seemsworthwhile to first see how things work in a simpler context. In particular, thediscussion of representations of N oK for M commutative is motivated by thecase of the Poincare group (see chapter 40), and that of intertwining operatorsby the case of symmetry groups acting on quantum fields (see chapter 36).

18.1 Intertwining operators and the metaplecticrepresentation

For a general semi-direct product N oK with non-commutative N , the repre-sentation theory can be quite complicated. For the Jacobi group case though, itturns out that things simplify dramatically because of the Stone-von Neumanntheorem which says that, up to unitary equivalence, we only have one irreduciblerepresentation of N = H2d+1.

In the general case, recall that for each k ∈ K the definition of the semi-directproduct comes with an automorphism φk : N → N satisfying φk1k2

= φk1φk2

.Given a representation π of N , for each k we can define a new representationπk of N by first acting with φk:

πk(n) = π(φk(n))

In the special case of the Heisenberg group and Schrodinger representation ΓS ,we can do this for each k ∈ K = Sp(2d,R), defining a new represention by

ΓS,k(n) = ΓS(φk(n))

The Stone-von Neumann theorem assures us that these must all be unitarilyequivalent, so there must exist unitary operators Uk satisfying

ΓS,k = UkΓSU−1k = ΓS(φk(n)) (18.1)

Operators like this that relate two representations are called “intertwiningoperators”.

Definition (Intertwining operator). If (π1, V1), (π2, V2) are two representationsof a group G, an intertwining operator between these two representations is anoperator U such that

π2(g)U = Uπ1(g) ∀g ∈ G

In our case V1 = V2 is the Schrodinger representation state space H and Uk :H → H is an intertwining operator between ΓS and ΓS,k for each k ∈ Sp(2d,R).Since

ΓS,k1k2= Uk1k2

ΓSU−1k1k2

204

the Uk should satisfy the group homomorphism property

Uk1k2= Uk1

Uk2

and give us a representation of the group Sp(2d,R) on H. This is what weexpect on general principles: a group action on the classical phase space afterquantization becomes a unitary representation on the quantum state space.

The problem with this argument is that the Uk are not uniquely defined.Schur’s lemma tells us that since the representation on H is irreducible, theoperators commuting with the representation operators are just the complexscalars. These give a phase ambiguity in the definition of the unitary operatorsUk, which then give a representation of Sp(2d,R) on H only up to a phase, i.e.

Uk1k2= Uk1

Uk2eiϕ(k1,k2)

for some real-valued function ϕ of pairs of group elements. In terms of corre-sponding Lie algebra representation operators U ′L, this ambiguity appears as anunknown constant times the identity operator.

The question then arises whether the phases of the Uk can be chosen soas to satisfy the homomorphism property (i.e. can one choose phases so thatϕ(k1, k2) = N2π for N integral?). It turns out that one can not quite do this,needing to allow N to be half-integral, so one gets the homomorphism propertyup to a sign. Just as in the SO(d) case where a similar sign ambiguity showedthe need to go to a double-cover Spin(d) to get a true representation, here onealso needs to go to a double cover of Sp(2d,R), called the metaplectic groupMp(2d,R). The nature of this sign ambiguity and double cover is subtle, fordetails see [43] or [27]. In section 18.2.1 we will show by computation one aspectof the double cover.

Since this is just a sign ambiguity, it does not appear infinitesimally: one canchoose the ambiguous constants in the Lie algebra representation operators sothat the Lie algebra homomorphism property is satisfied. However, this will nolonger be true for infinite dimensional phase spaces, a situation that is describedas an “anomaly” in the symmetry. This phenomenon will be examined in moredetail in chapter 37.

In the finite dimensional case, we can construct the Uk explicitly, by expo-nentiating the Lie algebra representation operators U ′L, which are constructedby quantization of the homogeneous order-two polynomials that give the Liealgebra sp(2d,R).

This quantization takes

µL ∈ sp(2d,R)→ U ′L

where µL is the polynomial corresponding to the matrix L to U ′L, which is thecorresponding polynomial in operators Qj , Pj , as determined by equation 15.1.This choice will satisfy the Lie algebra homomorphism property

[U ′L1, U ′L2

] = U ′[L1,L2] (18.2)

205

The Lie algebra relation (see 14.14)

µL,(

qp

) = LT

(qp

)(18.3)

becomes after quantization

[U ′L,

(QP

)] = LT

(QP

)(18.4)

Exponentiating this U ′L will give us our Uk, and thus the operators we want.Note that if we only need to satisfy equation 18.4 the U ′L can be changed by

a constant times the identity operator, but such a change would be inconsistentwith equation 18.2 (and thus called an “anomaly”). Equation 18.4 could bewritten

[U ′L,Γ′S(X)] = Γ′S(L ·X) (18.5)

where

L ·X =d

dtφetL(X)|t=0

It is the infinitesimal expression of the intertwining property 18.1.

18.2 Some examples

As a balance to the abstract discussion so far in this chapter, in this sectionwe’ll work out a couple of the simplest possible examples in great detail. Theseexamples will also make clear the conventions being chosen, and show the basicstructure of what the quadratic operators corresponding to symmetries look like,a structure that will reappear in the much more complicated infinite-dimensionalquantum field theory examples we will come to later.

18.2.1 The SO(2) action on the d = 1 phase space

In the case d = 1 one has elements g ∈ SO(2) ⊂ Sp(2,R) acting on cqq+ cpp ∈M by (

cqcp

)→ g

(cqcp

)=


)(cqcp

)so

g = eθL

where

L =

(0 1−1 0

)To find the intertwining operators, we first find the quadratic function µL

in q, p that satisfies

µL,(qp

) = LT

(qp

)=

(−pq

)206

By equation 14.5 this is

µL =1

2

(q p

)(1 00 1

)(qp

)=

1

2(q2 + p2)

Quantizing µL using the Schrodinger representation Γ′S , one has a unitaryLie algebra representation U ′ of so(2) with

U ′L = − i2

(Q2 + P 2)

satisfying

[U ′L,

(QP

)] =

(−PQ

)(18.6)

and intertwining operators

Ug = eθU′L = e−i

θ2 (Q2+P 2)

These give a representation of SO(2) only up to a sign. To see the problem,consider the state ψ(q) ⊂ H = L2(R) given by

ψ(q) = e−q2

2

One has

(Q2 + P 2)ψ(q) = (q2 − d2

dq2)ψ(q) = ψ(q)

so ψ(q) is an eigenvector of Q2 +P 2 with eigenvalue 1. As one goes around thegroup SO(2) once (taking θ from 0 to 2π), the representation acts by a phasethat only goes from 0 to π, demonstrating the same problem that occurs in thecase of the spinor representation.

Conjugating the Heisenberg Lie algebra representation operators by the uni-tary operators Ug intertwines the representations corresponding to rotations ofthe phase space plane by an angle θ.

e−iθ2 (Q2+P 2)

(QP

)eiθ2 (Q2+P 2) =


)(QP

)(18.7)

Note that this is a quite different calculation than in the spin case where wealso constructed a double cover of SO(2). Despite the quite different context(rotations acting on an infinite dimensional state space), again one sees thedouble cover here, as either Ug or −Ug will give the same rotation.

This example will be studied in much greater detail when we get to thetheory of the quantum harmonic oscillator in chapter 20. Note that the SO(2)group action here inherently requires using both coordinate and momentumvariables, it is not a symmetry that can be seen just by looking at the problemin configuration space.

207

18.2.2 The SO(2) action by rotations of the plane for d = 2

In the case d = 2 there is a another example of an SO(2) group which is asubgroup of the symplectic group, here Sp(4,R). This is the group of rotationsof the configuration space R2, with a simultaneous rotation of the momentumspace, leaving invariant the Poisson bracket. The group SO(2) acts on cq1q1 +cq2q2 + cp1p1 + cp2p2 ∈M by

cq1cq2cp1

cp2

→ g

cq1cq2cp1

cp2

=

cos θ − sin θ 0 0sin θ cos θ 0 0

0 0 cos θ − sin θ0 0 sin θ cos θ

cq1cq2cp1

cp2

so g = eθL where L ∈ sp(4,R) is given by

L =

0 −1 0 01 0 0 00 0 0 −10 0 1 0

L acts on phase space coordinate functions by

q1

q2

p1

p2

→ LT

q1

q2

p1

p2

=

q2

−q1

p2

−p1

By equation 14.14, with

A =

(0 −11 0

), B = C = 0

the quadratic function µL that satisfies

µL,

q1

q2

p1

p2

= LT

q1

q2

p1

p2

=

0 1 0 0−1 0 0 00 0 0 10 0 −1 0

q1

q2

p1

p2

=

q2

−q1

p2

−p1

is

µL = −q ·(

0 −11 0

)p = q1p2 − q2p1

This is just the formula for the angular momentum corresponding to rotationabout an axis perpendicular to the q1 − q2 plane

l = q1p2 − q2p1

Quantization gives a representation of the Lie algebra so(2) with

U ′L = −i(Q1P2 −Q2P1)

208

satisfying

[U ′L,

(Q1

Q2

)] =

(Q2

−Q1

), [U ′L,

(P1

P2

)] =

(P2

−P1

)Exponentiating gives a representation of SO(2)

UeθL = e−iθ(Q1P2−Q2P1)

with conjugation by UeθL rotating linear combinations of the Q1, Q2 (or theP1, P2) each by an angle θ.

UeθL(cq1Q1 + cq2Q2)U−1eθL

= c′q1Q1 + c′q2Q2

where (c′q1c′q2

)=


)(cq1cq2

)Note that for this SO(2) the double cover is trivial. As far as this subgroup

of Sp(4,R) is concerned, there is no need to consider the double cover Mp(4,R)to get a well-defined representation.

Replacing the matrix L by (A 00 A

)for A any real 2 by 2 matrix

A =

(a11 a12

a21 a22

)we get an action of the group GL(2,R) ⊂ Sp(4,R) onM, and after quantizationa Lie algebra representation

U ′A = i(Q1 Q2

)(a11 a12

a21 a22

)(P1

P2

)which will satisfy

[U ′A,

(Q1

Q2

)] = −A

(Q1

Q2

), [U ′A,

(P1

P2

)] = AT

(P1

P2

)Note that the action of A on the momentum operators is the dual of the actionon the position operators. Only in the case of an orthogonal action (the SO(2)earlier) are these the same, with AT = −A.

18.3 Representations of N oK, N commutative

The representation theory of semi-direct productsNoK will in general be rathercomplicated. However, when N is commutative things simplify considerably,and in this section we’ll survey some of the general features of this case. The

209

special cases of the Euclidean groups in 2 and 3 dimensions were covered inchapter 17 and the Poincare group case will be discussed in chapter 40.

For a general commutative group N , one does not have the simplifying fea-ture of the Heisenberg group, the uniqueness of its irreducible representation.For commuative groups on the other hand, while there are many irreduciblerepresentations, they are all one-dimensional. As a result, the set of represen-tations of N acquires its own group structure, also commutative, and one candefine

Definition (Character group). For N a commutative group, let N be the set ofcharacters of N , i.e. functions

α : N → C

that satisfy the homomorphism property

α(n1n2) = α(n1)α(n2)

The elements of N form a group, with multiplication

(α1α2)(n) = α1(n)α2(n)

We only will actually need the case N = Rd, where we have already seenthat the differentiable irreducible representations are one-dimensional and givenby

αp(a) = eip·a

where a ∈ N . So the character group in this case is N = Rd, with elementslabeled by the vector p.

For a semidirect product N oK, we will have an automorphism φk of N foreach k ∈ K. From this action on N , we get an induced action on functions onN , in particular on elements of N , by

φk : α ∈ N → φk(α) ∈ N

where φk(α) is the element of N satisfying

φk(α)(n) = α(φ−1k (n))

For the case of N = Rd, we have

φk(αp)(a) = eip·φ−1k (a) = ei(((φ

−1k )T (p))·a

soφk(αp) = α(φ−1

k )T (p)

When K acts by orthogonal transformations on N = Rd, φTk = φ−1k so

φk(αp) = αφk(p)

210

To analyze representations (π, V ) of N o K, one can begin by restrictingattention to the N action, decomposing V into subspaces Vα where N actsaccording to α. v ∈ V is in the subspace Vα when

π(n,1)v = α(n)v

Acting by K will take this subspace to another one according to

Theorem.v ∈ Vα =⇒ π(0, k)v ∈ Vφk(α)

Proof. Using the definition of the semi-direct product in chapter 16 one canshow that the group multiplication satisfies

(0, k−1)(n,1)(0, k) = (φk−1(n),1)

Using this, one has

π(n,1)π(0, k)v =π(0, k)π(0, k−1)π(n,1)π(0, k)v

=π(0, k)π(φk−1(n),1)v

=π(0, k)α(φk−1(n))v

=φk(α)(n)π(0, k)v

For each α ∈ N one can look at its orbit under the action of K by φk, whichwill give a subset Oα ⊂ N . From the above theorem, we see that if Vα 6= 0,then we will also have Vβ 6= 0 for β ∈ Oα, so one piece of information thatcharacterizes a representation V is the set of orbits one gets in this way.

α also defines a subgroup Kα ⊂ K consisting of group elements whose actionon N leaves α invariant:

Definition (Stabilizer group or little group). The subgroup Kα ⊂ K of elementsk ∈ K such that

φk(α) = α

for a given α ∈ N is called the stabilizer subgroup (by mathematicians) or littlesubgroup (by physicists).

The group Kα will act on the subspace Vα, and this representation of Kα isa second piece of information one can use to characterize a representation.

In the case of the Euclidean group E(2) we found that the non-zero orbitsOα were circles and the groups Kα were trivial. For E(3), the non-zero orbitswere spheres, with Kα an SO(2) subgroup of SO(3) (one that varies with α). Inthese cases we found that our construction of representations of E(2) or E(3) onspaces of solutions of the single-component Schrodinger equation correspondedunder Fourier transform to a representation on functions on the orbits Oα.We also found in the E(3) case that using multiple-component wavefunctions

211

gave new representations corresponding to a choice of orbit Oα and a choiceof irreducible representation of Kα = SO(2). We did not show this, but thisconstruction gives an irreducible representation when a single orbit Oα occurs(with a transitive K action), with an irreducible representation of Kα on Vα.

We will not further pursue the general theory here, but one can show thatdistinct irreducible representations of N o K will occur for each choice of anorbit Oα and an irreducible representation of Kα. One way to construct theserepresentations is as the solution space of an appropriate wave-equation, withthe wave-equation corresponding to the eigenvalue equation for a Casimir oper-ator. In general, other “subsidiary conditions” then need be imposed to pick outa subspace of solutions that give an representation of N oK, this correspondsto the existence of other Casimir operators.


For more on representations of semi-direct products, see section 3.8 of [65],chapter 5 of [73], [8], and [28]. The general theory was developed by Mackeyduring the late 1940s and 1950s, and his lecture notes on representation theory[45] are a good source for the details of this. The point of view taken here,that emphasizes constructing representations as solutions spaces of differentialequations, where the differential operators are Casimir operators, is explainedin more detail in [35].

The conventional derivation found in most physics textbooks of the opera-tors U ′L coming from an infinitesimal group action uses Lagrangian methods andNoether’s theorem. The purely Hamiltonian method used here treats configu-ration and momentum variables on the same footing, and is useful especiallyin the case of group actions that mix them. For another treatment of theseoperators along the lines of this chapter, see section 14 of [27].

The issue of the phase factor in the intertwining operators and the metaplec-tic double cover will be discussed in later chapters using a different realizationof the Heisenberg Lie algebra representation. For a discussion of this in termsof the Schrodinger representation, see part I of [43].

212

Chapter 19

Central Potentials and theHydrogen Atom

When the Hamiltonian function is invariant under rotations, we then expecteigenspaces of the corresponding Hamiltonian operator to carry representationsof SO(3). These spaces of eigenfunctions of a given energy break up into ir-reducible representations of SO(3), and we have seen that these are labeledby a integer l = 0, 1, 2, . . . and have dimension 2l + 1. One can use this tofind properties of the solutions of the Schrodinger equation whenever one hasa rotation-invariant potential energy. We will work out what happens for thecase of the Coulomb potential describing the hydrogen atom. This specific caseis exactly solvable because it has a second not-so-obvious SO(3) symmetry, inaddition to the one coming from rotations of R3.

19.1 Quantum particle in a central potential

In classical physics, to describe not free particles, but particles experiencingsome sort of force, one just needs to add a “potential energy” term to thekinetic energy term in the expression for the energy (the Hamiltonian function).In one dimension, for potential energies that just depend on position, one has

h =p2

2m+ V (q)

for some function V (q). In the physical case of three dimensions, this will be

h =1

2m(p2

1 + p22 + p2

3) + V (q1, q2, q3)

Quantizing and using the Schrodinger representation, the Hamiltonian op-

213

erator for a particle moving in a potential V (q1, q2, q3) will be

H =1

2m(P 2

1 + P 22 + P 3

3 ) + V (Q1, Q2, Q3)

=−~2

2m(∂2

∂q21

+∂2

∂q22

+∂2

∂q23

) + V (q1, q2, q3)

=−~2

2m∆ + V (q1, q2, q3)

We will be interested in so-called “central potentials”, potential functions thatare functions only of q2

1 + q22 + q2

3 , and thus only depend upon r, the radialdistance to the origin. For such V , both terms in the Hamiltonian will beSO(3) invariant, and eigenspaces of H will be representations of SO(3).

Using the expressions for the angular momentum operators in spherical co-ordinates derived in chapter 8 (including equation 8.8 for the Casimir operatorL2), one can show that the Laplacian has the following expression in sphericalcoordinates

∆ =∂2

∂r2+

2

r

∂

∂r− 1

r2L2

The Casmir operator L2 has eigenvalues l(l + 1) on irreducible representationsof dimension 2l+ 1 (integral spin l). So, restricted to such an irreducible repre-sentation, we have

∆ =∂2

∂r2+

2

r

∂

∂r− l(l + 1)

r2

To solve the Schrodinger equation, we want to find the eigenfunctions ofH. The space of eigenfunctions of energy E will be a sum of of irreduciblerepresentations of SO(3), with the SO(3) acting on the angular coordinatesof the wavefunctions, leaving the radial coordinate invariant. We have seen inchapter 8 that such representations on functions of angular coordinates can beexplicitly expressed in terms of the spherical harmonic functions Y ml (θ, φ). So,to find eigenfunctions of the Hamiltonian

H = − ~2

2m∆ + V (r)

we want to find functions glE(r) depending on l = 0, 1, 2, . . . and the energyeigenvalue E satisfying

(−~2

2m(d2

dr2+

2

r

d

dr− l(l + 1)

r2) + V (r))glE(r) = EglE(r)

Given such a glE(r) we will have

HglE(r)Y ml (θ, φ) = EglE(r)Y ml (θ, φ)

and theψ(r, θ, φ) = glE(r)Y ml (θ, φ)

214

will span a 2l+ 1 dimensional (since m = −l,−l+ 1, . . . , l−1, l) space of energyeigenfunctions for H of eigenvalue E.

For a general potential function V (r), exact solutions for the eigenvalues Eand corresponding functions glE(r) cannot be found in closed form. One specialcase where we can find such solutions is for the three-dimensional harmonic os-cillator, where V (r) = 1

2mω2r2. These are much more easily found though using

the creation and annihilation operator techniques to be discussed in chapter 20.

The other well-known and physically very important case is the case of a 1r

potential, called the Coulomb potential. This describes a light charged particlemoving in the potential due to the electric field of a much heavier chargedparticle, a situation that corresponds closely to that of a hydrogen atom. Inthis case we have

V = −e2

r

where e is the charge of the electron, so we are looking for solutions to

(−~2

2m(d2

dr2+

2

r

d

dr− l(l + 1)

r2)− e2

r)glE(r) = EglE(r)

Since having

d2

dr2(rg) = Erg

is equivalent to

(d2

dr2+

2

r)g = Eg

for any function g, glE(r) will satisfy

(−~2

2m(d2

dr2− l(l + 1)

r2)− e2

r)rglE(r) = ErglE(r)

The solutions to this equation can be found through a rather elaborate pro-cess described in most quantum mechanics textbooks, which involves lookingfor a power series solution. For E ≥ 0 there are non-normalizable solutions thatdescribe scattering phenomena that we won’t study here. For E < 0 solutionscorrespond to an integer n = 1, 2, 3, . . ., with n ≥ l+ 1. So, for each n we get nsolutions, with l = 0, 1, 2, . . . , n− 1, all with the same energy

En = − me4

2~2n2

A plot of the different energy eigenstates looks like this:

215

The degeneracy in the energy values leads one to suspect that there is someextra group action in the problem commuting with the Hamiltonian. If so, theeigenspaces of energy eigenfunctions will come in irreducible representationsof some larger group than SO(3). If the representation of the larger groupis reducible when one restricts to the SO(3) subgroup, giving n copies of ourSO(3) representation of spin l, that would explain the pattern observed here.In the next section we will see that this is the case, and there use representationtheory to derive the above formula for En.

We won’t go through the process of showing how to explicitly find the func-tions glEn(r) but just quote the result. Setting

a0 =~2

me2

(this has dimensions of length and is known as the “Bohr radius”), and defining

216

gnl(r) = glEn(r) the solutions are of the form

gnl(r) ∝ e−rna0 (

2r

na0)lL2l+1

n+l (2r

na0)

where the L2l+1n+l are certain polynomials known as associated Laguerre polyno-

mials.So, finally, we have found energy eigenfunctions

ψnlm(r, θ, φ) = gnl(r)Yml (θ, φ)

for

n = 1, 2, . . .

l = 0, 1, . . . , n− 1

m = −l,−l + 1, . . . , l − 1, l

The first few of these, properly normalized, are

ψ100 =1√πa3

0

e−ra0

(called the 1S state, “S” meaning l = 0)

ψ200 =1√

8πa30

(1− r

2a0)e−

r2a0

(called the 2S state), and the three dimensional l = 1 (called 2P , “P” meaningl = 1) states with basis elements

ψ211 = − 1

8√πa3

0

r

a0e−

r2a0 sin θeiφ

ψ211 = − 1

4√

2πa30

r

a0e−

r2a0 cos θ

ψ21−1 =1

8√πa3

0

r

a0e−

r2a0 sin θe−iφ

19.2 so(4) symmetry and the Coulomb potential

The Coulomb potential problem is very special in that it has an additionalsymmetry, of a non-obvious kind. This symmetry appears even in the classi-cal problem, where it is responsible for the relatively simple solution one canfind to the essentially identical Kepler problem. This is the problem of findingthe classical trajectories for bodies orbiting around a central object exerting agravitational force, which also has a 1

r potential. Kepler’s second law for such

217

motion comes from conservation of angular momentum, which corresponds tothe Poisson bracket relation

lj , h = 0

Here we’ll take the Coulomb version of the Hamiltonian that we need for thehydrogen atom problem

h =1

2m|p|2 − e2

r

One can read the relation lj , h = 0 in two ways:

• The hamiltonian h is invariant under the action of the group (SO(3))whose infinitesimal generators are lj .

• The components of the angular momentum (lj) are invariant under theaction of the group (R of time translations) whose infinitesimal generatoris h, so the angular momentum is a conserved quantity.

Kepler’s first and third laws have a different origin, coming from the existenceof a new conserved quantity for this special choice of Hamiltonian. This quantityis, like the angular momentum, a vector, often called the Lenz (or sometimesRunge-Lenz, or even Laplace-Runge-Lenz) vector.

Definition (Lenz vector). The Lenz vector is the vector-valued function on thephase space R6 given by

w =1

m(l× p) + e2 q

|q|

Simple manipulations of the cross-product show that one has

l ·w = 0

We won’t here explicitly calculate the various Poisson brackets involving thecomponents wj of w, since this is a long and unilluminating calculation, butwill just quote the results, which are

•wj , h = 0

This says that, like the angular momentum, the vector with componentswj is a conserved quantity under time evolution of the system, and itscomponents generate symmetries of the classical system.

•lj , wk = εjklwl

These relations say that the generators of the SO(3) symmetry act on wjthe way one would expect, since wj is a vector.

218

•wj , wk = εjklll(

−2h

m)

This is the most surprising relation, and it has no simple geometricalexplanation (although one can change variables in the problem to try andgive it one). It expresses a highly non-trivial relationship between the twosets of symmetries generated by the vectors l,w and the Hamiltonian h.

The wj are cubic in the q and p variables, so one would expect that theGroenewold-van Hove no-go theorem would tell one that there is no consistentway to quantize this system by finding operators Wj corresponding to the wjthat would satisfy the commutation relations corresponding to these Poissonbrackets. It turns out though that this can be done, although not for functionsdefined over the entire phase-space. One gets around the no-go theorem bydoing something that only works when the Hamiltonian h is negative (we’ll betaking a square root of −h).

The choice of operators Wj that works is

W =1

2m(L×P−P× L) + e2 Q

|Q|2

where the last term is the operator of multiplication by e2qj/|q|2. By elaborateand unenlightening computations the Wj can be shown to satisfy the commu-tation relations corresponding to the Poisson bracket relations of the wj :

[Wj , H] = 0

[Lj ,Wk] = i~εjklWl

[Wj ,Wk] = i~εjklLl(−2

mH)

as well asL ·W = W · L = 0

The first of these shows that energy eigenstates will be preserved not just bythe angular momentum operators Lj , but by a new set of non-trivial operators,the Wj , so will be representations of a larger Lie algebra thatn so(3).

In addition, one has the following relation between W 2, H and the Casimiroperator L2

W 2 = e41 +2

mH(L2 + ~21)

and it is this which will allow us to find the eigenvalues of H, since we knowthose for L2, and can find those of W 2 by changing variables to identify a secondso(3) Lie algebra.

To do this, first change normalization by defining

K =

√−m2E

W

219

where E is the eigenvalue of the Hamiltonian that we are trying to solve for.Note that it is at this point that we violate the conditions of the no-go theorem,since we must have E < 0 to get a K with the right properties, and this restrictsthe validity of our calculations to a subset of the energy spectrum. For E > 0one can proceed in a similar way, but the Lie algebra one gets is different (so(3, 1)instead of so(4)).

One then has the following relation between operators

2H(K2 + L2 + ~21) = −me41

and the following commutation relations

[Lj , Lk] = i~εjklLl

[Lj ,Kk] = i~εjklKl

[Kj ,Kk] = i~εjklLlDefining

M =1

2(L + K), N =

1

2(L−K)

one has[Mj ,Mk] = i~εjklMl

[Nj , Nk] = i~εjklNl[Mj , Nk] = 0

This shows that we have two commuting copies of so(3) acting on states, spannedrespectively by the Mj and Nj , with two corresponding Casimir operators M2

and N2.Using the fact that

L ·K = K · L = 0

one finds thatM2 = N2

Recall from our discussion of rotations in three dimensions that representa-tions of so(3) = su(2) correspond to representations of Spin(3) = SU(2), thedouble cover of SO(3) and the irreducible ones have dimension 2l + 1, with lhalf-integral. Only for l integral does one get representations of SO(3), and itis these that occur in the SO(3) representation on functions on R3. For four di-mensions, we found that Spin(4), the double cover of SO(4), is SU(2)×SU(2),and one thus has spin(4) = so(4) = su(2)×su(2) = so(3)×so(3). This is exactlythe Lie algebra we have found here, so one can think of the Coulomb problem ashaving an so(4) symmetry. The representations that will occur can include thehalf-integral ones, since neither so(3) is the so(3) of physical rotations in 3-space(those are generated by L = M + N, which will have integral eigenvalues of l).

The relation between the Hamiltonian and the Casimir operators M2 andN2 is

2H(K2 + L2 + ~21) = 2H(2M2 + 2N2 + ~21) = 2H(4M2 + ~21) = −me41

220

On irreducible representations of so(3) of spin µ, we will have

M2 = µ(µ+ 1)1

for some half-integral µ, so we get the following equation for the energy eigen-values

E = − −me4

2~2(4µ(µ+ 1) + 1)= − −me4

2~2(2µ+ 1)2

Letting n = 2µ + 1, for µ = 0, 12 , 1, . . . we get n = 1, 2, 3, . . . and precisely the

same equation for the eigenvalues described earlier

En = − me4

2~2n2

It is not hard to show that the irreducible representations of a product likeso(3) × so(3) are just tensor products of irreducibles, and in this case the twofactors of the product are identical due to the equality of the Casimirs M2 = N2.The dimension of the so(3)×so(3) irreducibles is thus (2µ+1)2 = n2, explainingthe multiplicity of states one finds at energy eigenvalue En.

19.3 The hydrogen atom

The Coulomb potential problem provides a good description of the quantumphysics of the hydrogen atom, but it is missing an important feature of thatsystem, the fact that electrons are spin 1

2 systems. To describe this, one reallyneeds to take as space of states two-component wavefunctions

|ψ〉 =

(ψ1(q)ψ2(q)

)(or, equivalently, replace our state space H of wavefunctions by the tensor prod-uct H⊗C2) in a way that we will examine in detail in chapter 31.

The Hamiltonian operator for the hydrogen atom acts trivially on the C2

factor, so the only effect of the additional wavefunction component is to doublethe number of energy eigenstates at each energy. Electrons are fermions, soantisymmetry of multi-particle wavefunctions implies the Pauli principle thatstates can only be occupied by a single particle. As a result, one finds that whenadding electrons to an atom described by the Coulomb potential problem, thefirst two fill up the lowest Coulomb energy eigenstate (the ψ100 or 1S state at n =1), the next eight fill up the n = 2 states ( two each for ψ200, ψ211, ψ210, ψ21−1),etc. This goes a long ways towards explaining the structure of the periodic tableof elements.

When one puts a hydrogen atom in a constant magnetic field B, the Hamil-tonian acquires a term that acts only on the C2 factor, of the form

2e

mcB · σ

221

This is exactly the sort of Hamiltonian we began our study of quantum mechan-ics with for a simple two-state system. It causes a shift in energy eigenvaluesproportional to ±|B| for the two different components of the wavefunction, andthe observation of this energy splitting makes clear the necessity of treating theelectron using the two-component formalism.


This is a standard topic in all quantum mechanics books. For example, seechapters 12 and 13 of [63]. The so(4) calculation is not in [63], but is in someof the other such textbooks, a good example is chapter 7 of [4]. For extensivediscussion of the symmetries of the 1

r potential problem, see [26] or [28].

222

Chapter 20

The Harmonic Oscillator

In this chapter we’ll begin the study of the most important exactly solvablephysical system, the harmonic oscillator. Later chapters will discuss extensionsof the methods developed here to the case of fermionic oscillators, as well as freequantum field theories, which are harmonic oscillator systems with an infinitenumber of degrees of freedom.

For a finite number of degrees of freedom, the Stone-von Neumann theo-rem tells us that there is essentially just one way to non-trivially represent the(exponentiated) Heisenberg commutation relations as operators on a quantummechanical state space. We have seen two unitarily equivalent constructionsof these operators: the Schrodinger representation in terms of functions on ei-ther coordinate space or momentum space. It turns out that there is anotherclass of quite different constructions of these operators, one that depends uponintroducing complex coordinates on phase space and then using properties ofholomorphic functions. We’ll refer to this as the Bargmann-Fock representation,although quite a few mathematicians have had their name attached to it for onegood reason or another (some of the other names one sees are Friedrichs, Segal,Shale, Weil, as well as the descriptive terms “holomorphic” and “oscillator”).

Physically the importance of this representation is that it diagonalizes theHamiltonian operator for a fundamental sort of quantum system: the harmonicoscillator. In the Bargmann-Fock representation the energy eigenstates of sucha system are the simplest states and energy eigenvalues are just integers. Theseintegers label the irreducible representations of the U(1) symmetry generatedby the Hamiltonian, and they can be interpreted as counting the number of“quanta” in the system. It is the ubiquity of this example that justifies the“quantum” in “quantum mechanics”. The operators on the state space can besimply understood in terms of basic operators called annihilation and creationoperators which increase or decrease by one the number of quanta.

223

20.1 The harmonic oscillator with one degree offreedom

An even simpler case of a particle in a potential than the Coulomb potential ofthe last chapter is the case of V (q) quadratic in q. This is also the lowest-orderapproximation when one studies motion near a local minimum of an arbitraryV (q), expanding V (q) in a power series around this point. We’ll write this as

h =p2

2m+

1

2mω2q2

with coefficients chosen so as to make ω the angular frequency of periodic motionof the classical trajectories. These satisfy Hamilton’s equations

p = −∂V∂q

= mω2q, q =p

m

so

q = −ω2q

which will have solutions with periodic motion of angular frequency ω. Thesesolutions can be written as

q(t) = c+eiωt + c−e

−iωt

for c+, c− ∈ C where, since q(t) must be real, we have c− = c+. The space ofsolutions of the equation of motion is thus two real-dimensional, and abstractlyone can think of this as the phase space of the system.

More conventionally, one can parametrize the phase space by initial valuesthat determine the classical trajectories, for instance by the position q(0) andmomentum p(0) at an initial time t(0). Since

p(t) = mq = mc+iωeiωt −mc−iωe−iωt = imω(c+e

iωt − c+e−iωt)

we have

q(0) = c+ + c− = 2 Re(c+), p(0) = imω(c+ − c−) = 2mω Im(c+)

so

c+ =1

2q(0) + i

1

2mωp(0)

The classical phase space trajectories are

q(t) = (1

2q(0) + i

1

2mωp(0))eiωt + (

1

2q(0)− i 1

2mωp(0))e−iωt

p(t) = (imω

2q(0)− 1

2p(0))eiωt + (

−imω2

q(0) +1

2p(0))e−iωt

224

Instead of using two real coordinates to describe points in the phase space(and having to introduce a reality condition when using complex exponentials),one can instead use a single complex coordinate

z(t) =1√2

(q(t)− i

mωp(t))

Then the equation of motion is a first-order rather than second-order differentialequation

z = iωz

with solutionsz(t) = z(0)eiωt (20.1)

The classical trajectories are then realized as complex functions of t, and paramet-rized by the complex number

z(0) =1√2

(q(0)− i

mωp(0))

Since the Hamiltonian is just quadratic in the p and q, we have seen that wecan construct the corresponding quantum operator uniquely using the Schroding-er representation. For H = L2(R) we have a Hamiltonian operator

H =P 2

2m+

1

2mω2Q2 = − ~2

2m

d2

dq2+

1

2mω2q2

To find solutions of the Schrodinger equation, as with the free particle, oneproceeds by first solving for eigenvectors of H with eigenvalue E, which meansfinding solutions to

HψE = (− ~2

2m

d2

dq2+

1

2mω2q2)ψE = EψE

Solutions to the Schrodinger equation will then be linear combinations of thefunctions

ψE(q)e−i~Et

Standard but somewhat intricate methods for solving differential equationslike this show that one gets solutions for E = En = (n+ 1

2 )~ω, n a non-negativeinteger, and the normalized solution for a given n (which we’ll denote ψn) willbe

ψn(q) = (mω

π~22n(n!)2)

14Hn(

√mω

~q)e−

mω2~ q

2

(20.2)

where Hn is a family of polynomials called the Hermite polynomials. Theψn provide an orthonormal basis for H (one does not need to consider non-normalizable wavefunctions as in the free particle case), so any initial wavefunc-tion ψ(q, 0) can be written in the form

ψ(q, 0) =

∞∑n=0

cnψn(q)

225

with

cn =

∫ +∞

−∞ψn(q)ψ(q, 0)dq

(note that the ψn are real-valued). At later times, the wavefunction will be

ψ(q, t) =

∞∑n=0

cnψn(q)e−i~Ent =

∞∑n=0

cnψn(q)e−i(n+ 12 )ωt

20.2 Creation and annihilation operators

It turns out that there is a quite easy method which allows one to explicitly findeigenfunctions and eigenvalues of the harmonic oscillator Hamiltonian (althoughit’s harder to show it gives all of them). This also leads to a new representationof the Heisenberg group (of course unitarily equivalent to the Schrodinger oneby the Stone-von Neumann theorem). Instead of working with the self-adjointoperators Q and P that satisfy the commutation relation

[Q,P ] = i~1

we define

a =

√mω

2~Q+ i

√1

2mω~P, a† =

√mω

2~Q− i

√1

2mω~P

which satisfy the commutation relation

[a, a†] = 1

To simplify calculations, from now on we will set ~ = m = ω = 1. Thiscorresponds to a specific choice of units for energy, distance and time, and thegeneral case of arbitrary constants can be recovered by rescaling the results ofour calculations. So, now

a =1√2

(Q+ iP ), a† =1√2

(Q− iP )

and

Q =1√2

(a+ a†), P =1

i√

2(a− a†)

The Hamiltonian operator is

H =1

2(Q2 + P 2) =

1

2(1

2(a+ a†)2 − 1

2(a− a†)2)

=1

2(aa† + a†a)

=a†a+1

2

226

Up to the constant 12 , H is given by the operator

N = a†a

which satisfies the commutation relations

[N, a] = [a†a, a] = a†[a, a] + [a†, a]a = −a

and

[N, a†] = a†

If |c〉 is a normalized eigenvector of N with eigenvalue c, one has

c = 〈c|a†a|c〉 = |a|c〉|2 ≥ 0

so eigenvalues of N must be non-negative. Using the commutation relations ofN, a, a† gives

Na|c〉 = ([N, a] + aN)|c〉 = a(N − 1)|c〉 = (c− 1)a|c〉

and

Na†|c〉 = ([N, a†] + a†N)|c〉 = a†(N + 1)|c〉 = (c+ 1)a†|c〉

This shows that a|c〉 will have eigenvalue c − 1 for N , and a normalized eigen-function for N will be

|c− 1〉 =1√ca|c〉

Similarly, since

|a†|c〉|2 = 〈c|aa†|c〉 = 〈c|(N + 1)|c〉 = c+ 1

we have

|c+ 1〉 =1√c+ 1

a†|c〉

We can find eigenfunctions for H by first solving

a|0〉 = 0

for |0〉 (the lowest energy or “vacuum” state) which will have energy eigenvalue12 , then acting by a† n-times on |0〉 to get states with energy eigenvalue n+ 1

2 .The equation for |0〉 is thus

a|0〉 =1√2

(Q+ iP )ψ0(q) =1√2

(q +d

dq)ψ0(q) = 0

One can check that all solutions to this are all of the form

ψ0(q) = Ce−q2

2

227

so there is a unique normalized lowest-energy eigenfunction

ψ0(q) =1

π14

e−q2

2

The rest of the energy eigenfunctions can be found by computing

|n〉 =a†√n· · · a

†√

2

a†√1|0〉 =

1

π14 2

n2

√n!

(q − d

dq)ne−

q2

2

which (after putting back in constants and consulting the definition of a Hermitepolynomial) can be shown to give the eigenfunctions claimed earlier in equation20.2.

In the physical interpretation of this quantum system, the state |n〉, withenergy ~ω(n + 1

2 ) is thought of as a state describing n “quanta”. The state|0〉 is the “vacuum state” with zero quanta, but still carrying a “zero-point”energy of 1

2~ω. The operators a† and a have somewhat similar properties tothe raising and lowering operators we used for SU(2) but their commutatoris different (just the identity operator), leading to simpler behavior. In thiscase they are called “creation” and “annihilation” operators respectively, dueto the way they change the number of quanta. The relation of such quanta tophysical particles like the photon is that quantization of the electromagnetic fieldinvolves quantization of an infinite collection of oscillators, with the quantumof an oscillator corresponding physically to a photon with a specific momentumand polarization. This leads to a well-known problem of how to handle theinfinite vacuum energy corresponding to adding up 1

2~ω for each oscillator.

The first few eigenfunctions are plotted below. The lowest energy eigenstateis a Gaussian centered at q = 0, with a Fourier transform that is also a Gaussiancentered at p = 0. Classically the lowest energy solution is an oscillator at rest atits equilibrium point (q = p = 0), but for a quantum oscillator one cannot havesuch a state with a well-defined position and momentum. Note that the plotgives the wavefunctions, which in this case are real and can be negative. Thesquare of this function is what has an intepretation as the probability densityfor measuring a given position.

228

20.3 The Bargmann-Fock representation

Working with the operators a and a† and their commutation relation

[a, a†] = 1

makes it clear that there is a simpler way to represent these operators thanthe Schrodinger representation as operators on position space functions that wehave been using, while the Stone-von Neumann theorem assures us that this willbe unitarily equivalent to the Schrodinger representation. This representationappears in the literature under a large number of different names, depending onthe context, all of which refer to the same representation:

Definition (Bargmann-Fock or oscillator or holomorphic or Segal-Shale-Weilrepresentation). The Bargmann-Fock (etc.) representation is given by taking asstate space H = F , where F is the space of holomorphic functions on C withfinite norm in the inner product

〈ψ1|ψ2〉 =1

π

∫C

ψ1(w)ψ2(w)e−|w|2

dudv (20.3)

229

where w = u + iv. The space F is sometimes called “Fock space”. We definethe following two operators acting on this space:

a =d

dw, a† = w

One has

[a, a†]wn =d

dw(wwn)− w d

dwwn = (n+ 1− n)wn = wn

so this commutator is the identity operator on polynomials

[a, a†] = 1

and

Theorem. The Bargmann-Fock representation has the following properties

• The elementswn√n!

of F for n = 0, 1, 2, . . . are orthornormal.

• The operators a and a† are adjoints with respect to the given inner producton F .

• The basiswn√n!

of F for n = 0, 1, 2, . . . is complete.

Proof. The proofs of the above statements are not difficult, in outline they are

• For orthonormality one can just compute the integrals∫C

wmwne−|w|2

dudv

in polar coordinates.

• To show that w and ddw are adjoint operators, use integration by parts.

• For completeness, assume 〈n|ψ〉 = 0 for all n. The expression for the |n〉as Hermite polynomials times a Gaussian implies that∫

F (q)e−q2

2 ψ(q)dq = 0

for all polynomials F (q). Computing the Fourier transform of ψ(q)e−q2

2

gives ∫e−ikqe−

q2

2 ψ(q)dq =

∫ ∞∑j=0

(−ikq)j

j!e−

q2

2 ψ(q)dq = 0

230

So ψ(q)e−q2

2 has Fourier transform 0 and must be 0 itself. Alternatively,one can invoke the spectral theorem for the self-adjoint operator H, whichguarantees that its eigenvectors form a complete and orthonormal set.

Since in this representation the number operator N = a†a satisfies

Nwn = wd

dwwn = nwn

the monomials in w diagonalize the number and energy operators, so one has

|n〉 =wn√n!

for the normalized energy eigenstate of energy ~ω(n+ 12 ).

Note that we are here taking the state space F to include infinite linearcombinations of the states |n〉, as long as the Bargmann-Fock norm is finite.We will sometimes want to restrict to the subspace of finite linear combinationsof the |n〉, which we will denote Ffin. This is just the space C[w] of polynomials,and F is its completion for the Bargmann-Fock norm.

20.4 The Bargmann transform

The Stone von-Neumann theorem implies the existence of

Definition. Bargmann transformThere is a unitary map called the Bargmann transform

B : F → HS

between the Bargmann-Fock and Schrodinger representations, with operatorssatisfying the relation

Γ′BF (X) = B−1Γ′S(X)B

for X ∈ h2d+1.

In practice, knowing B explicitly is often not needed, since one can use therepresentation independent relation

aj =1√2

(Qj + iPj)

to express operators either purely in terms of aj and a†j , which have a simpleexpression

aj =∂

∂wj, a†j = wj

231

in the Bargmann-Fock representation, or purely in terms of Qj and Pj whichhave a simple expression

Qj = qj , Pj = −i ∂∂qj

in the Schrodinger representation.To give an idea of what the Bargmann transform looks like explicitly, we’ll

just give the formula for the d = 1 case here, without proof. If ψ(q) is a statein HS = L2(R), then

(Bψ)(z) =1

π14

∫ +∞

−∞e−z

2− q2

2 +2qzψ(q)dq

One can check this equation for the case of the lowest energy state in theSchrodinger representation, where |0〉 has coordinate space representation

ψ(q) =1

π14

e−q2

2

and

(Bψ)(z) =1

π14

∫ +∞

−∞e−z

2− q2

2 +2qz 1

π14

e−q2

2 dq

=1

π12

∫ +∞

−∞e−z

2−q2+2qzdq

=1

π12

∫ +∞

−∞e−(q−z)2

dq

=1

π12

∫ +∞

−∞e−q

2

dq

=1

which is the expression for the state |0〉 in the Bargmann-Fock representation.

20.5 Multiple degrees of freedom

Up until now we have been working with the simple case of one physical degree offreedom, i.e. one pair (Q,P ) of position and momentum operators satisfying theHeisenberg relation [Q,P ] = i1, or one pair of adjoint operators a, a† satisfying[a, a†] = 1. We can easily extend this to any number d of degrees of freedomby taking tensor products of our state space F , and d copies of our operators,acting on each factor of the tensor product. Our new state space will be

H = Fd = F ⊗ · · · ⊗ F︸︷︷︸d times

232

and we will have operators

Qj , Pj j = 1, . . . d

satisfying[Qj , Pk] = iδjk1, [Qj , Qk] = [Pj , Pk] = 0

where Qj and Pj just act on the j’th term of the tensor product in the usualway.

We can now define annihilation and creation operators in the general case:

Definition (Annihilation and creation operators). The 2d operators

aj =1√2

(Qj + iPj), a†j =1√2

(Qj − iPj), j = 1, . . . , d

are called annihilation (the aj) and creation (the a†j) operators.

One can easily check that these satisfy:

Definition (Canonical commutation relations). The canonical commutation re-lations (often abbreviated CCR) are

[aj , a†k] = δjk1, [aj , ak] = [a†j , a

†k] = 0

Using the fact that tensor products of function spaces correspond to func-tions on the product space, in the Schrodinger representation we have

H = L2(Rd)

and in the Bargmann-Fock representation H = Fd is the space of holomorphicfunctions in d complex variables (with finite norm in the d-dimensional versionof 20.3).

The harmonic oscillator Hamiltonian for d degrees of freedom will be

H =1

2

d∑j=1

(P 2j +Q2

j ) =

d∑j=1

(a†jaj +1

2)

where one should keep in mind that one can rescale each degree of freedomseparately, allowing different parameters ωj for the different degrees of freedom.The energy and number operator eigenstates will be written

|n1, . . . , nd〉

wherea†jaj |n1, . . . , nd〉 = Nj |n1, . . . , nd〉 = nj |n1, . . . , nd〉

Note that for d = 3 the harmonic oscillator problem is an example of the cen-tral potential problems described in chapter 19. It has an SO(3) symmetry, withangular momentum operators that commute with the Hamiltonian, and space

233

of energy eigenstates that can be organized into irreducible SO(3) representa-tions. In the Schrodinger representation states are in H = L2(R3), decribedby wavefunctions that can be written in rectangular or spherical coordinates,and the Hamiltonian is a second order differential operator. In the Bargmann-Fock representation, states in F3 are described by holomorphic functions of 3complex variables, with operators given in terms of products of annihilationand creation operators. The Hamiltonian is, up to a constant, just the numberoperator, with energy eigenstates homogeneous polynomials (with eigenvalue ofthe number operator their degree).

Either the Pj , Qk or the aj , a†k together with the identity operator will give

a representation of the Heisenberg Lie algebra h2d+1 on H, and by exponentia-tion a representation of the Heisenberg group H2d+1. Quadratic combinationsof these operators will give a representation of sp(2d,R), the Lie algebra ofSp(2d,R). In the next chapters we will study these and other aspects of thequantum harmonic oscillator as a unitary representation.


All quantum mechanics books should have a similar discussion of the harmonicoscillator, with a good example the detailed one in chapter 7 of Shankar [63].One source for a detailed treatment of the Bargmann-Fock representation andBargmann transform is [20].

234

Chapter 21

The Harmonic Oscillator asa Representation of theHeisenberg Group

The quantum harmonic oscillator explicitly constructed in the previous chapterprovides new insight into the representation theory of the Heisenberg and meta-plectic groups, using the existence of not just the Schrodinger representationΓS , but the unitarily equivalent Bargmann-Fock version. In this chapter we’llexamine various aspects of the Heisenberg group H2d+1 part of this story thatthe formalism of annihilation and creation operators illuminates, going beyondthe understanding of this representation one gets from the use of position spacewavefunctions and the Schrodinger representation.

The Schrodinger representation ΓS of H2d+1 uses a specific choice of ex-tra structure on classical phase space: a decomposition of its coordinates intopositions qj and momenta pj . For the unitarily equivalent Bargmann-Fock rep-resentation a different sort of extra structure is needed, a decomposition ofcoordinates on phase space into complex coordinates zj and their complex con-jugates zj . Such a decomposition is called a “complex structure” J , and willcorrespond after quantization to a choice that distinguishes annihilation andcreation operators. This choice has physical significance, it is equivalent to aspecification of the lowest energy state |0〉 ∈ H (since this by definition is thestate satisfying aj |0〉 = 0 for all annihilation operators aj). In chapter 20 weused one particular standard choice of J . For other choices one gets differenteigenstates of the number operator, known to physicists as “squeezed states”.In later chapters on relativistic quantum field theory, we will see that the phe-nomenon of anti-particles is best understood in terms of a new possibility forthe choice of J that appears in that case.

The Heisenberg group representation constructed in this way is denoted ΓJ .It does not commute with the Hamiltonian H, so it is not a symmetry, whichwould take states to other states with the same energy. While it doesn’t com-

235

mute with the Hamiltonian, it does have physically important aspects. In par-ticular it takes the state |0〉 to a distinguished set of states known as “coherentstates”. These states are labeled by points of the phase space R2d and providethe closest analog possible in the quantum system of classical states (i.e. thosewith a well-defined value of position and momentum variables).

21.1 Complex structures and phase space

Quantization of phase space M = R2d using the Schrodinger representationgives a unitary Lie algebra representation Γ′S of the Heisenberg Lie algebra h2d+1

which takes the qj and pj coordinate functions on phase space to operators Qjand Pj on HS = L2(Rd). This involves a choice, that of taking states to befunctions of the qj , or (using the Fourier transform) of the pj . It turns out to bea general phenomenon that quantization involves choosing some extra structureon phase space, beyond the Poisson bracket.

For the case of the harmonic oscillator, we found in chapter 20 that quantiza-tion was most conveniently performed using annihilation and creation operators,which involve a different sort of choice of extra structure on phase space. Therewe introduced complex coordinates on phase space, making the choice

zj =1√2

(qj + ipj), zj =1√2

(qj − ipj)

The zj were then quantized using creation operators a†j , the zj using annihilationoperators aj . In the Bargmann-Fock representation, where the state space is aspace of functions of complex variables wj , we have

aj =∂

∂wj, a†j = wj

and there is a distinguished state, the constant function, which is annihilatedby all the aj .

In this section we’ll introduce the notion of a complex structure on a real vec-tor space, with such structures characterizing the possible ways of introducingcomplex coordinates zj , zj and thus annihilation and creation operators. Theabstract notion of a complex structure can be formalized as follows. Given anyreal vector space V = Rn, we have seen that taking complex linear combinationsof vectors in V gives a complex vector space V ⊗C, the complexification of V ,and this can be identified with Cn, a real vector space of twice the dimension.When n = 2d is even, one can get a complex vector space out of V = R2d

in a different way, but to do this one needs the following additional piece ofinformation:

Definition (Complex structure). A complex structure on a real vector space Vis a linear operator

J : V → V

236

such thatJ2 = −1

Given such a pair (V = R2d, J), one can break up complex linear combina-tions of vectors in V into those on which J acts as i and those on which it actsas −i (since J2 = −1, its eigenvalues must be ±i). Note that we have extendedthe action of J on V to an action on V ⊗C using complex linearity. One has

V ⊗C = V +J ⊕ V

−J

where V +J is the +i eigenspace of the operator J on V ⊗ C and V −J is the

−i eigenspace. Complex conjugation takes elements of V +J to V −J and vice-

versa. The choice of J has thus given us two complex vector spaces of complexdimension d, V +

J and V −J , related by this complex conjugation.Since

J(v − iJv) = i(v − iJv)

for any v ∈ V , one can identify the real vector space V with the complex vectorspace V +

J by the map

v ∈ V → 1√2

(v − iJv) ∈ V +J (21.1)

This allows one to think of the pair (V, J) as giving V the structure of a complexvector space, with J providing multiplication by i. Similarly, taking

v ∈ V → 1√2

(v + iJv) ∈ V −J

identifies V with V −J , with J now providing multiplication by −i. V +J and V −J

are interchanged by changing the complex structure J to −J .The real vector space we want to choose a complex structure on is the dual

phase space M = M∗. There will then be a decomposition

M⊗C =M+J ⊕M

−J

and quantization will take elements of M+J to linear combinations of creation

operators, M−J to linear combinations of annihilation operators. Note that, asin earlier chapters, it is elements not of M but of the dual phase space M thatwe want to work with, since it is these that have a Lie algebra structure, andbecome operators after quantization.

The standard choice of complex structure is to take J = J0, where J0 is thelinear operator that acts on basis vectors qj , pj of M by

J0qj = pj , J0pj = −qj

One can take as basis elements ofM−J0, the −i eigenspace of J0, the complex

coordinate functions in M⊗C given by

zj =1√2

(qj + ipj)

237

since one has

J0zj =1√2

(pj − iqj) = −izj

Basis elements of M+J0

are the complex conjugates

zj =1√2

(qj − ipj)

With respect to the chosen basis qj , pj , one can write a complex structureas a matrix. For the case of J0 and for d = 1, on an arbitrary element of Mone has

J0(cqq + cpp) = cqp− cpq

so J0 in matrix form with respect to the basis (q, p) is

J0

(cqcp

)=

(0 −11 0

)(cqcp

)=

(−cpcq

)(21.2)

21.2 Complex structures and quantization

Recall that the Heisenberg Lie algebra is just the Lie algebra of linear andconstant functions on M , so can be thought of as

h2d+1 =M⊕R

where the R component is the constant functions. The Lie bracket is just thePoisson bracket. Complexifying this, one has

h2d+1 ⊗C = (M⊕R)⊗C = (M⊗C)⊕C =M+J ⊕M

−J ⊕C

so one can write elements of h2d+1 ⊗C as pairs (u, c) = (u+ + u−, c) where

u ∈M⊗C, u+ ∈M+J , u− ∈M−J , c ∈ C

This complexified Lie algebra is still a Lie algebra, with the Lie bracket relationsextended from the real Lie algebra by complex linearity. One has

[(u1, c1), (u2, c2)] = (0,Ω(u1, u2)) (21.3)

where Ω is the antisymmetric bilinear form on M given by the Poisson bracketon linear functions (extended by complex linearity to a bilinear form onM⊗C).

For each J , we would like to find a quantization that takes elements ofM+

J to linear combinations of creation operators,M−J to linear combinations ofannihilation operators. This will give a representation of the complexified Liealgebra

Γ′J : (u, c) ∈ h2d+1 ⊗C→ Γ′J(u, c)

which must satisfy the Lie algebra homomorphism property

[Γ′J(u1, c1),Γ′J(u2, c2)] = Γ′J([(u1, c1), (u2, c2)]) = Γ′J(0,Ω(u1, u2)) (21.4)

238

Note that Γ′J will only be a unitary representation (with Γ′J(u, c) skew-adjointoperators) for (u, c) in the real Lie subalgebra h2d+1 (meaning u ∈M, c ∈ R).

Since we can write

(u, c) = (u+, 0) + (u−, 0) + (0, c)

where u+ ∈M+J and u− ∈M−J , we have

Γ′J(u, c) = Γ′J(u+, 0) + Γ′J(u−, 0) + Γ′J(0, c)

The last term must be

Γ′J(0, c) = −ic1

which is chosen so that for c real one has a skew-adjoint transformation andthus a unitary representation.

We would like to construct Γ′J(u+, 0) as a linear combination of creationoperators and Γ′J(u−, 0) as a linear combination of annihilation operators. Forthis to be possible, we need two conditions on J . The first is a compatibilitycondition between Ω and J that will ensure that linear combination of creationoperators commute with each other (and similarly for annihilation operators).The second condition will be discussed in section 21.3.

The necessary compatibility condition between J and Ω is that, for allv1, v2 ∈M

Ω(Jv1, Jv2) = Ω(v1, v2) (21.5)

From the definition of the symplectic group in chapter 14, this condition justsays that J ∈ Sp(2d,R). Since we are extending the action of J to M⊗C bycomplex linearity, this condition will remain true for u1, u2 ∈ M ⊗ C. Giventhis condition, the Γ′J(u+, 0) will commute, since if u+

1 , u+2 ∈ M

+J , by 21.4 we

have

[Γ′J(u+1 , 0),Γ′J(u+

2 , 0)] = Γ′J(0,Ω(u+1 , u

+2 ))

and

Ω(u+1 , u

+2 ) = Ω(Ju+

1 , Ju+2 ) = Ω(iu+

1 , iu+2 ) = −Ω(u+

1 , u+2 ) = 0

The Γ′J(u−, 0) will commute with each other by essentially the same argument.For the case of the standard complex structure J = J0, one can check this

compatibility condition by computing (treating the d = 1 case, which generalizeseasily, and using equations 14.1 and 21.2)

Ω(J0(cqq + cpp), J0(c′qq + c′pp)) =(

(0 −11 0

)(cqcp

))T(

0 1−1 0

)(

(0 −11 0

)(c′qc′p

))

=(cq cp

)( 0 1−1 0

)(0 1−1 0

)(0 −11 0

)(c′qc′p

)=(cq cp

)( 0 1−1 0

)(c′qc′p

)=Ω(cqq + cpp, c

′qq + c′pp)

239

More simply of course, one could just note that J0 ∈ SL(2,R) = Sp(2,R).For the case of J = J0 and arbitrary values of d, one can write out explicitly

Ω on basis elementszj ∈M−J0

, zj ∈M+J0

as

Ω(zj , zk) = zj , zk = 1√2

(qj + ipj),1√2

(qj + ipj) = 0

Ω(zj , zk) = (zj , zk = 1√2

(qj − ipj),1√2

(qj − ipj) = 0

Ω(zj , zk) = zj , zk = 1√2

(qj + ipj),1√2

(qj − ipj) = −iδjk

orzj , izk = δjk

Note that here the conjugate variable to zj with respect to the Poisson bracketis izk.

The Lie algebra representation is given in this case by

Γ′J0(zj , 0) = −ia = −i, ∂

∂wjΓ′J0

(zj , 0) = −ia†j = −iwj , Γ′J0(0, 1) = −i1

Note that this is precisely the Bargmann-Fock representation, i.e. we haveshown that Γ′J0

= Γ′BF .To check that Γ′J0

is a Lie algebra homomorphism, compute

[Γ′J0(zj , 0),Γ′J0

(zk, 0)] =[−iaj ,−ia†k] = −[aj , a†k]

= −δjk1 = δjkΓ′J0(0,−i)

which is correct since

[(zj , 0), (zk, 0)] = (0, zj , zk) = (0,−iδjk)

Note that the operators aj and a†j are not skew-adjoint, so Γ′J0is not unitary

on the full Lie algebra h2d+1 ⊗ C, but only on the real subspace h2d+1 of reallinear combinations of qj , pj , 1.

21.3 The positivity condition on J

We still need a second condition on J , a positivity condition. Recall that theannihilation and creation operators satisfy (for d = 1)

[a, a†] = 1

This condition on the commutator corresponds to the following condition onthe representation, for J = J0

[Γ′J0(z, 0),Γ′J0

(z, 0)] = [−ia†,−ia] = 1

240

By the homomorphism property 21.4 we have

[Γ′J0(z, 0),Γ′J0

(z, 0)] = Γ′J0(0,Ω(z, z)) = −iΩ(z, z)1

so positivity here corresponds to the fact that

−iΩ(z, z) = 1 > 0

Use of the opposite sign for the commutator

[a, a†] = −1

would correspond to interchanging the role of a and a†, with the lowest energystate now satisfying a†|0〉 = 0 and no state in the state space satisfying a|0〉 = 0.This is equivalent to a change of sign of J , interchangingM+

J andM−J . For thed = 1 case with the wrong sign, one can just make this interchange to constructthe state space, but in higher dimensions one needs to have the same sign forall [aj , a

†j ]. In order to have a state |0〉 that is annihilated by all annihilation

operators, we need all the commutators [aj , a†j ] to have the positive sign.

For the case of general J and arbitrary d, to get operators Γ′J(u, 0) with theright positivity properties, we will need the condition

−iΩ(u, u) > 0

for non-zero u ∈ M+J . Only if this condition is satisfied will we be able to

construct Γ′J in terms of conventional annihilation and creation operators forbasis elements of M+

J and M−J . Using the identification 21.1 of M and M+J ,

u ∈M+J can be written as

u =1√2

(v − iJv)

for some non-zero v ∈ M. So another way to write the positivity condition isas

−iΩ(u, u) =− i12

Ω(v − iJv, v + iJv)

=1

2(−iΩ(v, v)− iΩ(Jv, Jv) + Ω(v, Jv)− Ω(Jv, v))

=Ω(v, Jv) > 0

For the standard complex structure J0, we can check this using the matrixexpressions for Ω and J0

Ω(cqq + cpp, J0(cqq + cpp)) =(cq cp

)( 0 1−1 0

)(0 −11 0

)(cqcp

)=(cq cp

)(1 00 1

)(cqcp

)=c2q + c2p

241

Note that Ω(v, Jv) thus gives a positive-definite quadratic function onM. Usingthe isomorphismM = M provided by Ω, this corresponds to a positive-definitequadratic function on the phase space itself. For J0 this is just (twice) thestandard harmonic oscillator Hamiltonian function. One application of moregeneral J is to the case of more general quadratic (but still positive-definite)Hamiltonian functions.

We can give a name to the class of J for which we will have a formalism ofannihilation and creation operators:

Definition (Positive compatible complex structures). A complex structure Jon M is said to be positive and compatible with Ω if it satisfies the compatibil-ity condition 21.5 (i.e. is in Sp(2d,R)) and either of the equivalent positivityconditions

•−iΩ(u, u) > 0

for non-zero u ∈M+J .

•Ω(v, Jv) > 0

for non-zero v ∈M.

Given such a complex structure, we have:

Theorem. Given a positive compatible complex structure J onM, one can finda basis zJj of M−J such that a representation of h2d+1 ⊗C, unitary for the realsubalgebra h2d+1, is given by

Γ′J(zJj , 0) = −iaj , Γ′J(zJj , 0) = −ia†j , Γ′J(0, c) = −ic1

where aj , a†k satisfy the conventional commutation relations, and zJj is the com-

plex conjugate of zJj .

Proof. An outline of the construction goes as follows:

1. Define a norm on M by |v|2 = Ω(v, Jv). One then has a positive innerproduct 〈·, ·〉J on M and by Gram-Schmidt orthornormalization can finda basis of spanqj ⊂ M consisting of d vectors qJj satisfying

〈qJj , qJk 〉J = δjk

2. The vectors JqJj will also be orthonormal since

〈JqJj , JqJk 〉J = Ω(JqJj , J2qJj ) = Ω(qJj , Jq

Jj ) = 〈qJj , qJk 〉J

They will be orthogonal to the qJj since

〈qJj , JqJk 〉J = Ω(qJj , J2qJk ) = −Ω(qJj , q

Jk ) = 0

242

3. Define

zJj =1√2

(qJj + iJqJj )

The zJj give a complex basis ofM−J , their complex conjugates zJj a complex

basis of M+J .

4. One can check that

Γ′J(zJj , 0) = −iaj = −i ∂

∂wjwj , Γ′J(zJj , 0) = −ia†j = −iwj

satisfy the desired commutation relations and give a unitary representationon linear combinations of the (zJj , 0) and (zJj , 0) in the real subalgebrah2d+1.

21.4 Complex structures for d = 1 and squeezedstates

To get a better understanding of what happens for other complex structuresthan J0, in this section we’ll see explicitly what happens when d = 1. We cangeneralize the case J = J0, where a basis of M+

J0is given by

z =1√2

(q − ip)

by replacing the i by an arbitrary complex number τ . Then the condition thatq − τp be in M+

J and its conjugate in M−J is

J(q − τp) = J(q)− τJ(p) = i(q − τp)

J(q − τp) = J(q)− τJ(p) = −i(q − τp)

Subtracting the two equations one finds

J(p) = − 1

Im(τ)q +

Re(τ)

Im(τ)p

and adding them gives

J(q) = −Re(τ)

Im(τ)q + (Im(τ) +

(Re(τ))2

Im(τ))p

With respect to the basis (q, p) the matrix for J is

J =

(−Re(τ)

Im(τ) − 1Im(τ)

Im(τ) + (Re(τ))2

Im(τ)Re(τ)Im(τ)

)=

1

Im(τ)

(−Re(τ) −1|τ |2 Re(τ)

)(21.6)

243

One can easily check that det J = 1, so J ∈ SL(2,R) and is compatible with Ω.The positivity condition here is that the matrix(

0 1−1 0

)J =

1

Im(τ)

(|τ |2 −Re(τ)

Re(τ) 1

)give a positive quadratic form. This will be the case when Im(τ) > 0. Wehave thus constructed a set of J that are positive, compatible with Ω, andparametrized by an element τ of the upper half-plane, with J0 corresponding toτ = i.

To construct annihilation and creation operators satisfying the standardcommutation relations

[aτ , aτ ] = [a†τ , a†τ ] = 0, [aτ , a

†τ ] = 1

set

aτ =1√

2 Im(τ)(Q− τP ), a†τ =

1√2 Im(τ)

(Q− τP )

The Hamiltonian with eigenvalues n+ 12 for n = 0, 1, 2, · · · will be

Hτ =1

2(aτa

†τ + a†τaτ ) =

1

2 Im(τ)(Q2 + |τ |2P 2 − Re(τ)(QP + PQ)) (21.7)

The lowest energy state will satisfy

aτ |0〉τ = 0

which in the Schrodinger representation is the differential equation

(Q− τP )ψ(q) = (q + iτd

dq)ψ(q) = 0

which has as solutions

ψ(q) ∝ eiτ

2|τ|2q2

(21.8)

This will be a normalizable state for Im(τ) > 0, again showing the necessity ofthe positivity condition.

Eigenstates of Hτ for general τ are known as “squeezed states” in physics.Note that for τ pure imaginary, they correspond just to a rescaling of variables

q → 1√Im(τ)

q, p→√

Im(τ)p

Such states are “squeezed” in the sense that for Im(τ) large the position un-certainty in the state |0〉τ will become small (while the momentum uncertaintybecomes large).

The construction for d = 1 can be generalized to arbitrary d, with complexstructures J now parametrized by a d-dimensional complex matrix τ which mustbe symmetric (to be compatible with Ω), and such that Im(τ) is a positive-definite matrix. The space of such complex structures is known as the Siegelupper half-space.

Add a discussion in terms of usual Bogoliubov transformations

244

21.5 Coherent states and the Heisenberg groupaction

Since the Hamiltonian for the harmonic oscillator does not commute with theoperators aj or a†j which give the representation of the Lie algebra h2d+1 on thestate space F , the Heisenberg Lie group and its Lie algebra are not symmetries ofthe system. Energy eigenstates do not break up into irreducible representationsof the group but rather the entire state space makes up such an irreduciblerepresentation. The state space for the harmonic oscillator does however have adistinguished state, the lowest energy state |0〉, and one can ask what happensto this state under the Heisenberg group action. We’ll study this question forthe simplest case of d = 1 and the standard complex structure J0.

Considering the basis of operators for the Lie algebra representation 1, a, a†,we see that the first acts as a constant on |0〉, generating a phase tranformationof the state, while the second annihilates |0〉, so generates group transformationsthat leave the state invariant. It is only the third operator a†, that takes |0〉 toother non-zero states, and one could consider the family of states

eαa†|0〉

for α ∈ C. The transformations eαa†

are not unitary since αa† is not skew ad-joint. It is better to fix this by replacing αa† with the skew-adjoint combinationαa† − αa, defining

Definition (Coherent states). The coherent states in H are the states

|α〉 = eαa†−αa|0〉

where α ∈ C.

Since eαa†−αa is unitary, the |α〉 will be a family of distinct normalized states

in H, with α = 0 corresponding to the lowest energy state |0〉 (for α 6= 0 thesewill not be energy eigenstates). These are, up to phase transformation, preciselythe states one gets by acting on |0〉 with arbitrary elements of the Heisenberggroup H3.

Using the Baker-Campbell-Hausdorff formula gives

|α〉 = eαa†−αa|0〉 = eαa

†e−αae−

|α|22 |0〉

and since a|0〉 = 0 one has

|α〉 = e−|α|2

2 eαa†|0〉 = e−

|α|22

∞∑n=0

αn√n!|n〉 (21.9)

Since a|n〉 =√n|n− 1〉 one finds

a|α〉 = e−|α|2

2

∞∑n=1

αn√(n− 1)!

|n− 1〉 = α|α〉

245

and this property could be used as an equivalent definition of coherent states.Note that coherent states are superpositions of different states |n〉, so are

not eigenvectors of the number operator N . They are eigenvectors of

a =1√2

(Q+ iP )

with eigenvalue α so one can try and think of α as a complex number whosereal part gives the position and imaginary part the momentum. This does notlead to a violation of the Heisenberg uncertainty principle since this is not aself-adjoint operator, and thus not an observable. Such states are however veryuseful for describing certain sorts of physical phenomena, for instance the stateof a laser beam, where (for each momentum component of the electromagneticfield) one does not have a definite number of photons, but does have a definiteamplitude and phase.

One thing coherent states do provide is an alternate complete set of normone vectors in H, so any state can be written in terms of them. However, thesestates are not orthogonal (they are eigenvectors of a non-self-adjoint operator sothe spectral theorem for self-adjoint operators does not apply). One can easilycompute that

|〈β|α〉|2 = e−|α−β|2

One possible reason these states are given the name “coherent” is that theyremain coherent states as they evolve in time (for the harmonic oscillator Hamil-tonian), with α evolving in time along a classical phase space trajectory. If thestate at t = 0 is a coherent state labeled by α0 (|ψ(0)〉 = |α0〉), by 21.9, at latertimes one has (here ~ = ω = 1)

|ψ(t)〉 =e−iHt|α0〉

=e−iHte−|α0|

2

2

∞∑n=0

αn0√n!|n〉

=e−i(n+ 12 )te−

|α0|2

2

∞∑n=0

αn0√n!|n〉

=e−i12 te−

|e−itα0|2

2

∞∑n=0

(e−itα0)n√n!

|n〉

=e−i12 t|e−itα0〉

Up to the phase factor e−i12 t, this remains a coherent state, with label α given by

the classical time-dependence of the complex coordinate z(t) = 1√2(q(t) + ip(t))

for the harmonic oscillator (see 20.1) with z(0) = α0.

Digression (Spin coherent states). One can perform a similar constructionreplacing the group H3 by the group SU(2), and the state |0〉 by a highest weightvector of an irreducible representation (πn, V

n = Cn+1) of spin n2 . Writing |n2 〉

246

for a highest weight vector, we have

π′n(S3)|n2〉 =

n

2, π′n(S+)|n

2〉 = 0

and we can create a family of spin coherent states by acting on |n2 〉 by elementsof SU(2). If we identify states in this family that differ just by a phase, thestates are parametrized by a sphere.

By analogy with the Heisenberg group coherent states, with π′n(S+) playingthe role of the annihilation operator a and π′n(S−) playing the role of the creationoperator a†, we can define a skew-adjoint transformation

1

2θeiφπ′n(S−)− 1

2θe−iφπ′n(S+)

and exponentiate to get a family of unitary transformations parametrized by(θ, φ). Acting on the highest weight state we get a definition of the family ofspin coherent states as

|θ, φ〉 = e12 θe

iφπ′n(S−)− 12 θe−iφπ′n(S+)|n

2〉

One can show that the SU(2) group element used here corresponds, in terms ofits action on vectors, to a rotation by an angle θ about the axis (sinφ,− cosφ, 0),so one can associate the state |θ, φ〉 to the unit vector along the z-axis, rotatedby this transformation.


For more about complex structures symplectic vector spaces, including a discus-sion of the Siegel upper half-space of (positive compatible complex structuresfor arbitrary d), see chapter 1.4 of [7]. Coherent states and spin coherent statesare discussed in chapter 21 of [63]. Few quantum mechanics textbooks discusssqueezed states, for one that does, see chapter 12 of [85].

247

Chapter 22

The Harmonic Oscillatorand the MetaplecticRepresentation, d = 1

In the last chapter we examined those aspects of the harmonic oscillator quan-tum system and the Bargmann-Fock representation that correspond to quan-tization of phase space functions of order less than or equal to one, finding aunitary representation ΓJ of the Heisenberg group H2d+1 for each positive com-patible complex structure J on the dual phase spaceM. We’ll now turn to whathappens for order two functions, which will give a representation of Mp(2d,R)on the harmonic oscillator state space, extending ΓJ to a representation of thefull Jacobi group. In this chapter we will see what happens in some detail forthe d = 1 case, where the symplectic group is just SL(2,R).

The choice of complex structure J corresponds not only to a choice of |0〉J(since J determines which operators are annihilation operators), but also tothat of a specific subgroup U(1) ⊂ SL(2,R). The nature of the double-coverneeded for ΓJ to be a true representation (not just a representation up to sign)is best seen by considering the action of this U(1) on the harmonic oscillatorstate space. The Lie algebra of this U(1) acts on energy eigenstates with anextra 1

2 term, well-known to physicists as the non-zero energy of the vacuumstate, and this shows the need for the double-cover.

22.1 The metaplectic representation for d = 1

In the last chapter we saw that choosing a complex structure J on the dualphase space M = R2d allows one to break up complex linear combinations ofthe Qj , Pj into annihilation and creation operators, giving a unitary represen-tation Γ′J of h2d+1 on the Fock space Fd. For the standard choice of complexstructure J = J0 we have Γ′J0

= Γ′BF and this representation is the one used in

249

the standard description in terms of annihilation and creation operators of thequantum harmonic oscillator system with d degrees of freedom.

Recall from our discussion of the Schrodinger representation Γ′S in section18 that we can extend that representation from h2d+1 to include quadraticcombinations of the qj , pj , getting a unitary representation of the semi-directproduct h2d+1 o sp(2d,R). Restricting attention to the sp(2d,R) factor, we getthe metaplectic representation, and it is this that we will construct explicitlyusing Γ′BF instead of Γ′S . In this chapter, we’ll start with the case d = 1, wheresp(2,R) = sl(2,R).

One can readily compute the Poisson brackets of order two of z and z usingthe basic relation z, z = −i and the Leibniz rule, finding the following for thenon-zero cases

zz, z2 = 2iz2, zz, z2 = −2iz2, z2, z2 = 4izz

In the case of the Schrodinger representation, our quadratic combinations of pand q were real, and we could identify the Lie algebra they generated with theLie algebra sl(2,R) of traceless 2 by 2 real matrices with basis

E =

(0 10 0

), F =

(0 01 0

), G =

(1 00 −1

)Since we have complexified, our quadratic combinations of z and z are in the

complexification of sl(2,R), the Lie algebra sl(2,C) of traceless 2 by 2 complexmatrices. We can take as a basis of sl(2,C) over the complex numbers

Z = E − F, X± =1

2(G± i(E + F ))

which satisfy

[Z,X−] = −2iX−, [Z,X+] = 2iX+, [X+, X−] = −iZ

and then use as our isomorphism between quadratics in z, z and sl(2,C)

z2

2↔ X+,

z2

2↔ X−, zz ↔ Z

The element

zz =1

2(q2 + p2)↔ Z =

(0 1−1 0

)exponentiates to give a SO(2) = U(1) subgroup of SL(2,R) with elements ofthe form

eθZ =


)Note that h = 1

2 (p2 + q2) = zz is the classical Hamiltonian function for theharmonic oscillator.

250

We can now quantize quadratics in z and z using annihilation and creationoperators acting on the Fock space F . There is no operator ordering ambiguityfor

z2 → (a†)2 = w2, z2 → a2 =d2

dw2

For the case of zz (which is real), in order to get the sl(2,R) commutationrelations to come out right (in particular, the Poisson bracket z2, z2 = 4izz),we must take the symmetric combination

zz → 1

2(aa† + a†a) = a†a+

1

2= w

d

dw+

1

2

(which of course is just the standard Hamiltonian for the quantum harmonicoscillator).

Multiplying as usual by−i one can now define an extension of the Bargmann-Fock representation to an sl(2,C) representation by taking

Γ′BF (X+) = − i2a2, Γ′BF (X−) = − i

2(a†)2, Γ′BF (Z) = −i1

2(a†a+ aa†)

One can check that we have made the right choice of Γ′BF (Z) to get an sl(2,C)representation by computing

[Γ′BF (X+),Γ′BF (X−)] =[− i2a2,− i

2(a†)2] = −1

2(aa† + a†a)

=− iΓ′BF (Z) = Γ′BF ([X+, X−])

As a representation of the real sub-Lie algebra sl(2,R) of sl(2,C), one has(using the fact that G,E + F,E − F is a real basis of sl(2,R)):

Definition (Metaplectic representation of sl(2,R)). The representation Γ′BFon F given by

Γ′BF (G) = Γ′BF (X+ +X−) = − i2

((a†)2 + a2)

Γ′BF (E + F ) = Γ′BF (−i(X+ −X−)) = −1

2((a†)2 − a2) (22.1)

Γ′BF (E − F ) = Γ′BF (Z) = −i12

(a†a+ aa†)

is a representation of sl(2,R) called the metaplectic representation.

Note that one can explicitly see from these expressions that this is a unitaryrepresentation, since all the operators are skew-adjoint (using the fact that aand a† are each other’s adjoints).

This representation Γ′BF will be unitarily equivalent to the Schrodinger ver-sion Γ′S found earlier when quantizing q2, p2, pq as operators on H = L2(R).It is however much easier to work with since it can be studied as the state

251

space of the quantum harmonic oscillator, with the Lie algebra acting simplyby quadratic expressions in the annihilation and creation operators.

One thing that can now easily be seen is that this representation Γ′BF doesnot integrate to give a representation of the group SL(2,R). If the Lie algebrarepresentation Γ′BF comes from a Lie group representation ΓBF of SL(2,R), wehave

ΓBF (eθZ) = eθΓ′BF (Z)

where

Γ′BF (Z) = −i(a†a+1

2) = −i(N +

1

2)

so

ΓBF (eθZ)|n〉 = e−iθ(n+ 12 )|n〉

Taking θ = 2π, this gives an inconsistency

ΓBF (1)|n〉 = −|n〉

which has its origin in the physical phenomenon that the energy of the lowestenergy eigenstate |0〉 is 1

2 rather than 0, so not an integer.

This is precisely the same sort of problem we found when studying thespinor representation of the Lie algebra so(3). Just as in that case, the problemindicates that we need to consider not the group SL(2,R), but a double cover,the metaplectic group Mp(2,R). The behavior here is quite a bit more subtlethan in the Spin(3) double cover case, where Spin(3) was just the group SU(2),and topologically the only non-trivial cover of SO(3) was the Spin(3) one sinceπ1(SO(3)) = Z2. Here one has π1(SL(2,Z)) = Z, and each extra time one goesaround the U(1) subgroup we are looking at one gets a topologically differentnon-contractible loop in the group. As a result, SL(2,R) has lots of non-trivialcovering groups, of which only one interests us, the double cover Mp(2,R). In

particular, there is an infinite-sheeted universal cover ˜SL(2,R), but that playsno role here.

Digression. This group Mp(2,R) is quite unusual in that it is a finite-dim-ensional Lie group, but does not have any sort of description as a group offinite-dimensional matrices. This is related to the fact that its only interestingirreducible representation is the infinite-dimensional one we are studying. Thelack of any significant irreducible finite-dimensional representations correspondsto its not having a matrix description, which would give such a representation.Note that the lack of a matrix description means that this is a case where thedefinition we gave of a Lie algebra in terms of the matrix exponential does notapply. The more general geometric definition of the Lie algebra of a group interms of the tangent space at the identity of the group does apply, although to dothis one really needs a construction of the double cover Mp(2,R), which is quitenon-trivial. This is not actually a problem for purely Lie algebra calculations,since the Lie algebras of Mp(2,R) and SL(2,R) can be identified.

252

Another aspect of the metaplectic representation that is relatively easy tosee in the Bargmann-Fock construction is that the state space F is not anirreducible representation, but is the sum of two irreducible representations

F = Feven ⊕Fodd

where Feven consists of the even functions, Fodd of odd functions. On the sub-space Ffin ⊂ F of finite sums of the number eigenstates, these are just theeven and odd degree polynomials. Since the generators of the Lie algebra rep-resentation are degree two combinations of annihilation and creation operators,they will take even functions to even functions and odd to odd. The separateirreducibility of these two pieces is due to the fact that (when n and m havethe same parity), one can get from state |n〉 to any another |m〉 by repeatedapplication of the Lie algebra representation operators.

22.2 Complex structures and the SL(2,R) actionon M

Recall from the discussion in chapter 18 that the existence of the metaplecticrepresentation can be understood in terms of the fact that the action of SL(2,R)onM implies an action by automorphisms on the Heisenberg group H3, togetherwith the uniqueness (up unitary equivalence) of the irreducible representation ofH3. This SL(2,R) action on H3 was studied in section 14.2 where we saw thatit was given infinitesimally by Poisson brackets between order two (sl(2,R))and linear (h3) polynomials (see equation 14.8).

The Bargmann-Fock construction above of the metaplectic representationdepends on an extra choice, one that is not invariant under the SL(2,R) action.This is the choice of a complex structure J0 to provide the splitting M⊗C =M+

J0⊕M−J0

into ±i eigenspaces of J0. Recall that the group SL(2,R) acts onM by matrices (

α βγ δ

)satisfying αδ − βγ = 1 (with respect to the choice of a basis q, p). Only asubgroup of SL(2,R) will respect the splitting provided by J0: the subgroup ofmatrices commuting with J0. This will be this subgroup of SL(2,R) that actson M⊗ C (extending the action on M by complex linearity) taking M+

J0to

M+J0

and M−J0to M−J0

.This commutativity condition is explicitly(

α βγ δ

)(0 1−1 0

)=

(0 1−1 0

)(α βγ δ

)so (

β −αδ −γ

)=

(−γ −δα β

)253

which implies β = −γ and α = δ. The elements of SL(2,R) that we want willbe of the form (

α β−β α

)with unit determinant, so α2 + β2 = 1. This is the U(1) = SO(2) subgroup ofSL(2,R) of matrices of the form(

cos θ sin θ− sin θ cos θ

)= eθZ

The metaplectic representation restricted to this subgroup was studied usingthe Schrodinger representation in 18.2.1, and its role in showing the necessity ofthe metaplectic double cover was seen in the last section. Here we’ll study therepresentation on the same subgroup, but in the Bargmann-Fock version usingannihilation and creation operators.

Introducing the complex structure J0 and thus coordinates z, z on the com-plexification M⊗ C, the Poisson bracket relations 14.8 after complexificationbreak up into two sorts: those that preserve the complex structure and thosethat don’t. The ones that preserve the complex structure will be

zz, z = iz, zz, z = −iz (22.2)

since it is zz that corresponds to Z according to our isomorphism of order twopolynomials and sl(2,R). Note that this could be written as

zz,(zz

) =

(−i 00 i

)(zz

)which is a (complexified) example of equation 14.14 since µZ = zz and Z = −J0

acts as i on the coordinate function z, and −i on the coordinate function z.Quantization by the metaplectic representation gives (see 22.1)

Γ′BF (zz) = Γ′BF (Z) = − i2

(aa† + a†a)

and the quantized analogs of 22.2 are

[− i2

(aa† + a†a), a†] = −ia†, [− i2

(aa† + a†a), a] = ia (22.3)

Exponentiating, one has g = eθZ ∈ U(1) ⊂ SL(2,R) and unitary operators

Ug = ΓBF (eθZ) = e−iθ2 (aa†+a†a)

which satisfy

Uga†U−1

g = e−iθa†, UgaU−1g = eiθa (22.4)

254

Note that, using equation 5.1 one has

d

dθ(Uga

†U−1g )|θ=0 = [− i

2(aa† + a†a), a†]

so equation 22.3 is just the derivative at the identity of equation 22.4. Also notethat

We see that, on operators, conjugation by the action of the U(1) subgroupof SL(2,R) does not mix creation and annihilation operators. On the distin-guished state |0〉, Ug acts as the phase transformation

Ug|0〉 = e−i12 θ|0〉

Besides 22.2, there are also Poisson bracket relations corresponding to in-finitesimal sl(2,R) transformations that do not preserve the complex structure.They are

z2, z = −2iz, z2, z = 0, z2, z = 2iz, z2, z = 0 (22.5)

As an example of SL(2,R) transformations that change the complex structure,consider the subgroup of elements of the form

gα =

(eα 00 e−α

)For α > 0 these are “squeezing” transformations which expand vectors in the qdirection in M, and contract them in the p direction. One can think of theseas a change of basis in M, with the complex structure in the new basis

Jα = gαJ0g−1α =

(eα 00 e−α

)(0 −11 0

)(e−α 0

0 eα

)=

(0 e2α

e−2α 0

)Comparing to equation 21.6 we find that Jα is the positive compatible complexstructure with parameter τ = ie2α.

The relations 22.5 correspond to the following relations of annihilation andcreation operators:

[(a†)2, a] = −2a†, [(a†)2, a†] = 0, [a2, a†] = 2a, [a2, a] = 0

and one sees that conjugation by exponentials of linear combinations of a2 and(a†)2 will mix annihilation and creation operators (unlike the U(1) ⊂ SL(2,R)case of equations 22.3 and 22.4). For the example above

gα = eαG

and we find

ΓBF (gα) = ΓBF (eαG) = eαΓ′BF (G) = e−αi2 ((a†)2+a2)

by 22.1.

255

The ΓBF (gα) are unitary operators (since exponentials of skew-adjoint oper-ators) on the Fock space F that take |0〉 to a different state |0〉α, not proportionalto |0〉. This state |0〉α can be used to characterize the complex structure Jα,as the one corresponding to the change of variables that takes the annihilationoperator a to the linear combination of annihilation and creation operators thatannihilates |0〉α. One can generalize this for arbitrary positive compatible com-plex structures, and get a map that take τ in the upper half-plane to vectors inF (modulo scalar multiplication of the vectors). This map turns out to be quiteuseful in algebraic geometry, providing an embedding of the upper half-planein complex projective space (the same holds true for d > 1 for the Siegel upperhalf-space).

22.3 Normal Ordering

Whenever one has a product involving both z and z that one would like toquantize, the non-trivial commutation relation of a and a† means that one hasdifferent inequivalent possibilities, depending on the order one chooses for thea and a†. In this case, we chose to quantize h = zz using the symmetric choice

H =1

2(aa† + a†a) = a†a+

1

2

because quantization of order two polynomials then gave a representation ofsl(2,R). We could instead have chosen to use a†a, which is an example of a“normal-ordered” product.

Definition. Normal-ordered productGiven any product P of the a and a† operators, the normal ordered product

of P , written :P : is given by re-ordering the product so that all factors a† areon the left, all factors a on the right, for example

:a2a†a(a†)3: = (a†)4a3

The advantage of working with the normal-ordered choice

a†a = :1

2(aa† + a†a):

is that it acts trivially on |0〉 and has integer rather than half-integer eigenvalueson F . There is then no need to invoke a double-covering. The disadvantageis that one gets a representation of u(1) that does not extend to sl(2,R). Onealso needs to keep in mind that the definition of normal-ordering depends uponthe choice of J , or equivalently, the choice of distinguished state |0〉 annihilatedby the annihilation operators. A better notation would be something like :P :Jrather than just :P :.

We have now seen that the necessary choice of a complex structure J ina Bargmann-Fock type quantization shows up in the following distinguishedfeatures of the representation:

256

• The choice of the decompositionM⊗C =M+J ⊕M

−J . After quantization

this determines which operators are linear combinations of annihilationoperators and which are linear combinations of creation operators.

• The choice of normal-ordering prescription. SL(2,R) transformationsthat mix annihilation and creation operators change the definition of thenormal ordering symbol : :.

• The choice (up to a scalar factor) of a distinguished state |0〉 ∈ H, thestate annihilated by annihilation operators. Equation 21.8 gives this stateexplicitly in the Schrodinger representation, showing how it depe)nds onthe complex structure τ .

• The choice of quadratic Hamiltonian as half the symmetrized productof annihilation and creation operators, and thus with energy eigenstatesgiven by applying creation operators to |0〉. See equation 21.7 for theexplicit form of this as a function of τ .

• The choice of a distinguished subgroup U(1) ⊂ SL(2,R), the subgroupthat commutes with J .

We saw in chapter 21 that the space of positive, compatible complex struc-tures can be parametrized by the upper half-plane. An alternate way of charac-terizing this space is as the quotient space SL(2,R)/U(1), since SL(2,R) actstransitively on the complex structures (by conjugation of the matrix for J), withthe stabilizer of J a subgroup U(1) ⊂ SL(2,R).

SL(2,R) transformations that do not commute with the complex structureare known in the physics literature as “Bogoliubov transformations”. Insteadof describing such transformations as we have done, in terms of real matricesand the basis q, p of M, one can instead use complex matrices and the basisz, z of M⊗C. Transformations that preserve the Poisson brackets (and, afterquantization, the commutation relations of annihilation and creation operators)are given by complex matrices of the form(

α β

β α

), |α|2 − |β|2 = 1

The group of such matrices is called SU(1, 1) and is isomorphic to SL(2,R).


The metaplectic representation is not usually mentioned in the physics litera-ture, and the discussions in the mathematical literature tend to be aimed atan advanced audience. Two good examples of such detailed discussions canbe found in [20] and chapters 1 and 11 of [73]. For an example of the useof Bogoliubov transformations (change of complex structure) in the theory ofsuperfluidity, see chapter 10.3 of [67].

257

Chapter 23

The Harmonic Oscillator asa Representation of U(d)

As a representation of Mp(2d,R), ΓBF is unitarily equivalent (by the Barg-mann transform) to ΓS , the Schrodinger representation studied earlier. TheBargmann-Fock version though makes some aspects of the representation easierto study; in particular it makes clear the nature of the double-cover that appears.A subtle aspect of the Bargmann-Fock construction is that it depends on aspecific choice of complex structure J , a choice which corresponds to a choice of adistinguished state |0〉. This same choice picks out a subgroup U(d) ⊂ Sp(2d,R)of transformations which commute with J , and the harmonic oscillator statespace gives one a representation of a double-cover of this group. We will seethat normal-ordering the products of annihilation and creation operators turnsthis into a representation of U(d) itself. In this way, a U(d) action on the finite-dimensional phase space gives operators that provide an infinite dimensionalrepresentation of U(d) on the harmonic oscillator state space H. This methodfor turning symmetries of the classical phase space into unitary representationsof the symmetry group on a quantum state space is elaborated in great detailhere not just because of its application to these simple quantum systems, butbecause it will turn out to be fundamental in our later study of quantum fieldtheories.

We will see in detail how in the case d = 2 the group U(2) ⊂ Sp(4,R) com-mutes with the Hamiltonian, so acts as symmetries preserving energy eigenspaceson the harmonic oscillator state space. This gives the same construction of allSU(2) ⊂ U(2) irreducible representations that we studied in chapter 8. Thecase d = 3 corresponds to the physical example of a quadratic central potentialin three dimensions, with the rotation group acting on the state space as anSO(3) subgroup of the subgroup U(3) ⊂ Sp(6,R) of symmetries commutingwith the Hamiltonian.

259

23.1 Complex structures and the Sp(2d,R) ac-tion on M

We saw in chapter 21 that a generalization of the Bargmann-Fock representationcan be defined for any choice of a positive compatible complex structure J onM = R2d. J provides a decomposition

M⊗C =M+J ⊕M

−J

One can choose complex coordinates zj , j = 1, · · · , d which will be basis el-ements in M−J , while the zj , j = 1, · · · , d are basis elements in M+. Thestandard choice of complex structure J = J0 corresponds to the choice

zj =1√2

(qj + ipj), zj =1√2

(qj − ipj)

with more general choices of complex structures given by the linear combinationsof qj , pj described in detail for the d = 1 case in section 21.4. For the calculationsof this chapter, one can assume unless otherwise stated that either the choiceJ = J0 has been made, or the choice of complex structure does not matter.

The choice of complexified coordinate functions zj , zj gives a decompositionof the complexified Lie algebra sp(2d,C) into three sub-algebras as follows:

• A Lie subalgebra with basis elements zjzk (as usual, the Lie bracket is thePoisson bracket). There are 1

2 (d2 +d) distinct such basis elements. This isa commutative Lie subalgebra, since the Poisson bracket of any two basiselements is zero.

• A Lie subalgebra with basis elements zjzk. Again, it has dimension 12 (d2+

d) and is a commutative Lie subalgebra.

• A Lie subalgebra with basis elements zjzk, which has dimension d2. Com-puting Poisson brackets one finds

zjzk, zlzm =zjzk, zlzm+ zkzj , zlzm=− izjzmδkl + izlzkδjm (23.1)

The first two subalgebras correspond to complexified infinitesimal Sp(2,R)transformations that do not preserve the decomposition

M⊗C =M+J0⊕M−J0

These are the analogs for arbitrary d of the Bogoliubov transformations studiedin the case d = 1 and will be discussed in section 23.3. Here we’ll consider thethird subalgebra and the operators that arise by quantization of its elements.

Taking all complex linear combinations, this subalgebra can be identifiedwith the Lie algebra gl(d,C) of all d by d complex matrices, since if Ejk is thematrix with 1 at the j-th row and k-th column, zeros elsewhere, one has

[Ejk, Elm] = Ejmδkl − Elkδjm

260

and these provide a basis of gl(d,C). Identifying bases by

izjzk ↔ Ejk

gives the isomorphism of Lie algebras. This gl(d,C) is the complexification ofu(d), the Lie algebra of the unitary group U(d). Elements of u(d) will corre-spond to skew-adjoint matrices so real linear combinations of the real quadraticfunctions

zjzk + zjzk, i(zjzk − zjzk)

on M.The moment map here is

A ∈ gl(d,C)→ µA = i∑j,k

zjAjkzk

and we have

Theorem 23.1. One has the Poisson bracket relation

µA, µA′ = µ[A,A′]

so the moment map is a Lie algebra homomorphism.One also has (for column vectors z with components z1, . . . , zd)

µA, z = −Az, µA, z = AT z (23.2)

Proof. Using 23.1 one has

µA, µA′ = −∑j,k,l,m

zjAjkzk, zlA′lmzm

= −∑j,k,l,m

AjkA′lmzjzk, zlzm

= i∑j,k,l,m

AjkA′lm(zjzmδkl − zlzkδjm)

= i∑j,k

zj [A,A′]jkzk = µ[A,A′]

To show 23.2, compute

µA, zl =i∑j,k

zjAjkzk, zl = i∑j,k

zjAjkzk, zl

=∑j

zjAjl

and

µA, zl =i∑j,k

zjAjkzk, zl = i∑j,k

Ajkzj , zlzk

=−∑k

Alkzk

261

Note that here we have written formulas for A ∈ gl(d,C), an arbitrary com-

plex d by d matrix. It is only for A ∈ u(d), the skew-adjoint (AT = −A)matrices, that µA will be a real-valued moment map, lying in the real lie al-gebra sp(2d,R), and giving a unitary representation on the state space afterquantization. For such A we can write the relations 23.2 as a (complexified)example of 14.14

µA,(

zz

) =

(AT 0

0 AT

)(zz

)Recall that in chapter 14 we found the moment map µL = −q · Ap for

elements L ∈ sp(2d,R) of the block-diagonal form(A 00 −AT

)where A is a real d by d matrix and so in gl(d,R). That block decompositioncorresponded to the decomposition of basis vectors of M into the sets qj andpj . Here we have complexified, and are working with respect to a differentdecomposition, that of M⊗C =M+

J ⊕M−J . The matrices A in this case are

complex, skew-adjoint, and in a different Lie subalgebra, u(d) ⊂ sp(2d,R).

The standard Hamiltonian

h =

d∑j=1

zjzj

lies in this sub-algebra (it is the case A = −i1), and one can show that itsPoisson brackets with the rest of the sub-algebra are zero. It gives a basiselement of the one-dimensional u(1) subalgebra that commutes with the rest ofthe u(d) subalgebra.

In section 22.2, for the case d = 1 and J = J0, we found that there wasa U(1) ⊂ SL(2,R) group acting on M preserving Ω, and commuting with J0.Complexifying M this U(1) acted separately on M+

J0and M−J0

, and there wasa moment map taking Z = −J0 to the function µZ = zz on M . Here we havea U(d) ⊂ Sp(2d,R), again acting on M preserving Ω, and commuting with J ,so also acting separately on M+

J and M−J after complexification.

23.2 The metaplectic representation and U(d) ⊂Sp(2d,R)

Turning to the quantization problem, we would like to extend the quantization oflinear functions of zj , zj of chapter 21 to quadratic functions, using annihilationand creation operators. For any j, k one can take

zjzk → −iajak, zjzk → −ia†ja†k

262

There is no ambiguity in the quantization of the two subalgebras given by pairsof the z coordinates or pairs of the z coordinates since creation operators com-mute with each other, and annihilation operators commute with each other.

If j 6= k one can take

zjzk → −ia†jak = −iaka†j

and there is again no ordering ambiguity. If j = k, as in the d = 1 case there isa choice to be made. One possibility is to take

zjzj → −i1

2(aja

†j + a†jaj) = −i(a†jaj +

1

2)

which will have the proper sp(2d,R) commutation relations (in particular for

commutators of a2j with (a†j)

2), but require going to a double cover to get a truerepresentation of the group. The Bargmann-Fock construction thus gives us aunitary representation of u(d) on Fock space Fd, but after exponentiation this

is a representation not of the group U(d), but of a double cover we call U(d).

One could instead quantize using normal-ordered operators, taking

zjzj → −ia†jaj

The definition of normal ordering generalizes simply, since the order of anni-hilation and creation operators with different values of j is immaterial. If oneuses this normal-ordered choice, one has shifted the usual quantized operatorsof the Bargmann-Fock representation by a scalar 1

2 for each j, and after expo-nentiation the state space H = Fd provides a representation of U(d), with noneed for a double cover. As a u(d) representation however, this does not extend

to a representation of sp(2d,R), since commutation of a2j with (a†j)

2 can landone on the unshifted operators.

We saw above that the infinitesimal action of u(d) ⊂ sp(2d,R) preservesthe decomposition of M ⊗ C = M+ ⊕ M−, and this will be true after ex-ponentiating for U(d) ⊂ Sp(2d,R). We won’t show this here, but U(d) isthe maximal subgroup that preserves this decomposition. The analog of thed = 1 parametrization of possible distinguished states |0〉 by SL(2,R)/U(1)here would be a parametrization of such states (or, equivalently, of possiblechoices of J) by the space Sp(2d,R)/U(d), the Siegel upper half-space.

Since the normal-ordering doesn’t change the commutation relations obeyedby products of the form a†jak, one can quantize the quadratic expression for

µA and get quadratic combinations of the aj , a†k with the same commutation

relations as in theorem 23.1. Letting

U ′A =∑j,k

a†jAjkak (23.3)

we have

263

Theorem 23.2. For A ∈ gl(d,C) a d by d complex matrix one has

[U ′A, U′A′ ] = U[A,A′]

SoA ∈ gl(d,C)→ U ′A

is a Lie algebra representation of gl(d,C) on H = C[w1, . . . , wd], the harmonicoscillator state space in d degrees of freedom.

One also has (for column vectors a with components a1, . . . , ad)

[U ′A,a†] = ATa†, [U ′A,a] = −Aa (23.4)

Proof. Essentially the same proof as 23.1.

For A ∈ u(d) the Lie algebra representation U ′A of u(d) exponentiates to givea representation of U(d) on H = C[w1, . . . , wd] by operators

UeA = eU′A

These satisfy

UeAa†(UeA)−1 = eAT

a†, UeAa(UeA)−1 = eATa (23.5)

(the relations 24.1 are the derivative of these). This shows that the UeA areintertwining operators for a U(d) action on annihilation and creation operatorsthat preserves the canonical commutation relations (the relations that say the

aj , a†j give a representation of the complexified Heisenberg Lie algebra). Here

the use of normal-ordered operators means that U ′A is a representation of u(d)that differs by a constant from the metaplectic representation, and UeA differsby a phase-factor. This does not affect the commutation relations with U ′A orthe conjugation action of UeA . The representation one gets this way differsin two ways from the metaplectic representation. It acts on the same spaceH = Fd, but it is a true representation of U(d), no double-cover is needed. Italso does not extend to a representation of the larger group Sp(2d,R).

The operators U ′A and UeA commute with the Hamiltonian operator. Fromthe physics point of view, this is useful, as it provides a decomposition of en-ergy eigenstates into irreducible representations of U(d). From the mathematicspoint of view, the quantum harmonic oscillator state space provides a construc-tion of a large class of irreducible representations of U(d) (the energy eigenstatesof a given energy).

23.3 Bogoliubov transformations

For A ∈ u(d) ⊂ sp(2d,R) we have seen that we can construct the Lie algebraversion of the metaplectic representation as

U ′A =1

2

∑j,k

Ajk(a†jak + aka†j)

264

which gives a representation that extends to sp(2d,R), or we can normal-order,getting

U ′A = :U ′A: =∑j,k

a†jAjkak = U ′A −1

2

d∑j=1

Ajj1

To see that this normal-ordered version does not extend to sp(2d,R), observethat basis elements of sp(2d,R) that are not in u(d) are the linear combinationsof zjzk and zjzk that correspond to real-valued functions. These are given by

1

2

∑jk

(Bjkzjzk +Bjkzjzk)

for a complex symmetric matrix B with matrix entries Bjk. There is no normalordering ambiguity here, and quantization will give the unitary Lie algebrarepresentation operators

− i2

∑jk

(Bjka†ja†k +Bjkajak)

Exponentiating such operators will give Bogoliubov transformations, which take|0〉 to a distinct state (one not proportional to |0〉).

Using the canonical commutation relations one can show

[a†ja†k, alam] = −ala†jδkm − ala

†kδjm − a

†jamδkl − a

†kamδjl

and these relations can in turn be used to compute the commutator of two suchLie algebra representation operators, with the result

[− i2

∑jk

(Bjka†ja†k +Bjkajak),− i

2

∑lm

(Clma†l a†m + Clmalam)]

=1

2

∑jk

(BC − CB)jk(a†jak + aka†j) = U ′

BC−CB

Note that one has

U ′BC−CB = :U ′

BC−CB : = U ′BC−CB −

1

2tr(BC − CB)1 (23.6)

The normal-ordered operators fail to give a Lie algebra homomorphism whenextended to sp(2d,R), but this failure is just by a constant term. Recall fromsection 13.3 that even at the classical level, there was an ambiguity of a constantin the choice of a moment map which in principle could lead to an “anomaly”, asituation where the moment map failed to be a Lie algebra homomorphism by aconstant term. The situation here is that this potential “anomaly” is removable,just by the shift

U ′A → U ′A +1

2tr(A)1 = U ′A

which gives representation operators that satisfy the Lie algebra homomorphismproperty. We will see in chapter 37 that for an infinite number of degrees offreedom, the anomaly may not be removable, since the trace of the operatormay be divergent.

265

23.4 Examples in d = 2 and 3

23.4.1 Two degrees of freedom and SU(2)

In the case d = 2, the group U(2) ⊂ Sp(4,R) preserving the complex structureJ0 commutes with the standard harmonic oscillator Hamiltonian and so actsas symmetries on the quantum state space, preserving energy eigenspaces. Re-stricting to the subgroup SU(2) ⊂ U(2), we can recover our earlier constructionof SU(2) representations in terms of homogeneous polynomials, in a new con-text. This use of the energy eigenstates of a two-dimensional harmonic oscillatorappears in the physics literature as the “Schwinger boson method” for studyingrepresentations of SU(2).

The state space for the d = 2 Bargmann-Fock representation, restricting tofinite linear combinations of energy eigenstates, is

H = Ffin2 = C[w1, w2]

the polynomials in two complex variables w1, w2. Recall from our SU(2) dis-cussion that it was useful to organize these polynomials into finite dimensionalsets of homogeneous polynomials of degree n for n = 0, 1, 2, . . .

H = H0 ⊕H1 ⊕H2 ⊕ · · ·

There are four annihilation or creation operators

a†1 = w1, a†2 = w2, a1 =

∂

∂w1, a2 =

∂

∂w2

acting on H. These are the quantizations of complexified phase space coordi-nates z1, z2, z1, z2 with quantization just the Bargmann-Fock construction ofthe representation Γ′BF of h2d

Γ′BF (1) = −i1, Γ′BF (zj) = −ia†j , Γ′BF (zj) = −iaj

Our original dual phase space wasM = R4, with a group Sp(4,R) acting onit, preserving the antisymmetric bilinear form Ω. When picking the coordinatesz1, z2, we made a standard choice of complex structure J0 onM. Complexifying,we have


= C2 ⊕C2

where z1, z2 are coordinates onM−J0, z1, z2 are coordinates onM+

J0. This choice

of J = J0 picks out a distinguished subgroup U(2) ⊂ Sp(4,R).The quadratic combinations of the creation and annihilation operators give

representations on H of three subalgebras of the complexification sp(4,C) ofsp(4,R):

• A three dimensional commutative Lie sub-algebra spanned by z1z2, z21 , z

22 ,

with quantization

Γ′BF (z1z2) = −ia1a2, Γ′BF (z21) = −ia2

1, Γ′BF (z22) = −ia2

2

266

• A three dimensional commutative Lie sub-algebra spanned by z1z2, z21, z

22,

with quantization

Γ′BF (z1z2) = −ia†1a†2, Γ′BF (z2

1) = −i(a†1)2, Γ′BF (z22) = −i(a†2)2

• A four dimensional Lie subalgebra isomorphic to gl(2,C) with basis

z1z1, z2z2, z2z1, z1z2

and quantization

Γ′BF (z1z1) = − i2

(a†1a1 + a1a†1), Γ′BF (z2z2) = − i

2(a†2a2 + a2a

†2)

Γ′BF (z2z1) = −ia†2a1, Γ′BF (z1z2) = −ia†1a2

Real linear combinations of

z1z1, z2z2, z1z2 + z2z1, i(z1z2 − z2z1)

span the Lie algebra u(2) ⊂ sp(4,R), and Γ′BF applied to these gives aunitary Lie algebra representation by skew-adjoint operators.

Inside this last subalgebra, there is a distinguished element h = z1z1 + z2z2

that Poisson-commutes with the rest of the subalgebra (but not with elementsin the first two subalgebras). Quantization of h gives the Hamiltonian operator

H =1

2(a1a

†1 + a†1a1 + a2a

†2 + a†2a2) = N1 +

1

2+N2 +

1

2= w1

∂

∂w1+w2

∂

∂w2+ 1

This operator will just multiply a homogeneous polynomial by its degree plusone, so it acts just by multiplication by n + 1 on Hn. Exponentiating thisoperator (multiplied by −i) one gets a representation of a U(1) subgroup of themetaplectic cover Mp(4,R). Taking instead the normal-ordered version

:H: = a†1a1 + a†2a2 = N1 +N2 = w1∂

∂w1+ w2

∂

∂w2

one gets a representation of a U(1) subroup of Sp(4,R). Neither H nor :H:commutes with operators coming from quantization of the first two subalgebras.These change the eigenvalue of H or :H: by ±2 so take

Hn → Hn±2

in particular taking |0〉 to either 0 or a state in H2.h is a basis element for the u(1) in u(2) = u(1)⊕ su(2). For the su(2) part a

correspondence to our basis Xj = −iσj2 in terms of 2 by 2 matrices is

X1 ↔1

2(z1z2 + z2z1), X2 ↔

i

2(z2z1 − z1z2), X3 ↔

1

2(z1z1 − z2z2)

267

This relates two different but isomorphic ways of describing su(2): as 2 by 2matrices with Lie bracket the commutator, or as quadratic polynomials, withLie bracket the Poisson bracket.

Quantizing using normal-ordering of operators give a representation of su(2)on H

Γ′(X1) = − i2

(a†1a2 + a†2a1), Γ′(X2) =1

2(a†2a1 − a†1a2)

Γ′(X3) = − i2

(a†1a1 − a†2a2)

Comparing this to the representation π′ of su(2) on homogeneous polynomialsdiscussed in chapter 8, one finds that they are isomorphic, although they act ondual spaces, so Γ′(X) = π′(−XT ) for all X ∈ su(2).

We see that, up to this change from a vector space to its dual, and thenormal-ordering (which only affects the u(1) factor, shifting the Hamiltonian bya constant), the Bargmann-Fock representation on polynomials and the SU(2)representation on homogeneous polynomials are identical. The inner productthat makes the representation unitary is the one of equation 8.2. The Bargmann-Fock representation extends this SU(2) representation as a unitary representa-tion to a much larger group (H5 oMp(4,R)), with all polynomials in w1, w2

now making up a single irreducible representation.

The fact that we have an SU(2) group acting on the state space of the d = 2harmonic oscillator and commuting with the action of the Hamiltonian H meansthat energy eigenstates can be organized as irreducible representations of SU(2).In particular, one sees that the space Hn of energy eigenstates of energy n+ 1will be a single irreducible SU(2) representation, the spin n

2 representation ofdimension n+ 1 (so n+ 1 will be the multiplicity of energy eigenstates of thatenergy).

Another physically interesting subgroup here is the SO(2) ⊂ SU(2) ⊂Sp(4,R) consisting of simultaneous rotations in the position and momentumplanes, which was studied in detail using the coordinates q1, q2, p1, p2 in section18.2.2. There we found that the moment map was given by

µL = l = q1p2 − q2p1

and quantization by the Schrodinger representation gave a representation of theLie algebra so(2) with

U ′L = −i(Q1P2 −Q2P1)

Note that this is a different SO(2) action than the one with moment map theHamiltonian, it acts separately on positions and momenta rather than mixingthem.

To see what happens if one instead uses the Bargmann-Fock representation,note that

qj =1√2

(zj + zj), pj = i1√2

(zj − zj)

268

so the moment map is

µL =i

2((z1 + z1)(z2 − z2)− (z2 + z2)(z1 − z1))

=i(z2z1 − z1z2)

Quantizing, Bargmann-Fock gives a unitary representation of so(2)

U ′L = a†2a1 − a†1a2

which is Γ′(2X2). The factor of two here reflects the fact that exponentiationgives a representation of SO(2) ⊂ Sp(4,R), with no need for a double cover.

23.4.2 Three degrees of freedom and SO(3)

The case d = 3 corresponds physically to the so-called isotropic quantum har-monic oscillator system, and it is an example of the sort of central poten-tial problem we studied in chapter 19 (since the potential just depends onr2 = q2

1 + q22 + q2

3). For such problems, we saw that since the classical Hamilto-nian is rotationally invariant, the quantum Hamiltonian will commute with theaction of SO(3) on wavefunctions and energy eigenstates can be decomposedinto irreducible representations of SO(3).

Here the Bargmann-Fock representation gives an action of H7oMp(6,R) onthe state space, with a U(3) subgroup commuting with the Hamiltonian (moreprecisely one has a double cover of U(3), but by normal-ordering one can get anactual U(3)). The eigenvalue of the U(1) corresponding to the Hamiltonian givesthe energy of a state, and states of a given energy will be sums of irreduciblerepresentations of SU(3). This works much like in the d = 2 case, although hereour irreducible representations are the spaces Hn of homogeneous polynomialsof degree n in three variables rather than two. These spaces have dimension12 (n + 1)(n + 2). A difference with the SU(2) case is that one does not get allirreducible representations of SU(3) this way.

The rotation group SO(3) will be a subgroup of this U(3) and one can askhow the SU(3) irreducible Hn decomposes into a sum of irreducibles of thesubgroup (which will be characterized by an integral spin l = 0, 1, 2, · · · ). Onecan show that for even n one gets all even values of l from 0 to n, and for oddn one gets all odd values of l from 1 to n. A derivation can be found in somequantum mechanics textbooks, see for example pgs. 456-460 of [47].

To construct the angular momentum operators in the Bargmann-Fock rep-resentation, recall that in the Schrodinger representation these were

L1 = Q2P3 −Q3P2, L2 = Q3P1 −Q1P3, L3 = Q1P2 −Q2P1

and one can rewrite these operators in terms of annihilation and creation op-erators. Alternatively, one can use theoem 23.2, for Lie algebra basis elementslj ∈ so(3) ⊂ u(3) ⊂ gl(3,C) which are (see chapter 6)

l1 =

0 0 00 0 −10 1 0

, l2 =

0 0 10 0 0−1 0 0

, l3 =

0 −1 01 0 00 0 0

269

to calculate

−iLj = U ′lj =

3∑m,n=1

a†m(lj)mnan

This gives

U ′l1 = a†3a2 − a†2a3, U ′l2 = a†1a3 − a†3a1, U ′l3 = a†2a1 − a†1a2

Exponentiating these operators gives a representation of the rotation groupSO(3) on the state space F3, commuting with the Hamiltonian, so acting onenergy eigenspaces (which will be the homogeneous polynomials of fixed degree).


The references from chapter 22 ([20], [73]) also contain the general case dis-cussed here. The construction of the metaplectic representation restricted toU(d) ⊂ Sp(2d,R) using annihilation and creation operators is a standard topicin quantum field theory textbooks, although there in an infinite rather thanfinite-dimensional context (and not explicitly in the language used here). Wewill encounter the quantum field theory version in later chapters.

270

Chapter 24

The Fermionic Oscillator

In this chapter we’ll introduce a new quantum system by using a simple varia-tion on techniques we used to study the harmonic oscillator, that of replacingcommutators by anticommutators. This variant of the harmonic oscillator willbe called a “fermionic oscillator”, with the original sometimes called a “bosonicoscillator”. The terminology of “boson” and “fermion” refers to the principleenunciated in chapter 9 that multiple identical particles are described by tensorproduct states that are either symmetric (bosons) or antisymmetric (fermions).

The bosonic and fermionic oscillator systems are single-particle systems, de-scribing the energy states of a single particle, so the usage of the bosonic/fermion-ic terminology is not obviously relevant. In later chapters we will study quantumfield theories, which can be treated as infinite-dimensional oscillator systems.In that context, multiple particle states will automatically be symmetric or an-tisymmetric, depending on whether the field theory is treated as a bosonic orfermionic oscillator system, thus justifying the terminology.

24.1 Canonical anticommutation relations andthe fermionic oscillator

Recall that the Hamiltonian for the quantum harmonic oscillator system in ddegrees of freedom (setting ~ = m = ω = 1) is

H =

d∑j=1

1

2(Q2

j + P 2j )

and that it can be diagonalized by introducing number operators Nj = a†jajdefined in terms of operators

aj =1√2

(Qj + iPj), a†j =1√2

(Qj − iPj)

271

that satisfy the so-called canonical commutation relations (CCR)

[aj , a†k] = δjk1, [aj , ak] = [a†j , a

†k] = 0

The simple change in the harmonic oscillator problem that takes one frombosons to fermions is the replacement of the bosonic annihilation and creationoperators (which we’ll now denote aB and aB

†) by fermionic annihilation andcreation operators called aF and aF

†, and replacement of the commutator

[A,B] ≡ AB −BA

of operators by the anticommutator

[A,B]+ ≡ AB +BA

The commutation relations are now

[aF , a†F ]+ = 1, [aF , aF ]+ = 0, [a†F , a

†F ]+ = 0

with the last two relations implying that a2F = 0 and (a†F )2 = 0

The fermionic number operator

NF = a†FaF

now satisfies

N2F = a†FaFa

†FaF = a†F (1− a†FaF )aF = NF − a†F

2a2F = NF

(using the fact that a2F = a†F

2= 0). So one has

N2F −NF = NF (NF − 1) = 0

which implies that the eigenvalues of NF are just 0 and 1. We’ll denote eigen-vectors with such eigenvalues by |0〉 and |1〉. The simplest representation of the

operators aF and a†F on a complex vector space HF will be on C2, and choosingthe basis

|0〉 =

(01

), |1〉 =

(10

)the operators are represented as

aF =

(0 01 0

), a†F =

(0 10 0

), NF =

(1 00 0

)Since

H =1

2(a†FaF + aFa

†F )

is just 12 the identity operator, to get a non-trivial quantum system, instead we

make a sign change and set

H =1

2(a†FaF − aFa

†F ) = NF −

1

21 =

(12 00 − 1

2

)272

The energies of the energy eigenstates |0〉 and |1〉 will then be ± 12 since

H|0〉 = −1

2|0〉, H|1〉 =

1

2|1〉

Note that the quantum system we have constructed here is nothing but ourold friend the two-state system of chapter 3. Taking complex linear combinationsof the operators

aF , a†F , NF ,1

we get all linear transformations of HF = C2 (so this is an irreducible represen-tation of the algebra of these operators). The relation to the Pauli matrices isjust

a†F =1

2(σ1 + iσ2), aF =

1

2(σ1 − iσ2), H =

1

2σ3

24.2 Multiple degrees of freedom

For the case of d degrees of freedom, one has this variant of the canonicalcommutation relations (CCR) amongst the bosonic annihilation and creation

operators aBj and aB†j :

Definition (Canonical anticommutation relations). A set of 2d operators

aF j , aF†j , j = 1, . . . , d

is said to satisfy the canonical anticommutation relations (CAR) when one has

[aF j , aF†k]+ = δjk1, [aF j , aF k]+ = 0, [aF

†j , aF

†k]+ = 0

In this case one may choose as the state space the tensor product of N copiesof the single fermionic oscillator state space

HF = (C2)⊗d = C2 ⊗C2 ⊗ · · · ⊗C2︸︷︷︸d times

The dimension of HF will be 2d. On this space the operators aF j and aF†j can

be explicitly given by

aF j = σ3 ⊗ σ3 ⊗ · · · ⊗ σ3︸︷︷︸j−1 times

⊗(

0 01 0

)⊗ 1⊗ · · · ⊗ 1

aF†j = σ3 ⊗ σ3 ⊗ · · · ⊗ σ3︸︷︷︸

j−1 times

⊗(

0 10 0

)⊗ 1⊗ · · · ⊗ 1

The factors of σ3 ensure that the canonical anticommutation relations

[aF j , aF k]+ = [aF†j , aF

†k]+ = [aF j , aF

†k]+ = 0

273

are satisfied for j 6= k since in these cases one will get in the tensor productfactors

[σ3,

(0 01 0

)]+ = 0 or [σ3,

(0 10 0

)]+ = 0

While this sort of tensor product construction is useful for discussing thephysics of multiple qubits, in general it is easier to not work with large tensorproducts, and the Clifford algebra formalism we will describe in chapter 25avoids this.

The number operators will be

NF j = aF†jaF j

These will commute with each other, so can be simultaneously diagonalized,with eigenvalues nj = 0, 1. One can take as an orthonormal basis of HF the 2d

states|n1, n2, · · · , nd〉

As an example, for the case d = 3 the pattern of states and their energylevels for the bosonic and fermionic cases looks like this

274

In the bosonic case the lowest energy state is at positive energy and there arean infinite number of states of ever increasing energy. In the fermionic case thelowest energy state is at negative energy, with the pattern of energy eigenvaluesof the finite number of states symmetric about the zero energy level.

Just as in the bosonic case, we can consider quadratic combinations of cre-ation and annihilation operators of the form

U ′A =∑j,k

a†F jAjkaF k

and we have

Theorem 24.1. For A ∈ gl(d,C) a d by d complex matrix one has

[U ′A, U′A′ ] = U[A,A′]

SoA ∈ gl(d,C)→ U ′A

is a Lie algebra representation of gl(d,C) on HFOne also has (for column vectors aF with components aF 1, . . . , aF d)

[U ′A,a†F ] = ATa†F , [U ′A,aF ] = −AaF (24.1)

Proof. The proof is similar to that of 23.1, except besides the relation

[AB,C] = A[B,C] + [A,B]C

we also use the relation

[AB,C] = A[B,C]+ − [A,B]+C

For example

[U ′A, aF†l ] =

∑j,k

[aF†jAjkaF k, aF

†l ]

=∑j,k

aF†jAjk[aF k, aF

†l ]+

=∑j

aF†jAjl

The Hamiltonian is

H =∑j

(NF j −1

21)

which (up to the constant 12 that doesn’t contribute to commutation relations) is

just U ′B for the case B = 1. Since this commutes with all other d by d matrices,we have

[H,U ′A] = 0

for all A ∈ gl(d,C), so these are symmetries and we have a representation ofthe Lie algebra gl(d,C) on each energy eigenspace. Only for A ∈ u(d) (A askew-adjoint matrix) will the representation turn out to be unitary.

275


Most quantum field theory books and a few quantum mechanics books containsome sort of discussion of the fermionic oscillator, see for example Chapter21.3 of [63] or Chapter 5 of [11]. The standard discussion often starts withconsidering a form of classical analog using anticommuting “fermionic” variablesand then quantization to get the fermionic oscillator. Here we are doing thingsin the opposite order, starting in this chapter with the quantized oscillator, thenconsidering the classical analog in a later chapter.

276

Chapter 25

Weyl and Clifford Algebras

We have seen that just changing commutators to anticommutators takes theharmonic oscillator quantum system to a very different one (the fermionic os-cillator), with this new system having in many ways a parallel structure. Itturns out that this parallelism goes much deeper, with every aspect of the har-monic oscillator story having a fermionic analog. We’ll begin in this chapter bystudying the operators of the corresponding quantum systems.

25.1 The Complex Weyl and Clifford algebras

In mathematics, a “ring” is a set with addition and multiplication laws that areassociative and distributive (but not necessarily commutative), and an “algebra”is a ring that is also a vector space over some field of scalars. The canonicalcommutation and anticommutation relations define interesting algebras, calledthe Weyl and Clifford algebras respectively. The case of complex numbers asscalars is simplest, so we’ll start with that, before moving on to the real numbercase.

25.1.1 One degree of freedom, bosonic case

Starting with the one degree of freedom case (corresponding to two operatorsQ,P , which is why the notation will have a 2) we can define

Definition (Complex Weyl algebra, one degree of freedom). The complex Weylalgebra in the one degree of freedom case is the algebra Weyl(2,C) generated by

the elements 1, aB , a†B, satisfying the canonical commutation relations:

[aB , a†B ] = 1, [aB , aB ] = [a†B , a

†B ] = 0

In other words, Weyl(2,C) is the algebra one gets by taking arbitrary prod-ucts and complex linear combinations of the generators. By repeated use of thecommutation relation

aBa†B = 1 + a†BaB

277

any element of this algebra can be written as a sum of elements in normal order,of the form

cl,m(a†B)lamB

with all annihilation operators aB on the right, for some complex constants cl,m.As a vector space over C, Weyl(2,C) is infinite-dimensional, with a basis

1, aB , a†B , a

2B , a

†BaB , (a†B)2, a3

B , a†Ba

2B , (a†B)2aB , (a†B)3, . . .

This algebra is isomorphic to a more familiar one. Setting

a†B = w, aB =∂

∂w

one sees that Weyl(2,C) can be identified with the algebra of polynomial coeffi-cient differential operators on functions of a complex variable w. As a complexvector space, the algebra is infinite dimensional, with a basis of elements

wl∂m

∂wm

In our study of quantization and the harmonic oscillator we saw that thesubset of such operators consisting of complex linear combinations of

1, w,∂

∂w, w2,

∂2

∂w2, w

∂

∂w

is closed under commutators, so it forms a Lie algebra of complex dimension 6.This Lie algebra includes as subalgebras the Heisenberg Lie algebra h3⊗C (firstthree elements) and the Lie algebra sl(2,C) = sl(2,R)⊗C (last three elements).Note that here we are allowing complex linear combinations, so we are gettingthe complexification of the real six-dimensional Lie algebra that appeared inour study of quantization.

Since the aB and a†B are defined in terms of P and Q, one could of coursealso define the Weyl algebra as the one generated by 1, P,Q, with the Heisenbergcommutation relations, taking complex linear combinations of all products ofthese operators.

25.1.2 One degree of freedom, fermionic case

Changing commutators to anticommutators, one gets a different algebra, theClifford algebra

Definition (Complex Clifford algebra, one degree of freedom). The complexClifford algebra in the one degree of freedom case is the algebra Cliff(2,C) gen-

erated by the elements 1, aF , a†F , subject to the canonical anticommutation rela-

tions

[aF , a†F ]+ = 1, [aF , aF ]+ = [a†F , a

†F ]+ = 0

278

This algebra is a four dimensional algebra over C, with basis

1, aF , a†F , a†FaF

since higher powers of the operators vanish, and one can use the anticommu-tation relation betwee aF and a†F to normal order and put factors of aF onthe right. We saw in the last chapter that this algebra is isomorphic with thealgebra M(2,C) of 2 by 2 complex matrices, using

1↔(

1 00 1

), aF ↔

(0 01 0

), a†F ↔

(0 10 0

), a†FaF ↔

(1 00 0

)We will see later on that there is also a way of identifying this algebra with“differential operators in fermionic variables”, analogous to what happens inthe bosonic (Weyl algebra) case.

Recall that the bosonic annihilation and creation operators were originallydefined in term of the P and Q operators by

aB =1√2

(Q+ iP ), a†B =1√2

(Q− iP )

Looking for the fermionic analogs of the operators Q and P , we use a slightlydifferent normalization, and set

aF =1

2(γ1 + iγ2), a†F =

1

2(γ1 − iγ2)

so

γ1 = aF + a†F , γ2 =1

i(aF − a†F )

and the CAR imply that the operators γj satisfy the anticommutation relations

[γ1, γ1]+ = [aF + a†F , aF + a†F ]+ = 2

[γ2, γ2]+ = −[aF − a†F , aF − a†F ]+ = 2

[γ1, γ2]+ =1

i[aF + a†F , aF − a

†F ]+ = 0

From this we see that

• One could alternatively have defined Cliff(2,C) as the algebra generatedby 1, γ1, γ2, subject to the relations

[γj , γk]+ = 2δjk

• Using just the generators 1 and γ1, one gets an algebra Cliff(1,C), gener-ated by 1, γ1, with the relation

γ21 = 1

This is a two-dimensional complex algebra, isomorphic to C⊕C.

279

25.1.3 Multiple degrees of freedom

For a larger number of degrees of freedom, one can generalize the above anddefine Weyl and Clifford algebras as follows:

Definition (Complex Weyl algebras). The complex Weyl algebra for d degrees

of freedom is the algebra Weyl(2d,C) generated by the elements 1, aBj , aB†j,

j = 1, . . . , d satisfying the CCR

[aBj , aB†k] = δjk1, [aBj , aBk] = [aB

†j , aB

†k] = 0

Weyl(2d,C) can be identified with the algebra of polynomial coefficient dif-ferential operators in m complex variables w1, w2, . . . , wd. The subspace ofcomplex linear combinations of the elements

1, wj ,∂

∂wj, wjwk,

∂2

∂wj∂wk, wj

∂

∂wk

is closed under commutators and is isomorphic to the complexification of the Liealgebra h2d+1 o sp(2d,R) built out of the Heisenberg Lie algebra in 2d variablesand the Lie algebra of the symplectic group Sp(2d,R). Recall that this is theLie algebra of polynomials of degree at most 2 on the phase space R2d, with thePoisson bracket as Lie bracket.

One could also define the complex Weyl algebra by taking complex linearcombinations of products of generators 1, Pj , Qj , subject to the Heisenberg com-mutation relations.

For Clifford algebras one has

Definition (Complex Clifford algebras, using annilhilation and creation op-erators). The complex Clifford algebra for d degrees of freedom is the algebra

Cliff(2d,C) generated by 1, aF j , aF†j for j = 1, 2, . . . , d satisfying the CAR

[aF j , aF†k]+ = δjk1, [aF j , aF k]+ = [aF

†j , aF

†k]+ = 0

or, alternatively, one has the following more general definition that also worksin the odd-dimensional case

Definition (Complex Clifford algebras). The complex Clifford algebra in n vari-ables is the algebra Cliff(n,C) generated by 1, γj for j = 1, 2, . . . , n satisfyingthe relations


We won’t try and prove this here, but one can show that, abstractly asalgebras, the complex Clifford algebras are something well-known. Generalizingthe case d = 1 where we saw that Cliff(2,C) was isomorphic to the algebra of 2by 2 complex matrices, one has isomorphisms

Cliff(2d,C)↔M(2d,C)

280

in the even-dimensional case, and in the odd-dimensional case

Cliff(2d+ 1,C)↔M(2d,C)⊕M(2d,C)

Two properties of Cliff(n,C) are

• As a vector space over C, a basis of Cliff(n,C) is the set of elements

1, γj , γjγk, γjγkγl, . . . , γ1γ2γ3 · · · γn−1γn

for indices j, k, l, · · · ∈ 1, 2, . . . , n, with j < k < l < · · · . To show this,consider all products of the generators, and use the commutation relationsfor the γj to identify any such product with an element of this basis. Therelation γ2

j = 1 shows that one can remove repeated occurrences of aγj . The relation γjγk = −γkγj can then be used to put elements of theproduct in the order of a basis element as above.

• As a vector space over C, Cliff(n,C) has dimension 2n. One way to seethis is to consider the product

(1 + γ1)(1 + γ2) · · · (1 + γn)

which will have 2n terms that are exactly those of the basis listed above.

25.2 Real Clifford algebras

We can define real Clifford algebras Cliff(n,R) just as for the complex case, bytaking only real linear combinations:

Definition (Real Clifford algebras). The real Clifford algebra in n variables isthe algebra Cliff(n,R) generated over the real numbers by 1, γj for j = 1, 2, . . . , nsatisfying the relations


For reasons that will be explained in the next chapter, it turns out that amore general definition is useful. We write the number of variables as n = r+s,for r, s non-negative integers, and now vary not just r + s, but also r − s, theso-called “signature”.

Definition (Real Clifford algebras, arbitrary signature). The real Clifford al-gebra in n = r + s variables is the algebra Cliff(r, s,R) over the real numbersgenerated by 1, γj for j = 1, 2, . . . , n satisfying the relations

[γj , γk]+ = ±2δjk1

where we choose the + sign when j = k = 1, . . . , r and the − sign when j = k =r + 1, . . . , n.

281

In other words, as in the complex case different γj anticommute, but onlythe first r of them satisfy γ2

j = 1, with the other s of them satisfying γ2j = −1.

Working out some of the low-dimensional examples, one finds:

• Cliff(0, 1,R). This has generators 1 and γ1, satisfying

γ21 = −1

Taking real linear combinations of these two generators, the algebra onegets is just the algebra C of complex numbers, with γ1 playing the role ofi =√−1.

• Cliff(0, 2,R). This has generators 1, γ1, γ2 and a basis

1, γ1, γ2, γ1γ2

with

γ21 = −1, γ2

2 = −1, (γ1γ2)2 = γ1γ2γ1γ2 = −γ21γ

22 = −1

This four-dimensional algebra over the real numbers can be identified withthe algebra H of quaternions by taking

γ1 ↔ i, γ2 ↔ j, γ1γ2 ↔ k

• Cliff(1, 1,R). This is the algebra M(2,R) of real 2 by 2 matrices, withone possible identification as follows

1↔(

1 00 1

), γ1 ↔

(0 11 0

), γ2 ↔

(0 −11 0

), γ1γ2 ↔

(−1 00 1

)Note that one can construct this using the aF , a

†F for the complex case

Cliff(2,C)

γ1 = aF + a†F , γ2 = aF − a†Fsince these are represented as real matrices.

• Cliff(3, 0,R). This is the algebra M(2,C) of complex 2 by 2 matrices,with one possible identification using Pauli matrices given by

1↔(

1 00 1

)

γ1 ↔ σ1 =

(0 11 0

), γ2 ↔ σ2 =

(0 −ii 0

), γ3 ↔ σ3 =

(1 00 −1

)γ1γ2 ↔ iσ3 =

(i 00 −i

), γ2γ3 ↔ iσ1 =

(0 ii 0

), γ1γ3 ↔ −iσ2 =

(0 −11 0

)γ1γ2γ3 ↔

(i 00 i

)

282

It turns out that Cliff(r, s,R) is always one or two copies of matrices of real,complex or quaternionic elements, of dimension a power of 2, but this requiresa rather intricate algebraic argument that we will not enter into here. For thedetails of this and the resulting pattern of algebras one gets, see for instance[42]. One special case where the pattern is relatively simple is when one hasr = s. Then n = 2r is even-dimensional and one finds

Cliff(r, r,R) = M(2r,R)

We will see in the next chapter that just as quadratic elements of the Weylalgebra give a basis of the Lie algebra of the symplectic group, quadratic ele-ments of the Clifford algebra give a basis of the Lie algebra of the orthogonalgroup.


A good source for more details about Clifford algebras and spinors is Chapter12 of the representation theory textbook [73]. For the details of what happensfor all Cliff(r, s,R), another good source is Chapter 1 of [42].

283

Chapter 26

Clifford Algebras andGeometry

The definitions given in last chapter of Weyl and Clifford algebras were purelyalgebraic, based on a choice of generators. These definitions do though have amore geometrical formulation, with the definition in terms of generators corre-sponding to a specific choice of coordinates. For the Weyl algebra, the geometryinvolved is known as symplectic geometry, and we have already seen that in thebosonic case quantization of a phase space R2d depends on the choice of a non-degenerate antisymmetric bilinear form Ω which determines the Poisson brack-ets and thus the Heisenberg commutation relations. Such a Ω also determines agroup Sp(2d,R), which is the group of linear transformations of R2d preservingΩ. The Clifford algebra also has a coordinate invariant definition, based on amore well-known structure on a vector space Rn, that of a non-degenerate sym-metric bilinear form, i.e. an inner product. In this case the group that preservesthe inner product is an orthogonal group. In the symplectic case antisymmetricforms require an even number of dimensions, but this is not true for symmetricforms, which also exist in odd dimensions.

26.1 Non-degenerate bilinear forms

In the case of M = R2d, the dual phase space, for two vectors u, u′ ∈M

u = cq1q1 + cp1p1 + · · ·+ cqdqd + cpdpd ∈M

u′ = c′q1q1 + c′p1p1 + · · ·+ c′qdqd + c′pdpd ∈M

the Poisson bracket determines an antisymmetric bilinear form on M, givenexplicitly by

285

Ω(u, u′) =cq1c′p1− cp1

c′q1 + · · ·+ cqdc′pd− cpdc′qd

=(cq1 cp1

. . . cqd cpd)

0 1 . . . 0 0−1 0 . . . 0 0...

......

...0 0 . . . 0 10 0 . . . −1 0

c′q1c′p1

...c′qdc′pd

Matrices g ∈M(2d,R) such that

gT

0 1 . . . 0 0−1 0 . . . 0 0...

......

...0 0 . . . 0 10 0 . . . −1 0

g =

0 1 . . . 0 0−1 0 . . . 0 0...

......

...0 0 . . . 0 10 0 . . . −1 0

make up the group Sp(2d,R) and preserve Ω, satisfying

Ω(gu, gu′) = Ω(u, u′)

This choice of Ω is much less arbitrary than it looks. One can show thatgiven any non-degenerate antisymmetric bilinear form on R2d a basis can befound with respect to which it will be the Ω given here (for a proof, see [7]).This is also true if one complexifies, taking (q,p) ∈ C2d and using the sameformula for Ω, which is now a bilinear form on C2d. In the real case the groupthat preserves Ω is called Sp(2d,R), in the complex case Sp(2d,C).

To get a fermionic analog of this, it turns out that all we need to do is replace“non-degenerate antisymmetric bilinear form Ω(·, ·)” with “non-degenerate sym-metric bilinear form 〈·, ·〉”. Such a symmetric bilinear form is actually somethingmuch more familiar from geometry than the antisymmetric case analog: it isjust a notion of inner product. Two things are different in the symmetric case:

• The underlying vector space does not have to be even dimensional, onecan take M = Rn for any n, including n odd. To get a detailed analog ofthe bosonic case though, we will mostly consider the even case n = 2d.

• For a given dimension n, there is not just one possible choice of 〈·, ·〉 up tochange of basis, but one possible choice for each pair of integers r, s suchthat r + s = n. Given r, s, any choice of 〈·, ·〉 can be put in the form

〈u,u′〉 =u1u′1 + u2u

′2 + · · ·uru′r − ur+1u

′r+1 − · · · − unu′n

=(u1 . . . un

)

1 0 . . . 0 00 1 . . . 0 0...

......

...0 0 . . . −1 00 0 . . . 0 −1

︸︷︷︸

r + signs, s - signs

u′1u′2...

u′n−1

u′n

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For a proof by Gram-Schmidt orthogonalization, see [7].

We can thus extend our definition of the orthogonal group as the group oftransformations g preserving an inner product

〈gu, gu′〉 = 〈u, u′〉

to the case r, s arbitrary by

Definition (Orthogonal group O(r, s,R)). The group O(r, s,R) is the group ofreal r + s by r + s matrices g that satisfy

gT

1 0 . . . 0 00 1 . . . 0 0...

......

...0 0 . . . −1 00 0 . . . 0 −1

︸︷︷︸


g =

1 0 . . . 0 00 1 . . . 0 0...

......

...0 0 . . . −1 00 0 . . . 0 −1

︸︷︷︸


SO(r, s,R) ⊂ O(r, s,R) is the subgroup of matrices of determinant +1.

If one complexifies, taking components of vectors to be in Cn, using thesame formula for 〈·, ·〉, one can change basis by multiplying the s basis elementsby a factor of i, and in this new basis all basis vectors ej satisfy 〈ej , ej〉 = 1.One thus sees that on Cn, as in the symplectic case, up to change of basis thereis only one non-degenerate bilinear form. The group preserving this is calledO(n,C). Note that on Cn 〈·, ·〉 is not the Hermitian inner product (which isantilinear on the first variable), and it is not positive definite.

26.2 Clifford algebras and geometry

As defined by generators in the last chapter, Clifford algebras have no obviousgeometrical significance. It turns out however that they are powerful tools inthe study of the geometry of linear spaces with an inner product, includingespecially the study of linear transformations that preserve the inner product,i.e. rotations. To see the relation between Clifford algebras and geometry,consider first the positive definite case Cliff(n,R). To an arbitrary vector

v = (v1, v2, . . . , vn) ∈ Rn

we associate the Clifford algebra element /v = γ(v) where γ is the map

v ∈ Rn → γ(v) = v1γ1 + v2γ2 + · · ·+ vnγn ∈ Cliff(n,R)

Using the Clifford algebra relations for the γj , given two vectors v, w theproduct of their associated Clifford algebra elements satisfies

/v /w + /w/v = [v1γ1 + v2γ2 + · · ·+ vnγn, w1γ1 + w2γ2 + · · ·+ wnγn]+

= 2(v1w1 + v2w2 + · · ·+ vnwn)

= 2〈v,w〉

287

where 〈·, ·〉 is the symmetric bilinear form on Rn corresponding to the standardinner product of vectors. Note that taking v = w one has

/v2 = 〈v,v〉 = ||v||2

The Clifford algebra Cliff(n,R) contains Rn as the subspace of linear combi-nations of the generators γj , and one can think of it as a sort of enhancement ofthe vector space Rn that encodes information about the inner product. In thislarger structure one can multiply as well as add vectors, with the multiplicationdetermined by the inner product.

In general one can define a Clifford algebra whenever one has a vector spacewith a symmetric bilinear form:

Definition (Clifford algebra of a symmetric bilinear form). Given a vector spaceV with a symmetric bilinear form 〈·, ·〉, the Clifford algebra Cliff(V, 〈·, ·〉) is thealgebra generated by 1 and elements of V , with the relations

/v /w + /w/v = 2〈v, w〉

Note that different people use different conventions here, with

/v /w + /w/v = −2〈v, w〉

another common choice. One also sees variants without the factor of 2.

For n dimensional vector spaces over C, we have seen that for any non-degenerate symmetric bilinear form a basis can be found such that 〈·, ·〉 has thestandard form

〈z,w〉 = z1w1 + z2w2 + · · ·+ znwn

As a result, there is just one complex Clifford algebra in dimension n, the onewe defined as Cliff(n,C).

For n dimensional vector spaces over R with a non-degenerate symmetricbilinear forms of type r, s such that r+s = n, the corresponding Clifford algebrasCliff(r, s,R) are the ones defined in terms of generators in the last chapter.

In special relativity, space-time is a real 4-dimensional vector space with anindefinite inner product corresponding to (depending on one’s choice of conven-tion) either the case r = 1, s = 3 or the case s = 1, r = 3. The group of lineartransformations preserving this inner product is called the Lorentz group, andits orientation preserving component is written as SO(3, 1) or SO(1, 3) depend-ing on the choice of convention. In later chapters we will consider what happensto quantum mechanics in the relativistic case, and there encounter the corre-sponding Clifford algebras Cliff(3, 1,R) or Cliff(1, 3,R). The generators γj ofsuch a Clifford algebra are well-known in the subject as the “Dirac γ- matrices”.

For now though, we will restrict attention to the positive definite case, sojust will be considering Cliff(n,R) and seeing how it is used to study the groupO(n) of n-dimensional rotations in Rn.

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26.2.1 Rotations as iterated orthogonal reflections

We’ll consider two different ways of seeing the relationship between the Cliffordalgebra Cliff(n,R) and the group O(n) of rotations in Rn. The first is basedupon the geometrical fact (known as the Cartan-Dieudonne theorem) that onecan get any rotation by doing multiple orthogonal reflections in different hy-perplanes. Orthogonal reflection in the hyperplane perpendicular to a vector wtakes a vector v to the vector

v′ = v − 2〈v,w〉〈w,w〉

w

something that can easily be seen from the following picture

From now on we identify vectors v,v′,w with the corresponding Cliffordalgebra elements by the map γ. The linear transformation given by reflectionin w is

/v→ /v′ =/v − 2

〈v,w〉〈w,w〉 /

w

=/v − (/v /w + /w/v)/w

〈w,w〉

Since

/w/w

〈w,w〉=〈w,w〉〈w,w〉

= 1

we have (for non-zero vectors w)

/w−1 =

/w

〈w,w〉

and the reflection transformation is just conjugation by /w times a minus sign

/v→ /v′ = /v − /v − /w/v /w

−1 = − /w/v /w−1

So, thinking of vectors as lying in the Clifford algebra, the orthogonal trans-formation that is the result of one reflection is just a conjugation (with a minus

289

sign). These lie in the group O(n), but not in the subgroup SO(n), since theychange orientation. The result of two reflections in hyperplanes orthogonal tow1,w2 will be a conjugation by /w2 /w1

/v→ /v′ = − /w2(− /w1/v /w

−11 ) /w

−12 = ( /w2 /w1)/v( /w2 /w1)−1

This will be a rotation preserving the orientation, so of determinant one and inthe group SO(n).

This construction not only gives an efficient way of representing rotations(as conjugations in the Clifford algebra), but it also provides a construction ofthe group Spin(n) in arbitrary dimension n. One can define

Definition (Spin(n)). The group Spin(n) is the set of invertible elements ofthe Clifford algebra Cliff(n) of the form

/w1 /w2 · · · /wk

where the vectors wj for j = 1, · · · , k are vectors in Rn satisfying |wj |2 = 1and k is even. Group multiplication is just Clifford algebra multiplication.

The action of Spin(n) on vectors v ∈ Rn will be given by conjugation

/v→ ( /w1 /w2 · · · /wk)/v( /w1 /w2 · · · /wk)−1

and this will correspond to a rotation of the vector v. This construction gen-eralizes to arbitrary n the one we gave in chapter 6 of Spin(3) in terms of unitlength elements of the quaternion algebra H. One can see here the characteristicfact that there are two elements of the Spin(n) group giving the same rotationin SO(n) by noticing that changing the sign of the Clifford algebra element/w1 /w2 · · · /wk does not change the conjugation action, where signs cancel.

26.2.2 The Lie algebra of the rotation group and quadraticelements of the Clifford algebra

For a second approach to understanding rotations in arbitrary dimension, onecan use the fact that these are generated by taking products of rotations in thecoordinate planes. A rotation by an angle θ in the j−k coordinate plane (j < k)will be given by

v→ eθεjkv

where εjk is an n by n matrix with only two non-zero entries: jk entry −1 andkj entry +1 (see equation 5.2.1). Restricting attention to the j − k plane, eθεjk

acts as the standard rotation matrix in the plane(vjvk

)→(


)(vjvk

)In the SO(3) case we saw that there were three of these matrices

ε23 = l1, ε13 = −l2, ε12 = l3

290

providing a basis of the Lie algebra so(3). In n dimensions there will be 12 (n2−n)

of them, providing a basis of the Lie algebra so(n).Just as in the case of SO(3) where unit length quaternions were used, we can

use elements of the Clifford algebra to get these same rotation transformations,but as conjugations in the Clifford algebra. To see how this works, consider thequadratic Clifford algebra element γjγk for j 6= k and notice that

(γjγk)2 = γjγkγjγk = −γjγjγkγk = −1

so one has

eθ2 γjγk =(1− (θ/2)2

2!+ · · · ) + γjγk(θ/2− (θ/2)3

3!+ · · · )

= cos(θ

2) + γjγk sin(

θ

2)

Conjugating a vector vjγj + vkγk in the j − k plane by this, one can showthat

e−θ2 γjγk(vjγj + vkγk)e

θ2 γjγk = (vj cos θ − vk sin θ)γj + (vj sin θ + vk cos θ)γk

which is just a rotation by θ in the j − k plane. Such a conjugation will alsoleave invariant the γl for l 6= j, k. Thus one has

e−θ2 γjγkγ(v)e

θ2 γjγk = γ(eθεjkv) (26.1)

and the infinitesimal version

[−1

2γjγk, γ(v)] = γ(εjkv) (26.2)

Note that these relations are closely analogous to equations 18.7 and 18.6, whichin the symplectic case show that a rotation in the Q,P plane is given by con-jugation by the exponential of an operator quadratic in Q,P . We will examinethis analogy in greater detail in chapter 28.

One can also see that, just as in our earlier calculations in three dimensions,one gets a double cover of the group of rotations, with here the elements e

θ2 γjγk

of the Clifford algebra giving a double cover of the group of rotations in thej − k plane (as θ goes from 0 to 2π). General elements of the spin group canbe constructed by multiplying these for different angles in different coordinateplanes. One sees that the Lie algebra spin(n) can be identified with the Liealgebra so(n) by

εjk ↔ −1

2γjγk

Yet another way to see this would be to compute the commutators of the − 12γjγk

for different values of j, k and show that they satisfy the same commutationrelations as the corresponding matrices εjk.

Recall that in the bosonic case we found that quadratic combinations of theQj , Pk (or of the aBj , aB

†j) gave operators satisfying the commutation relations

291

of the Lie algebra sp(2n,R). This is the Lie algebra of the group Sp(2n,R),the group preserving the non-degenerate antisymmetric bilinear form Ω(·, ·) onthe phase space R2n. The fermionic case is precisely analogous, with the role ofthe antisymmetric bilinear form Ω(·, ·) replaced by the symmetric bilinear form〈·, ·〉 and the Lie algebra sp(2n,R) replaced by so(n) = spin(n).

In the bosonic case the linear functions of the Qj , Pj satisfied the commuta-tion relations of another Lie algebra, the Heisenberg algebra, but in the fermioniccase this is not true for the γj . In chapter 27 we will see that one can define anotion of a “Lie superalgebra” that restores the parallelism.


Some more detail about the relationship between geometry and Clifford algebrascan be found in [42], and an exhaustive reference is [54].

292

Chapter 27

Anticommuting Variablesand Pseudo-classicalMechanics

The analogy between the algebras of operators in the bosonic (Weyl algebra) andfermionic (Clifford algebra) cases can be extended by introducing a fermionicanalog of phase space and the Poisson bracket. This gives a fermionic ana-log of classical mechanics, sometimes called “pseudo-classical mechanics”, thequantization of which gives the Clifford algebra as operators, and spinors asstate spaces. In this chapter we’ll intoduce “anticommuting variables” ξj thatwill be the fermionic analogs of the variables qj , pj . These objects will becomegenerators of the Clifford algebra under quantization, and will later be used inthe construction of fermionic state spaces, by analogy with the Schrodinger andBargmann-Fock constructions in the bosonic case.

27.1 The Grassmann algebra of polynomials onanticommuting generators

Given a phase space M = R2d, one gets classical observables by taking poly-nomial functions on M . These are generated by the linear functions qj , pj , j =1, . . . , d, which lie in the dual space M = M∗. One can instead start with areal vector space V = Rn with n not necessarily even, and again consider thespace V ∗ of linear functions on V , but with a different notion of multiplication,one that is anticommutative on elements of V ∗. Using such a multiplication,one can generate an anticommuting analog of the algebra of polynomials on Vin the following manner, beginning with a choice of basis elements ξj of V ∗:

Definition (Grassmann algebra). The algebra over the real numbers generated

293

by ξj , j = 1, . . . , n, satisfying the relations

ξjξk + ξkξj = 0

is called the Grassmann algebra.

Note that these relations imply that generators satisfy ξ2j = 0. Also note

that sometimes the Grassmann algebra product of ξj and ξk is denoted ξj ∧ ξk.We will not use a different symbol for the product in the Grassmann algebra,relying on the notation for generators to keep straight what is a generator ofa conventional polynomial algebra (e.g. qj or pj) and what is a generator of aGrassman algebra (e.g. ξj).

The Grassman algebra is just the algebra Λ∗(V ∗) of antisymmetric multi-linear forms on V discussed in section 9.6, except that we have chosen a basisof V and have written out the definition in terms of the dual basis ξj of V ∗. Itis sometimes also called the “exterior algebra”. This algebra behaves in manyways like the polynomial algebra on Rn, but it is finite dimensional as a realvector space, with basis

1, ξj , ξjξk, ξjξkξl, · · · , ξ1ξ2 · · · ξn

for indices j < k < l < · · · taking values 1, 2, . . . , n. As with polynomials,monomials are characterized by a degree (number of generators in the product),which in this case takes values from 0 only up to n. Λk(Rn) is the subspace ofΛ∗(Rn) of linear combinations of monomials of degree k.

Digression (Differential forms). Readers may have already seen the Grass-man algebra in the context of differential forms on Rn. These are known tophysicists as “antisymmetric tensor fields”, and given by taking elements of theexterior algebra Λ∗(Rn) with coefficients not constants, but functions on Rn.This construction is important in the theory of manifolds, where at a point xin a manifold M , one has a tangent space TxM and its dual space (TxM)∗. Aset of local coordinates xj on M gives basis elements of (TxM)∗ denoted by dxjand differential forms locally can be written as sums of terms of the form

f(x1, x2, · · · , xn)dxj ∧ · · · ∧ dxk ∧ · · · ∧ dxl

where the indices j, k, l satisfy 1 ≤ j < k < l ≤ n.

A fundamental principle of mathematics is that a good way to understanda space is in terms of the functions on it. One can think of what we have donehere as creating a new kind of space out of Rn, where the algebra of functionson the space is Λ∗(Rn), generated by coordinate functions ξj with respect to abasis of Rn. The enlargement of conventional geometry to include new kindsof spaces such that this makes sense is known as “supergeometry”, but we willnot attempt to pursue this subject here. Spaces with this new kind of geometryhave functions on them, but do not have conventional points since we have seenthat one can’t ask what the value of an anticommuting function at a point is.

294

Remarkably, one can do calculus on such unconventional spaces, introducinganalogs of the derivative and integral for anticommuting functions. For the casen = 1, an arbitrary function is

F (ξ) = c0 + c1ξ

and one can take∂

∂ξF = c1

For larger values of n, an arbitrary function can be written as

F (ξ1, ξ2, . . . , ξn) = FA + ξjFB

where FA, FB are functions that do not depend on the chosen ξj (one gets FBby using the anticommutation relations to move ξj all the way to the left). Thenone can define

∂

∂ξjF = FB

This derivative operator has many of the same properties as the conventionalderivative, although there are unconventional signs one must keep track of. Anunusual property of this derivative that is easy to see is that one has

∂

∂ξj

∂

∂ξj= 0

Taking the derivative of a product one finds this version of the Leibniz rulefor monomials F and G

∂

∂ξj(FG) = (

∂

∂ξjF )G+ (−1)|F |F (

∂

∂ξjG)

where |F | is the degree of the monomial F .A notion of integration (often called the “Berezin integral”) with many of the

usual properties of an integral can also be defined. It has the peculiar featureof being the same operation as differentiation, defined in the n = 1 case by∫

(c0 + c1ξ)dξ = c1

and for larger n by∫F (ξ1, ξ2, · · · , ξn)dξ1dξ2 · · · dξn =

∂

∂ξn

∂

∂ξn−1· · · ∂

∂ξ1F = cn

where cn is the coefficient of the basis element ξ1ξ2 · · · ξn in the expression of Fin terms of basis elements.

This notion of integration is a linear operator on functions, and it satisfiesan analog of integration by parts, since if one has

F =∂

∂ξjG

295

then ∫Fdξj =

∂

∂ξjF =

∂

∂ξj

∂

∂ξjG = 0

using the fact that repeated derivatives give zero.

27.2 Pseudo-classical mechanics and the fermionicPoisson bracket

The basic structure of Hamiltonian classical mechanics depends on an evendimensional phase space M = R2d with a Poisson bracket ·, · on functions onthis space. Time evolution of a function f on phase space is determined by

d

dtf = f, h

for some Hamiltonian function h. This says that taking the derivative of anyfunction in the direction of the velocity vector of a classical trajectory is thelinear map

f → f, h

on functions. As we saw in chapter 12, since this linear map is a derivative, thePoisson bracket will have the derivation property, satisfying the Leibniz rule

f1, f2f3 = f2f1, f3+ f1, f2f3

for arbitrary functions f1, f2, f3 on phase space. Using the Leibniz rule and an-tisymmetry, one can calculate Poisson brackets for any polynomials, just fromknowing the Poisson bracket on generators qj , pj (or, equivalently, the antisym-metric bilinear form Ω(·, ·)), which we chose to be

qj , qk = pj , pk = 0, qj , pk = −pk, qj = δjk

Notice that we have a symmetric multiplication on generators, while the Poissonbracket is antisymmetric.

To get pseudo-classical mechanics, we think of the Grassmann algebra Λ∗(Rn)as our algebra of classical observables, an algebra we can think of as functionson a “fermionic” phase space V = Rn (note that in the fermionic case, the phasespace does not need to be even dimensional). We want to find an appropriatenotion of fermionic Poisson bracket operation on this algebra, and it turns outthat this can be done. While the standard Poisson bracket is an antisymmetricbilinear form Ω(·, ·) on linear functions, the fermionic Poisson bracket will bebased on a choice of symmetric bilinear form on linear functions, equivalently,a notion of inner product 〈·, ·〉.

Denoting the fermionic Poisson bracket by ·, ·+, for a multiplication anti-commutative on generators one has to adjust signs in the Leibniz rule, and the

296

derivation property analogous to the derivation property of the usual Poissonbracket is

F1F2, F3+ = F1F2, F3+ + (−1)|F2||F3|F1, F3+F2

where |F2| and |F3| are the degrees of F2 and F3. It will also have the symmetryproperty

F1, F2+ = −(−1)|F1||F2|F2, F1+and one can use these properties to compute the fermionic Poisson bracket forarbitrary functions in terms of the relations for generators.

One can think of the ξj as the “anti-commuting coordinate functions” withrespect to a basis ei of V = Rn. We have seen that the symmetric bilinearforms on Rn are classified by a choice of positive signs for some basis vectors,negative signs for the others. So, on generators ξj one can choose

ξj , ξk+ = ±δjk

with a plus sign for j = k = 1, · · · , r and a minus sign for j = k = r+ 1, · · · , n,corresponding to the possible inequivalent choices of non-degenerate symmetricbilinear forms.

Taking the case of a positive-definite inner product for simplicity, one cancalculate explicitly the fermionic Poisson brackets for linear and quadratic com-binations of the generators. One finds

ξjξk, ξl+ = ξjξk, ξl+ − ξj , ξl+ξk = δklξj − δjlξk (27.1)

and

ξjξk, ξlξm+ =ξjξk, ξl+ξm + ξlξjξk, ξm+=δklξjξm − δjlξkξm + δkmξlξj − δjmξlξk (27.2)

The second of these equations shows that the quadratic combinations of thegenerators ξj satisfy the relations of the Lie algebra of the group of rotations inn dimensions (so(n) = spin(n)). The first shows that the ξkξl acts on the ξj asinfinitesimal rotations in the k − l plane.

In the case of the conventional Poisson bracket, the antisymmetry of thebracket and the fact that it satisfies the Jacobi identity implies that it is aLie bracket determining a Lie algebra (the infinite dimensional Lie algebra offunctions on a phase space R2d). The fermionic Poisson bracket provides anexample of something called a Lie superalgebra. These can be defined for vectorspaces with some usual and some fermionic coordinates:

Definition (Lie superalgebra). A Lie superalgebra structure on a real or com-plex vector space V is given by a Lie superbracket [·, ·]±. This is a bilinear mapon V which on generators X,Y, Z (which may be usual coordinates or fermionicones) satisfies

[X,Y ]± = −(−1)|X||Y |[Y,X]±

297

and a super-Jacobi identity

[X, [Y,Z]±]± = [[X,Y ]±, Z]± + (−1)|X||Y |[Y, [X,Z]±]±

where |X| takes value 0 for a usual generator, 1 for a fermionic generator.

Analogously to the bosonic case, on polynomials in generators with order ofthe poynomial less than or equal to two, the fermionic Poisson bracket ·, ·+ is aLie superbracket, giving a Lie superalgebra of dimension 1+n+ 1

2 (n2−n) (sincethere is one constant, n linear terms ξj and 1

2 (n2 − n) quadratic terms ξjξk).On functions of order two this Lie superalgebra is a Lie algebra, so(n). We willsee in chapter 28 that one can generalize the definition of a representation toLie superalgebras, and quantization will give a distinguished representation ofthis Lie superalgebra, in a manner quite parallel to that of the Schrodinger orBargmann-Fock constructions of a representation in the bosonic case.

The relation between between the quadratic and linear polynomials in thegenerators is parallel to what happens in the bosonic case. Here we have thefermionic analog of the bosonic theorem 14.1:

Theorem 27.1. The Lie algebra so(n,R) is isomorphic to the Lie algebraΛ2(V ∗) (with Lie bracket ·, ·+) of order two anticommuting polynmials onV = Rn, by the isomorphism

L↔ µL

where L ∈ so(n,R) is an antisymmetric n by n real matrix, and

µL =1

2ξ · Lξ =

1

2

∑j,k

Ljkξjξk

The so(n,R) action on anticommuting coordinate functions is

µL, ξk+ =∑j

Ljkξj

orµL, ξ+ = LT ξ

Proof. The theorem follows from equations 27.1 and 27.2, or one can proceedby analogy with the proof of theorem 14.1 as follows. First prove the secondpart of the theorem by computing

1

2

∑j,k

ξjLjkξk, ξl+ =1

2

∑j,k

Ljk(ξjξk, ξl+ − ξj , ξl+ξk)

=1

2(∑j

Ljlξj −∑k

Llkξk)

=∑j

Ljlξj (since L = −LT )

298

For the first part of the theorem, the map

L→ µL

is a vector space isomorphism of the space of antisymmetric matrices andΛ2(Rn). To show that it is a Lie algebra isomorphism, one can use an analogousargument to that of the proof of 14.1. Here one considers the action

ξ → µL, ξ

of µL ∈ so(n,R) on an arbitrary

ξ =∑j

cjξj

and uses the super-Jacobi identity relating the fermionic Poisson brackets ofµL, µL′ , ξ.

27.3 Examples of pseudo-classical mechanics

In pseudo-classical mechanics, the dynamics will be determined by choosing aHamiltonian h in Λ∗(Rn). Observables will be other functions F ∈ Λ∗(Rn),and they will satisfy the analog of Hamilton’s equations

d

dtF = F, h+

We’ll consider two of the simplest possible examples.

27.3.1 The pseudo-classical spin degree of freedom

Using pseudo-classical mechanics, one can find a “classical” analog of somethingthat is quintessentially quantum: the degree of freedom that appears in thequbit or spin 1/2 system that we have seen repeatedly in this course. TakingV = R3 with the standard inner product as fermionic phase space, we havethree generators ξ1, ξ2, ξ3 ∈ V ∗ satisfying the relations

ξj , ξk+ = δjk

and an 8 dimensional space of functions with basis

1, ξ1, ξ2, ξ3, ξ1ξ2, ξ1ξ3, ξ2ξ3, ξ1ξ2ξ3

If we want the Hamiltonian function to be non-trivial and of even degree, itwill have to be a linear combination

h = B12ξ1ξ2 +B13ξ1ξ3 +B23ξ2ξ3

299

for some constants B12, B13, B23. This can be written

h =1

2

3∑j,k=1

Ljkξjξk

where Ljk are the entries of the matrix

L =

0 B12 B13

−B12 0 B23

−B13 −B23 0

The equations of motion on generators will be

d

dtξj(t) = ξj , h+ = −h, ξj+

which, since L = −LT , by theorem 27.1 can be written

d

dtξj(t) = Lξj(t)

with solutionξj(t) = etLξj(0)

This will be a time-dependent rotation of the ξj in the plane perpendicular to

B = (B23,−B13, B12)

at a constant speed proportional to |B|.

27.3.2 The pseudo-classical fermionic oscillator

We have already studied the fermionic oscillator as a quantum system, and onecan ask whether there is a corresponding pseudo-classical system. Such a systemis given by taking an even dimensional fermionic phase space V = R2d, with abasis of coordinate functions ξ1, · · · , ξ2d that generate Λ∗(R2d). On generatorsthe fermionic Poisson bracket relations come from the standard choice of positivedefinite symmetric bilinear form

ξj , ξk+ = δjk

As shown in theorem 27.1, quadratic products ξjξk act on the generators byinfinitesimal rotations in the j− k plane, and satisfy the commutation relationsof so(2d).

To get a pseudo-classical system corresponding to the fermionic oscillatorone makes the choice

h =1

2

d∑j=1

(ξ2jξ2j−1 − ξ2j−1ξ2j) =

d∑j=1

ξ2jξ2j−1

300

This makes h the moment map for a simultaneous rotation in the 2j − 1, 2jplanes, corresponding to a matrix in so(2d) given by

L =

d∑j=1

ε2j−1,2j

As in the bosonic case, we can make the standard choice of complex structureJ = J0 on R2d and get a decomposition

V ∗ ⊗C = R2d ⊗C = Cd ⊕Cd

into eigenspaces of J of eigenvalue ±i. This is done by defining

θj =1√2

(ξ2j−1 + iξ2j), θj =1√2

(ξ2j−1 − iξ2j)

for j = 1, . . . , d. These satisfy the fermionic Poisson bracket relations

θj , θk+ = θj , θk+ = 0, θj , θk+ = δjk

In terms of the θj , the Hamiltonian is

h = − i2

d∑j=1

(θjθj − θjθj) = −id∑j=1

θjθj

Using the derivation property of ·, ·+ one finds

h, θj+ = −id∑k=1

(θkθk, θj+ − θk, θj+θk) = −iθj

and, similarly,h, θj+ = iθj

so one sees that h is just the generator of U(1) ⊂ U(d) phase rotations on thevariables θj . The equations of motion are

d

dtθj = θj , h+ = −iθj ,

d

dtθj = θj , h+ = iθj

with solutionsθj(t) = e−itθj(0), θj(t) = eitθj(0)


For more details on pseudo-classical mechanics, a very readable original ref-erence is [6], and there is a detailed discussion in the textbook [71], chapter7.

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Chapter 28

Fermionic Quantization andSpinors

In this chapter we’ll begin by investigating the fermionic analog of the notionof quantization, which takes functions of anticommuting variables on a phasespace with symmetric bilinear form 〈·, ·〉 and gives an algebra of operators withgenerators satisfying the relations of the corresponding Clifford algebra. Wewill then consider analogs of the constructions used in the bosonic case whichthere gave us the Schrodinger and Bargmann-Fock representations of the Weylalgebra on a space of states.

We know that for a fermionic oscillator with d degrees of freedom, the alge-bra of operators will be Cliff(2d,C), the algebra generated by annihilation and

creation operators aF j , aF†j . These operators will act on HF = F+

d , a complex

vector space of dimension 2d, and this will be our fermionic analog of the bosonicΓ′. Since the spin group consists of invertible elements of the Clifford algebra,it has a representation on F+

d . This is known as the “spinor representation”,and it can be constructed by analogy with the construction of the metaplecticin the bosonic case. We’ll also consider the analog in the fermionic case of theSchrodinger representation, which turns out to have a problem with unitarity,but finds a use in physics as “ghost” degrees of freedom.

28.1 Quantization of pseudo-classical systems

In the bosonic case, quantization was based on finding a representation of theHeisenberg Lie algebra of linear functions on phase space, or more explicitly,for basis elements qj , pj of this Lie algebra finding operators Qj , Pj satisfyingthe Heisenberg commutation relations. In the fermionic case, the analog ofthe Heisenberg Lie algebra is not a Lie algebra, but a Lie superalgebra, withbasis elements 1, ξj , j = 1, . . . , n and a Lie superbracket given by the fermionic

303

Poisson bracket, which on basis elements is

ξj , ξk+ = ±δjk, ξj , 1+ = 0, 1, 1+ = 0

Quantization is given by finding a representation of this Lie superalgebra. Onecan generalize the definition of a Lie algebra representation to that of a Liesuperalgebra representation by

Definition (Representation of a Lie superalgebra). A representation of a Liesuperalgebra is a homomorphism Φ preserving the superbracket

[Φ(X),Φ(Y )]± = Φ([X,Y ]±)

and taking values in a Lie superalgebra of linear operators, with |Φ(X)| = |X|and

[Φ(X),Φ(Y )]± = Φ(X)Φ(Y )− (−)|X||Y |Φ(Y )Φ(X)

A representation of the pseudo-classical Lie superalgebra (and thus a quan-tization of the pseudo-classical system) will be given by finding a linear map Γ+

that takes basis elements ξj to operators Γ+(ξj) satisfying the anticommutationrelations

[Γ+(ξj),Γ+(ξk)]+ = ±δjkΓ+(1), [Γ+(ξj),Γ

+(1)] = [Γ+(1),Γ+(1)] = 0

One can satisfy these relations by taking

Γ+(ξj) =1√2γj , Γ+(1) = 1

since then

[Γ+(ξj),Γ+(ξk)]+ =

1

2[γj , γk]+ = ±δjk

are exactly the Clifford algebra relations. This can be extended to a represen-tation of the functions of the ξj of order two or less by

Theorem. A representation of the Lie superalgebra of anticommuting functionsof coordinates ξj on Rn of order two or less is given by

Γ+(1) = 1, Γ+(ξj) =1√2γj , Γ+(ξjξk) =

1

2γjγk

Proof. We have already seen that this is a representation on polynomials ofdegee zero and one. For simplicity just considering the case s = 0, in degree twothe fermionic Poisson bracket relations are given by equations 27.1 and 27.2.For 27.1, one can show that the products of Clifford algebra generators

Γ+(ξjξk) =1

2γjγk

satisfy

[1

2γjγk, γl] = δklγj − δjlγk

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by using the Clifford algebra relations, or by noting that this is the special caseof equation 26.2 for v = el. That equation shows that commuting by − 1

2γjγkacts by the infinitesimal rotation εjk in the j − k coordinate plane.

For 27.2, one can again just use the Clifford algebra relations to show

[1

2γjγk,

1

2γlγm] = δkl

1

2γjγm − δjl

1

2γkγm + δkm

1

2γlγj − δjm

1

2γlγk

One could also instead use the commutation relations for the so(n) Lie algebrasatisfied by the basis elements εjk corresponding to infinitesimal rotations. Onemust get identical commutation relations for the − 1

2γjγk and can show thatthese are the relations needed for commutators of Γ+(ξjξk) and Γ+(ξlξm).

Note that here we are not introducing the factors of i into the definition ofquantization that in the bosonic case were necessary to get a unitary represen-tation of the Lie group corresponding to the real Heisenberg Lie algebra h2d+1.In the bosonic case we worked with all complex linear combinations of powersof the Qj , Pj (the complex Weyl algebra Weyl(2d,C)), and thus had to identifythe specific complex linear combinations of these that gave unitary represen-tations of the Lie algebra h2d+1 o sp(2d,R). Here we are not complexifyingfor now, but working with the real Clifford algebra Cliff(r, s,R), and it is theirreducible representations of this algebra that provide an analog of the uniqueinteresting irreducible representation of h2d+1. In the Clifford algebra case, therepresentations of interest are not Lie algebra representations and may be onreal vector spaces. There is no analog of the unitarity property of the h2d+1

representation.In the bosonic case we found that Sp(2d,R) acted on the bosonic dual phase

space, preserving the antisymmetric bilinear form Ω that determined the Lie al-gebra h2d+1, so it acted on this Lie algebra by automorphisms. We saw (seechapter 18) that intertwining operators there gave us a representation of thedouble cover of Sp(2d,R) (the metaplectic representation), with the Lie alge-bra representation given by the quantization of quadratic functions of the qj , pjphase space coordinates. There is a closely analogous story in the fermioniccase, where SO(r, s,R) acts on the fermionic phase space V , preserving thesymmetric bilinear form 〈·, ·〉 that determines the Clifford algebra relations.Here one constructs a representation of the spin group Spin(r, s,R) doublecovering SO(r, s,R) using intertwining operators, with the Lie algebra repre-sentation given by quadratic combinations of the quantizations of the fermioniccoordinates ξj .

The fermionic analog of 18.1 is

UkΓ+(ξ)U−1k = Γ+(φk0

(ξ)) (28.1)

Here k0 ∈ SO(r, s,R), ξ ∈ V ∗ = Rn (n = r + s), φk0is the action of k0 on V ∗.

The Uk for k = Φ−1(k0) ∈ Spin(r, s) (Φk is the 2-fold covering map) are theintertwining operators we are looking for. The fermionic analog of 18.5 is

[U ′L,Γ+(ξ)] = Γ+(L · ξ)

305

where L ∈ so(r, s,R) and L acts on V ∗ as an infinitesimal orthogonal transfor-mation. In terms of basis vectors of V ∗

ξ =

ξ1...ξn

this says

[U ′L,Γ+(ξ)] = Γ+(LT ξ)

Just as in the bosonic case, the U ′L can be found by looking first at thepseudo-classical case, where one has theorem 27.1 which says

µL, ξ+ = LT ξ

where

µL =1

2ξ · Lξ =

1

2

∑j,k

Ljkξjξk

One then takes

U ′L = Γ+(µL) =1

4

∑j,k

Ljkγjγk

For the case s = 0 and a rotation in the j − k plane, with L = εjk onerecovers formulas 26.1 and 26.2 from chapter 26, with

[−1

2γjγk, γ(v)] = γ(εjkv)

the infinitesimal action of a rotation on the γ matrices, and

γ(v)→ e−θ2 γjγkγ(v)e

θ2 γjγk = γ(eθεjkv)

the group version. Just as in the symplectic case, exponentiating the U ′L onlygives a representation up to sign, and one needs to go to the double cover ofSO(n) to get a true representation. As in that case, the necessity of the doublecover is best seen by use of a complex structure and an analog of the Bargmann-Fock construction, a topic we will address in a later section.

In order to have a full construction of a quantization of a pseudo-classicalsystem, we need not just the abstract Clifford algebra elements given by themap Γ+, but also a realization of the Clifford algebra as linear operators on astate space. As mentioned in chapter 25, it can be shown that the real Cliffordalgebras Cliff(r, s,R) are isomorphic to either one or two copies of the matrixalgebras M(2l,R),M(2l,C), or M(2l,H), with the power l depending on r, s.The irreducible representations of such a matrix algebra are just the columnvectors of dimension 2l, and there will be either one or two such irreduciblerepresentations for Cliff(r, s,R) depending on the number of copies of the matrixalgebra. This is the fermionic analog of the Stone-von Neumann uniquenessresult in the bosonic case.

306

28.1.1 Quantization of the pseudo-classical spin

As an example, one can consider the quantization of the pseudo-classical spindegree of freedom of section 27.3.1. In that case Γ+ takes values in Cliff(3, 0,R),for which an explicit identification with the algebra M(2,C) of two by twocomplex matrices was given in section 25.2. One has

Γ+(ξj) =1√2γj =

1√2σj

and the Hamiltonian operator is

−iH = Γ+(h) =Γ+(B12ξ1ξ2 +B13ξ1ξ3 +B23ξ2ξ3)

=1

2(B12σ1σ2 +B13σ1σ3 +B23σ2σ3)

=i1

2(B1σ1 +B2σ2 +B3σ3)

This is nothing but our old example from chapter 7 of a fixed spin one-halfparticle in a magnetic field.

The pseudo-classical equation of motion

d

dtξj(t) = −h, ξj

after quantization becomes the Heisenberg picture equation of motion for thespin operators (see equation 7.2)

d

dtSH(t) = −i[SH ·B,SH ]

for the case of HamiltonianH = −µ ·B

(see equation 7.1) and magnetic moment operator

µ = S

Here the state space is H = C2, with an explicit choice of basis given byour chosen identification of Cliff(3, 0,R) with two by two complex matrices. Inthe next sections we will just consider the case of an even-dimensional fermionicphase space, but there provide a basis-independent construction of the statespace and the action of the Clifford algebra on it.

28.2 The Schrodinger representation for fermions:ghosts

We would like to construct representations of Cliff(r, s,R) and thus fermionicstate spaces by using analogous constructions to the Schrodinger and Bargmann-Fock ones in the bosonic case. The Schrodinger construction took the state

307

space H to be a space of functions on a subspace of the classical phase spacewhich had the property that the basis coordinate functions Poisson-commuted.Two examples of this are the position coordinates qj , since qj , qk = 0, or themomentum coordinates pj , since pj , pk = 0. Unfortunately, for symmetricbilinear forms 〈·, ·〉 of definite sign, such as the positive definite case Cliff(n,R),the only subspace the bilinear form is zero on is the zero subspace.

To get an analog of the bosonic situation, one needs to take the case ofsignature (d, d). The fermionic phase space will then be 2d dimensional, withd-dimensional subspaces on which 〈·, ·〉 and thus the fermionic Poisson bracketis zero. Quantization will give the Clifford algebra

Cliff(d, d,R) = M(2d,R)

which has just one irreducible representation, R2d . One can complexify this toget a complex state space

HF = C2d

This state space will come with a representation of Spin(d, d,R) from expo-nentiating quadratic combinations of the generators of Cliff(d, d,R). However,this is a non-compact group, and one can show that on general grounds it can-not have unitary finite-dimensional representations, so there must be a problemwith unitarity.

To see what happens explicitly, consider the simplest case d = 1 of one degreeof freedom. In the bosonic case the classical phase space is R2, and quantizationgives operators Q,P which in the Schrodinger representation act on functionsof q, with Q = q and P = −i ∂∂q . In the fermionic case with signature (1, 1),basis coordinate functions on phase space are ξ1, ξ2, with

ξ1, ξ1+ = 1, ξ2, ξ2+ = −1, ξ1, ξ2+ = 0

Defining

η =1√2

(ξ1 + ξ2), π =1√2

(ξ1 − ξ2)

we get objects with fermionic Poisson bracket analogous to those of q and p

η, η+ = π, π+ = 0, η, π+ = 1

Quantizing, we get analogs of the Q,P operators

η = Γ+(η) =1√2

(Γ+(ξ1) + Γ+(ξ2)), π = Γ+(π) =1√2

(Γ+(ξ1)− Γ+(ξ2))

which satisfy anticommutation relations

η2 = π2 = 0, ηπ + πη = 1

and can be realized as operators on the space of functions of one fermionicvariable η as

η = multiplication by η, π =∂

∂η

308

This state space is two complex dimensional, with an arbitrary state

f(η) = c11 + c2η

with cj complex numbers. The inner product on this space is given by thefermionic integral

(f1(η), f2(η)) =

∫f∗1 (η)f2(η)dη

withf∗(ξ) = c11 + c2η

With respect to this inner product, one has

(1, 1) = (η, η) = 0, (1, η) = (η, 1) = 1

This inner product is indefinite and can take on negative values, since

(1− η, 1− η) = −2

Having such negative-norm states ruins any standard interpretation of thisas a physical system, since this negative number is supposed to the probability offinding the system in this state. Such quantum systems are called “ghosts”, anddo have applications in the description of various quantum systems, but onlywhen a mechanism exists for the negative-norm states to cancel or otherwise beremoved from the physical state space of the theory.

28.3 Spinors and the Bargmann-Fock construc-tion

While the fermionic analog of the Schrodinger construction does not give a uni-tary representation of the spin group, it turns out that the fermionic analog ofthe Bargmann-Fock construction does, on the fermionic oscillator state spacediscussed in chapter 24. This will work for the case of a positive definite sym-metric bilinear form 〈·, ·〉. Note though that n must be even since this willrequire choosing a complex structure on the fermionic phase space Rn.

The corresponding pseudo-classical system will be the classical fermionicoscillator studied in section 27.3.2. Recall that this uses a choice of complexstructure J on the fermionic phase space R2d, with the standard choice J = J0

giving the relations

θj =1√2

(ξ2j−1 + iξ2j), θj =1√2

(ξ2j−1 − iξ2j)

for j = 1, . . . , d between real and complex coordinates. Here 〈·, ·〉 is positive-definite, and the ξj are coordinates with respect to an orthonormal basis, so wehave the standard relation ξj , ξk+ = δjk and the θj , θj satisfy

θj , θk+ = θj , θk+ = 0, θj , θk+ = δjk

309

To quantize this system we need to find operators Γ+(θj) and Γ+(θj) thatsatisfy

[Γ+(θj),Γ+(θk)]+ = [Γ+(θj),Γ

+(θk)]+ = 0

[Γ+(θj),Γ+(θk)]+ = δjk1

but these are just the CAR satisfied by fermionic annihilation and creationoperators. We can choose

Γ+(θj) = aF†j , Γ+(θj) = aF j

and realize these operators as

aF j =∂

∂χj, aF

†j = multiplication by χj

on the state space Λ∗Cd of polynomials in the anticommuting variables χj . Thisis a complex vector space of dimension 2d, isomorphic with the state space HFof the fermionic oscillator in d degrees of freedom, with the isomorphism givenby

1↔ |0〉Fχj ↔ aF

†j |0〉F

χjχk ↔ aF†jaF

†k|0〉

· · ·

χ1 . . . χd ↔ aF†1aF

†2 · · · aF

†d|0〉F

where the indices j, k, . . . take values 1, 2, . . . , d and satisfy j < k < · · · .If one defines a Hermitian inner product (·, ·) on HF by taking these basis

elements to be orthonormal, the operators aF j and a†F j will be adjoints withrespect to this inner product. This same inner product can also be definedusing fermionic integration by analogy with the Bargmann-Fock definition inthe bosonic case as

(f1(χ1, · · · , χd), f2(χ1, · · · , χd)) =

∫e−

∑dj=1 χjχjf1f2dχ1dχ1 · · · dχddχd

where f1 and f2 are complex linear combinations of the powers of the anticom-muting variables χj . For the details of the construction of this inner product,see chapter 7.2 of [71] or chapters 7.5 and 7.6 of [86].

We will denote this state space as F+d and refer to it as the fermionic Fock

space. Since it is finite dimensional, there is no need for a completion as in thebosonic case.

The quantization using fermionic annihilation and creation operators givenhere provides an explicit realization of a representation of the Clifford algebraCliff(2d,R) on the complex vector space F+

d . The generators of the Cliffordalgebra are identified as operators on F+

d by

γ2j−1 =√

2Γ+(ξ2j−1) =√

2Γ+(1√2

(θj + θj)) = aF j + a†F j

310

γ2j =√

2Γ+(ξ2j) =√

2Γ+(i√2

(θj − θj)) = i(a†F j − aF j)

Quantization of the pseudo-classical fermionic oscillator Hamiltonian h ofsection 27.3.2 gives

Γ+(h) = Γ+(− i2

d∑j=1

(θjθj − θjθj)) = − i2

d∑j=1

(a†F jaF j − aF ja†F j) = −iH (28.2)

where H is the Hamiltonian operator for the fermionic oscillator used in chapter24.

Taking quadratic combinations of the operators γj , γk provides a represen-tation of the Lie algebra so(2d) = spin(2d). This representation exponentiatesto a representation up to sign of the group SO(2d), and a true representationof its double-cover Spin(2d). The representation that we have constructed hereon the fermionic oscillator state space F+

d is called the spinor representation ofSpin(2d), and we will sometimes denote F+

d with this group action as S.In the bosonic case,H = Fd is an irreducible representation of the Heisenberg

group, but as a representation of Mp(2d,R), it has two irreducible components,corresponding to even and odd polynomials. The fermionic analog is that F+

d

is irreducible under the action of the Clifford algebra Cliff(2d,C). One wayto show this is to show that Cliff(2d,C) is isomorphic to the matrix algebra

M(2d,C) and its action on HF = C2d is isomorphic to the action of matriceson column vectors.

While F+d is irreducible as a representation of the Clifford algebra, it is the

sum of two irreducible representations of Spin(2d), the so-called “half-spinor”representations. Spin(2d) is generated by quadratic combinations of the Cliffordalgebra generators, so these will preserve the subspaces

S+ = span|0〉F , aF†jaF

†k|0〉F , · · · ⊂ S = F+

d

andS− = spanaF †j |0〉F , aF

†jaF

†kaF

†l |0〉F , · · · ⊂ S = F+

d

corresponding to the action of an even or odd number of creation operators on|0〉F . This is because quadratic combinations of the aF j , aF

†j preserve the parity

of the number of creation operators used to get an element of S by action on|0〉F .

28.4 Complex structures, U(d) ⊂ SO(2d) and thespinor representation

The construction of the spinor representation here has involved making a specificchoice of relation between the Clifford algebra generators and the fermionicannihilation and creation operators. This corresponds to a standard choice ofcomplex structure J0, which appears in a manner closely parallel to that of

311

the Bargmann-Fock case of section 21.1. The difference here is that for theanalogous construction of spinors a complex structure J must be chosen topreserve not an antisymmetric bilinear form Ω, but the inner product, so onehas

〈J(·), J(·)〉 = 〈·, ·〉

We will here restrict to the case of 〈·, ·〉 positive definite, and unlike in thebosonic case, no additional positivity condition on J will be required.

J splits the complexification of the real dual phase space V ∗ = V = R2d withits coordinates ξj into a d-dimensional complex vector space and a conjugatecomplex vector space. As in the bosonic case one has

V ⊗C = V+J ⊕ V

−J

and quantization of vectors in V+J gives linear combinations of creation op-

erators, while vectors in V−J are taken to linear combinations of annihilationoperators. The choice of J is reflected in the existence of a distinguished di-rection |0〉F in the spinor space S = F+

d which is determined (up to phase) bythe condition that it is annihilated by all linear combinations of annihilationoperators.

The choice of J also picks out a subgroup U(d) ⊂ SO(2d) of those orthogonaltransformations that commute with J . Just as in the bosonic case, two differentrepresentations of the Lie algebra u(d) of U(d) are used:

• The restriction of u(d) ⊂ so(2d) of the spinor representation describedabove. This exponentiates to give a representation not of U(d), but of adouble cover of U(d) that is a subgroup of Spin(2d).

• By normal-ordering operators, one shifts the spinor representation of u(d)by a constant and gets a representation that exponentiates to a true rep-resentation of U(d). This representation is reducible, with irreduciblecomponents the Λk(Cd) for k = 0, 1, . . . , d.

In both cases the representation of u(d) is constructed using quadratic combina-tions of annihilation and creation operators involving one annihilation operatorand one creation operator. Non-zero pairs of two creation operators will give“Bogoliubov transformations”, changing |0〉F .

Given any group element

g0 = eA ⊂ U(d)

acting on the fermionic dual phase space preserving J and the inner product, wecan use exactly the same method as in theorems 23.1 and 23.2 to construct itsaction on the fermionic state space by the second of the above representations.For A a skew-adjoint matrix we have a fermionic moment map

A ∈ u(d)→ µA =∑j,k

θjAjkθk

312

satisfying

µA, µA′+ = µ[A,A′]

and

µA,θ+ = ATθ, µA,θ+ = ATθ

The Lie algebra representation operators are the

U ′A =∑j,k

a†F jAjkaF k

which satisfy (see theorem 24.1)

[U ′A, U′A′ ] = U[A,A′]

and

[U ′A,a†F ] = ATa†F , [U ′A,aF ] = ATaF

Exponentiating these gives the intertwining operators, which act on the an-nihilation and creation operators as

UeAa†F (UeA)−1 = eAT

a†F , UeAaF (UeA)−1 = eATaF

For the simplest example, consider the U(1) ⊂ U(d) ⊂ SO(2d) that acts by

θj → eiφθj , θj → e−iφθj

corresponding to A = −iφ1. The moment map will be

µA = φh

where

h = −id∑j=1

θjθj

is the Hamiltonian for the classical fermionic oscillator. Quantizing h (see equa-tion 28.2) will give the Hamiltonian operator

H =1

2

d∑j=1

(aF†jaF j − aF jaF

†j) =

d∑j=1

(aF†jaF j −

1

2)

and a Lie algebra representation of u(1) with half-integral eigenvalues (±i 12 ).

Exponentiation will give a representation of a double cover of U(1) ⊂ U(d).Quantizing h instead using normal-ordering

:H: =

d∑j=1

aF†jaF j

313

will give a true representation of U(1) ⊂ U(d), with

U ′A = −iφd∑j=1

aF†jaF j

satisfying[U ′A,a

†F ] = −iφa†F , [U ′A,aF ] = iφaF

Exponentiating, the action on annihilation and creation operators is

e−iφ∑dj=1 aF

†jaF ja†F e

iφ∑dj=1 aF

†jaF j = e−iφa†F

e−iφ∑dj=1 aF

†jaF jaF e

iφ∑dj=1 aF

†jaF j = eiφaF

28.5 An example: spinors for SO(4)

We saw in chapter 6 that the spin group Spin(4) was isomorphic to Sp(1) ×Sp(1) = SU(2)×SU(2). Its action on R4 was then given by identifying R4 = Hand acting by unit quaternions on the left and the right (thus the two copies ofSp(1)). While this constructs the representation of Spin(4) on R4, it does notprovide the spin representation of Spin(4).

A conventional way of defining the spin representation is to choose an explicitmatrix representation of the Clifford algebra (in this case Cliff(4, 0,R)), forinstance

γ0 =

(0 11 0

), γ1 = −i

(0 σ1

−σ1 0

), γ2 = −i

(0 σ2

−σ2 0

), γ3 = −i

(0 σ3

−σ3 0

)where we have written the matrices in 2 by 2 block form, and are indexing thefour dimensions from 0 to 3. One can easily check that these satisfy the Cliffordalgebra relations: they anticommute with each other and

γ20 = γ2

1 = γ22 = γ2

3 = 1

The quadratic Clifford algebra elements − 12γjγk for j < k satisfy the com-

mutation relations of so(4) = spin(4). These are explicitly

−1

2γ0γ1 = − i

2

(σ1 00 −σ1

), −1

2γ2γ3 = − i

2

(σ1 00 σ1

)

−1

2γ0γ2 = − i

2

(σ2 00 −σ2

), −1

2γ1γ3 = − i

2

(σ2 00 σ2

)−1

2γ0γ3 = − i

2

(σ3 00 −σ3

)− 1

2γ1γ2 = − i

2

(σ3 00 σ3

)The Lie algebra spin representation is just matrix multiplication on S = C4,

and it is obviously a reducible representation on two copies of C2 (the upper

314

and lower two components). One can also see that the Lie algebra spin(4) =su(2) + su(2), with the two su(2) Lie algebras having bases

−1

4(γ0γ1 + γ2γ3), −1

4(γ0γ2 + γ1γ3), −1

4(γ0γ3 + γ1γ2)

and

−1

4(γ0γ1 − γ2γ3), −1

4(γ0γ2 − γ1γ3), −1

4(γ0γ3 − γ1γ2)

The irreducible spin representations of Spin(4) are just the spin one-half repre-sentations of the two copies of SU(2).

In the fermionic oscillator construction, we have

S = S+ + S−, S+ = span1, η1η2, S− = spanη1, η2

and the Clifford algebra action on S is given for the generators as (indexingdimensions from 1 to 4)

γ1 =∂

∂η1+ η1, γ2 = i(

∂

∂η1− η1)

γ3 =∂

∂η2+ η2, γ4 = i(

∂

∂η2− η2)

Note that in this construction there is a choice of complex structure J = J0.This gives a distinguished vector |0〉 = 1 ∈ S+, as well as a distinguished sub-Liealgebra u(2) ⊂ so(4) of transformations that act trivially on |0〉, given by linearcombinations of

η1∂

∂η1, η2

∂

∂η2, η1

∂

∂η2, η2

∂

∂η1,

There is also a distinguished sub-Lie algebra u(1) ⊂ u(2) given by

η1∂

∂η1+ η2

∂

∂η2

Bogoliubov transformations that are unitary transformations on the spinorstate space, but change |0〉 and correspond to a change in complex structure,are given by exponentiating the Lie algebra representation operators

i(aF†1aF

†2 + aF 2aF 1), aF

†1aF

†2 − aF 2aF 1

These act on |0〉 by multiplication by a complex number times η1η2. The possiblechoices of complex structure are parametrized by SO(4)/U(2), which can beidentified with the complex projective sphere CP 1 = S2.

The construction in terms of matrices is well-suited to calculations, butit is inherently dependent on a choice of coordinates. The fermionic versionof Bargmann-Fock is given here in terms of a choice of basis, but, like theclosely analogous bosonic construction, only actually depends on a choice ofinner product and a choice of compatible complex structure J , producing arepresentation on the coordinate-independent object F+

d = Λ∗V +J .

315

In chapter 39 we will consider explicit matrix representations of the Cliffordalgebra for the case of Spin(3, 1). One could also use the fermionic oscillatorconstruction, complexifying to get a representation of

so(4)⊗C = sl(2,C)× sl(2,C)

and then restricting to the subalgebra

so(3, 1) ⊂ so(3, 1)⊗C = so(4)⊗C

This will give a representation of Spin(3, 1) in terms of quadratic combinationsof Clifford algebra generators, but unlike the case of Spin(4), it will not beunitary. The lack of positivity for the inner product causes the same sort ofwrong-sign problems with the CAR that were found in the bosonic case for theCCR when J and Ω gave a non-positive symmetric bilinear form. In the fermioncase the wrong-sign problem does not stop one from constructing a non-trivialrepresentation, but it will not be a unitary representation.


For more about pseudo-classical mechanics and quantization, see [71] Chapter 7.The fermionic quantization map, Clifford algebras, and the spinor representationare discussed in detail in [46]. For another discussion of the spinor representationfrom a similar point of view to the one here, see chapter 12 of [73]. Chapter 12 of[52] contains an extensive discussion of the role of different complex structuresin the construction of the spinor representation.

316

Chapter 29

A Summary: ParallelsBetween Bosonic andFermionic Quantization

To summarize much of the material we have covered, it may be useful to con-sider the following table, which explicitly gives the correspondence between theparallel constructions we have studied in the bosonic and fermionic cases.

Bosonic Fermionic

Dual phase space M = R2d Dual phase space V = Rn

Non-degenerate antisymmetricbilinear form Ω(·, ·) on M

Non-degenerate symmetricbilinear form 〈·, ·〉 on V

Poisson bracket ·, · onfunctions on M = R2d

Poisson bracket ·, ·+ onanticommuting functions on V = Rn

Lie algebra of polynomials ofdegree 0, 1, 2

Lie superalgebra of anticommutingpolynomials of degree 0, 1, 2

Coordinates qj , pj , basis of M Coordinates ξj , basis of V

Quadratics in qj , pj , basis for sp(2d,R) Quadratics in ξj , basis for so(n)

Sp(2d,R) preserves Ω(·, ·) SO(n,R) preserves 〈·, ·〉

Weyl algebra Weyl(2d,C) Clifford algebra Cliff(n,C)

Momentum, position operators Pj , Qj Clifford algebra generators γj

Quadratics in Pj , Qj providerepresentation of sp(2d,R)

Quadratics in γj providerepresentation of so(2d)

Metaplectic representation Spinor representation

317

Stone-von NeumannUniqueness of h2d+1 representation

Uniqueness of Cliff(2d,C) representa-tion

Mp(2d,R) double-cover of Sp(2d,R) Spin(n) double-cover of SO(n)

J : J2 = −1, Ω(Ju, Jv) = Ω(u, v) J : J2 = −1, 〈Ju, Jv〉 = 〈u, v〉

M⊗C =M+J ⊕M

−J V ⊗C = V+

J ⊕ V−J

Coordinates zj ∈M−J , zj ∈M−J Coordinates θj ∈ V−J , θj ∈ V

+J

U(d) ⊂ Sp(2d,R) commutes with J U(d) ⊂ SO(2d,R) commutes with J

Compatible J ∈ Sp(2d,R)/U(d) Compatible J ∈ O(2d)/U(d)

aj , a†j satisfying CCR ajF , aj

†F satisfying CAR

aj |0〉 = 0, |0〉 depends on J ajF |0〉 = 0, |0〉 depends on J

Bogoliubov transformations generatedby symmetric quadratics in a†j , a

†k

Bogoliubov transformations generatedby anti-symmetric quadratics inaF†j , aF

†k

H = Ffind = C[w1, . . . , wd] = S∗(Cd) H = F+d = Λ∗(Cd)

318

Chapter 30

Supersymmetry, SomeSimple Examples

If one considers fermionic and bosonic quantum system that each separatelyhave operators coming from Lie algebra or superalgebra representations on theirstate spaces, when one combines the systems by taking the tensor product, theseoperators will continue to act on the combined system. In certain special casesnew operators with remarkable properties will appear that mix the fermionicand bosonic systems and commute with the Hamiltonian (often by giving somesort of “square root” of the Hamiltonian). These are generically known as“supersymmetries” and provide new information about energy eigenspaces. Inthis chapter we’ll examine in detail some of the simplest such quantum systems,examples of “superymmetric quantum mechanics”.

30.1 The supersymmetric oscillator

In the previous chapters we discussed in detail

• The bosonic harmonic oscillator in d degrees of freedom, with state spaceFd generated by applying d creation operators aBj an arbitrary numberof times to a lowest energy state |0〉B . The Hamiltonian is

H =1

2~ω

d∑j=1

(aB†jaBj + aBjaB

†j) =

d∑j=1

(NBj +1

2)~ω

where NBj is the number operator for the j’th degree of freedom, witheigenvalues nBj = 0, 1, 2, · · · .

• The fermionic oscillator in d degrees of freedom, with state space F+d

generated by applying d creation operators aF j to a lowest energy state

319

|0〉F . The Hamiltonian is

H =1

2~ω

d∑j=1

(aF†jaF j − aF jaF

†j) =

d∑j=1

(NF j −1

2)~ω

where NF j is the number operator for the j’th degree of freedom, witheigenvalues nF j = 0, 1.

Putting these two systems together we get a new quantum system with statespace

H = Fd ⊗F+d

and Hamiltonian

H =

d∑j=1

(NBj +NF j)~ω

Notice that the lowest energy state |0〉 for the combined system has energy 0,due to cancellation between the bosonic and fermionic degrees of freedom.

For now, taking for simplicity the case d = 1 of one degree of freedom, theHamiltonian is

H = (NB +NF )~ω

with eigenvectors |nB , nF 〉 satisfying

H|nB , nF 〉 = (nB + nF )~ω

Notice that while there is a unique lowest energy state |0, 0〉 of zero energy, allnon-zero energy states come in pairs, with two states

|n, 0〉 and |n− 1, 1〉

both having energy n~ω.

This kind of degeneracy of energy eigenvalues usually indicates the existenceof some new symmetry operators commuting with the Hamiltonian operator.We are looking for operators that will take |n, 0〉 to |n − 1, 1〉 and vice-versa,and the obvious choice is the two operators

Q+ = aBa†F , Q− = a†BaF

which are not self adjoint, but are each other’s adjoints ((Q−)† = Q+).

The pattern of energy eigenstates looks like this

320

Computing anticommutators using the CCR and CAR for the bosonic andfermionic operators (and the fact that the bosonic operators commute with thefermionic ones since they act on different factors of the tensor product), onefinds that

Q2+ = Q2

− = 0

and

(Q+ +Q−)2 = [Q+, Q−]+ = H

One could instead work with self-adjoint combinations

Q1 = Q+ +Q−, Q2 =1

i(Q+ −Q−)

which satisfy

[Q1, Q2]+ = 0, Q21 = Q2

2 = H

Notice that the Hamiltonian H is a square of the self-adjoint operator Q+ +Q−, and this fact alone tells us that the energy eigenvalues will be non-negative.It also tells us that energy eigenstates of non-zero energy will come in pairs

|ψ〉, (Q+ +Q−)|ψ〉

321

with the same energy. To find states of zero energy, instead of trying to solvethe equation H|0〉 = 0 for |0〉, one can look for solutions to

Q1|0〉 = 0 or Q2|0〉 = 0

These operators don’t correspond to a Lie algebra representation as H does,but do come from a Lie superalgebra representation, so are described as gen-erators of a “supersymmetry” transformation. In more general theories withoperators like this with the same relation to the Hamiltonian, one may or maynot have solutions to

Q1|0〉 = 0 or Q2|0〉 = 0

If such solutions exist, the lowest energy state has zero energy and is describedas invariant under the supersymmetry. If no such solutions exist, the lowestenergy state will have a non-zero, positive energy, and satisfy

Q1|0〉 6= 0 or Q2|0〉 6= 0

In this case one says that the supersymmetry is “spontaneously broken”, sincethe lowest energy state is not invariant under supersymmetry.

There is an example of a physical quantum mechanical system that has ex-actly the behavior of this supersymmetric oscillator. A charged particle confinedto a plane, coupled to a magnetic field perpendicular to the plane, can be de-scribed by a Hamiltonian that can be put in the bosonic oscillator form (toshow this, we need to know how to couple quantum systems to electromagneticfields, which we will come to later in the course). The equally spaced energylevels are known as “Landau levels”. If the particle has spin one-half, there willbe an additional term in the Hamiltonian coupling the spin and the magneticfield, exactly the one we have seen in our study of the two-state system. Thisadditional term is precisely the Hamiltonian of a fermionic oscillator. For thecase of gyromagnetic ratio g = 2, the coefficients match up so that we haveexactly the supersymmetric oscillator described above, with exactly the patternof energy levels seen there.

30.2 Supersymmetric quantum mechanics witha superpotential

The supersymmetric oscillator system can be generalized to a much wider classof potentials, while still preserving the supersymmetry of the system. In thissection we’ll introduce a so-called “superpotential” W (q), with the harmonicoscillator the special case

W (q) =q2

2For simplicity, we will here choose constants ~ = ω = 1

Recall that our bosonic annihilation and creation operators were defined by

aB =1√2

(Q+ iP ), a†B =1√2

(Q− iP )

322

Introducing an arbitrary superpotential W (q) with derivative W ′(q) we candefine new annihilation and creation operators:

aB =1√2

(W ′(Q) + iP ), a†B =1√2

(W ′(Q)− iP )

Here W ′(Q) is the multiplication operator W ′(q) in the Schrodinger representa-tion on functions of q, defined by conjugation with a unitary operator in otherunitarily equivalent representations. We keep our definition of the operators

Q+ = aBa†F , Q− = a†BaF

These satisfy

Q2+ = Q2

− = 0

for the same reason as in the oscillator case: repeated factors of aF or a†F vanish.Taking as the Hamiltonian the same square as before, we find

H =(Q+ +Q−)2

=1

2(W ′(Q) + iP )(W ′(Q)− iP )a†FaF +

1

2(W ′(Q)− iP )(W ′(Q) + iP )aFa

†F

=1

2(W ′(Q)2 + P 2)(a†FaF + aFa

†F ) +

1

2(i[P,W ′(Q)])(a†FaF − aFa

†F )

=1

2(W ′(Q)2 + P 2) +

1

2(i[P,W ′(Q)])σ3

But iP is the operator corresponding to infinitesimal translations in Q, so wehave

i[P,W ′(Q)] = W ′′(Q)

and

H =1

2(W ′(Q)2 + P 2) +

1

2W ′′(Q)σ3

which gives a large class of quantum systems, all with state space

H = HB ⊗F+d = L2(R)⊗C2

(using the Schrodinger representation for the bosonic factor).The energy eigenvalues will be non-negative, and energy eigenvectors with

positive energy will occur in pairs

|ψ〉, (Q+ +Q−)|ψ〉

.There may or may not be a state with zero energy, depending on whether

or not one can find a solution to the equation

(Q+ +Q−)|0〉 = Q1|0〉 = 0

323

If such a solution does exist, thinking in terms of Lie superalgebras, one callsQ1 the generator of the action of a supersymmetry on the state space, anddescribes the ground state |0〉 as invariant under supersymmetry. If no suchsolution exists, one has a theory with a Hamiltonian that is invariant under su-persymmetry, but with a ground state that isn’t. In this situation one describesthe supersymmetry as “spontaneously broken”. The question of whether a givensupersymmetric theory has its supersymmetry spontaneously broken or not isone that has become of great interest in the case of much more sophisticated su-persymmetric quantum field theories. There, hopes (so far unrealized) of makingcontact with the real world rely on finding theories where the supersymmetryis spontaneously broken.

In this simple quantum mechanical system, one can try and explicitly solvethe equation Q1|ψ〉 = 0. States can be written as two-component complexfunctions

|ψ〉 =

(ψ+(q)ψ−(q)

)and the equation to be solved is

(Q+ +Q−)|ψ〉 =1√2

((W ′(Q) + iP )a†F + (W ′(Q)− iP )aF )

(ψ+(q)ψ−(q)

)=

1√2

((W ′(Q) +d

dq)

(0 10 0

)+ (W ′(Q)− d

dq)

(0 01 0

))

(ψ+(q)ψ−(q)

)=

1√2

(W ′(Q)

(0 11 0

)+

d

dq

(0 1−1 0

))

(ψ+(q)ψ−(q)

)=

1√2

(0 1−1 0

)(d

dq−W ′(Q)σ3)

(ψ+(q)ψ−(q)

)= 0

which has general solution(ψ+(q)ψ−(q)

)= eW (q)σ3

(c+c−

)=

(c+e

W (q)

c−e−W (q

)for complex constants c+, c−. Such solutions will only be normalizable if

c+ = 0, limq→±∞

W (q) = +∞

or

c− = 0, limq→±∞

W (q) = −∞

If, for example, W (q) is an odd polynomial, one will not be able to satisfy eitherof these conditions, so there will be no solution, and the supersymmetry will bespontaneously broken.

324

30.3 Supersymmetric quantum mechanics anddifferential forms

If one considers supersymmetric quantum mechanics in the case of d degrees offreedom and in the Schrodinger representation, one has

H = L2(Rd)⊗ Λ∗(Cd)

the tensor product of complex-valued functions on Rd (acted on by the Weylalgebra Weyl(2d,C)) and anticommuting functions on Cd (acted on by theClifford algebra Cliff(2d,C)). There are two operators Q+ and Q−, adjointsof each other and of square zero. If one has studied differential forms, thisshould look familiar. This space H is well-known to mathematicians, as thecomplex valued differential forms on Rd, often written Ω∗(Rd), where here the∗ denotes an index taking values from 0 (the 0-forms, or functions) to d (thed-forms). In the theory of differential forms, it is well known that one has anoperator d on Ω∗(Rd) with square zero, called the de Rham differential. Usingthe inner product on Rd, one can put a Hermitian inner product on Ω∗(Rd)by integration, and then d has an adjoint δ, also of square zero. The Laplacianoperator on differential forms is

= (d+ δ)2

The supersymmetric quantum system we have been considering correspondsprecisely to this, once one conjugates d, δ as follows

Q+ = e−W (q)deW (q), Q− = eW (q)δe−W (q)

In mathematics, the interest in differential forms mainly comes from thefact that one can construct them not just on Rd, but on a general differentiablemanifold M , with a corresponding construction of d, δ, operators. In Hodgetheory, one studies solutions of

ψ = 0

(these are called “harmonic forms”) and finds that the dimension of the spaceof such solutions can be used to get topological invariants of the manifold M .


For a reference at the level of these notes, see [23]. For more details aboutsupersymmetric quantum mechanics see the quantum mechanics textbook ofTahktajan [71], and lectures by Orlando Alvarez [1]. These references alsodescribe the relation of these systems to the calculation of topological invariants,a topic pioneered in Witten’s 1982 paper on supersymmetry and Morse theory[81].

325

Chapter 31

The Pauli Equation and theDirac Operator

In chapter 30 we considered supersymmetric quantum mechanical systems whereboth the bosonic and fermionic variables that get quantized take values in aneven dimensional space R2d = Cd. There are then two different operators Q1

and Q2 that are square roots of the Hamiltonian operator. It turns out thatthere are much more interesting quantum mechanics systems that one can get byquantizing bosonic variables in phase space R2d, but fermionic variables in Rd.The operators appearing in such a theory will be given by the tensor productof the Weyl algebra in 2d variables and the Clifford algebra in d variables,and there will be one distinguished operator that provides a square root of theHamiltonian.

This means that the Casimir operator −|P|2 for the Euclidean group E(3)will have a square root, which is the Dirac operator /∂. This existence of asquare root of the Casimir operator provides a new way to construct irreduciblerepresentations of the group of spatial symmetries, using a new sort of quantumfree particle, one carrying an internal “spin” degree of freedom due to the useof the Clifford algebra.. Remarkably, fundamental matter particles are well-described in exactly this way, both in the non-relativistic theory we study inthis chapter as well as in the relativistic theory to be studied later.

31.1 The Pauli equation and free spin 12 particles

in d = 3

We have so far seen two quite different quantum systems based on three dimen-sional space:

• The free particle of chapter 17. This had classical phase space R6 withcoordinates q1, q2, q3, p1, p2, p3 and Hamiltonian 1

2m |p|2. Quantization us-

ing the Schrodinger representation gave operators Q1, Q2, Q3, P1, P2, P3

327

on the space HB = L2(R3) of square-integrable functions of the positioncoordinates. The Hamiltonian operator is

H =1

2m|P|2 = − 1

2m(∂2

∂q21

+∂2

∂q22

+∂2

∂q23

)

• The spin 12 quantum system, discussed first in chapter 7 and later in

section 28.1.1. This had a pseudo-classical fermionic phase space R3 withcoordinates ξ1, ξ2, ξ3 which after quantization became the operators

1√2σ1,

1√2σ2,

1√2σ3

on the state space HF = C2. For this system we considered the Hamilto-nian describing its interaction with a constant background magnetic field

H = −1

2(B1σ1 +B2σ2 +B3σ3)

It turns out to be an experimental fact that fundamental matter particles aredescribed by a quantum system that is the tensor product of these two systems,with state space

H = HB ⊗HF = L2(R3)⊗C2

which one can think of as two-component complex wavefunctions. This sys-tem has a pseudo-classical description using a phase space with six conventionalcoordinates qj , pj and three fermionic coordinates ξj . On functions of thesecoordinates one has a generalized Poisson bracket which provides a Lie superal-gebra structure on such functions. On generators, the non-zero bracket relationsare just

qj , pk = δjk, ξj , ξk = δjk

For now we will take the background magnetic field B = 0. In chapter 43we will see how to generalize the free particle to the case of a free particle in ageneral background electromagnetic field, and then the Hamiltonian term aboveinvolving the B field will appear. In the absence of electromagnetic fields theclassical Hamiltonian function will still just be

h =1

2m(p2

1 + p22 + p2

3)

but now this can be written in the following form (using the Leibniz rule for aLie superbracket)

h =1

2m

3∑j=1

pjξj ,

3∑k=1

pkξk =1

2m

3∑j,k=1

pjξj , ξkpk =1

2m

3∑j=1

p2j

Note the appearance of the function p1ξ1 + p2ξ2 + p3ξ3 which now plays a roleeven more fundamental than that of the Hamiltonian (which can be expressed

328

in terms of it). In this pseudo-classical theory p1ξ1 + p2ξ2 + p3ξ3 is a “super-symmetry”, Poisson commuting with the Hamiltonian, while at the same timeplaying the role of a sort of “square root” of the Hamiltonian, providing a newsort of symmetry that can be thought of as a “square root” of an infinitesimaltime translation.

Quantization takes

p1ξ1 + p2ξ2 + p3ξ3 →1√2σ ·P

and the Hamiltonian operator can now be written as an anticommutator and asquare

H =1

2m[

1√2σ ·P, 1√

2σ ·P]+ =

1

2m(σ ·P)2 =

1

2m(P 2

1 + P 22 + P 2

3 )

(using the fact that the σj satisfy the Clifford algebra relations for Cliff(3, 0,R)).We will define the three-dimensional Dirac operator as

/∂ = σ1∂

∂q1+ σ2

∂

∂q2+ σ3

∂

∂q3= σ ·∇

It operates on two-component wavefunctions(ψ1(q)ψ2(q)

)Using this Dirac operator (often called in this context the “Pauli operator”) wecan write a two-component version of the Schrodinger equation (often called the“Pauli equation” or “Schrodinger-Pauli equation”)

i∂

∂t

(ψ1(q)ψ2(q)

)=− 1

2m(σ1

∂

∂q1+ σ2

∂

∂q2+ σ3

∂

∂q3)2

(ψ1(q)ψ2(q)

)(31.1)

=− 1

2m(∂2

∂q21

+∂2

∂q22

+∂2

∂q23

)

(ψ1(q)ψ2(q)

)This free-particle version of the equation is just two copies of the standard free-particle Schrodinger equation, so physically just corresponds to two independentquantum free particles. It becomes much more non-trivial when a coupling toan electromagnetic field is introduced, as will be seen in chapter 43.

The introduction of a two-component wavefunction does allow us to findmore interesting irreducible representations of the group E(3), beyond the onesstudied in chapter 17. These have eigenvalue ± 1

2p (where p2 is the eigenvalueof the first Casimir operator |P|2) for the second Casimir operator J · P, asopposed to the zero eigenvalue case of single wavefunctions.

These representations will as before be on the space of solutions of the time-independent equation, and irreducible for fixed choice of the energy E. Theequation for the energy eigenfunctions of energy eigenvalue E will be

1

2m(σ ·P)2

(ψ1(q)ψ2(q)

)= E

(ψ1(q)ψ2(q)

)329

In terms of the inverse Fourier transform

ψ1,2(q) =1

(2π)32

∫∫∫R3

eip·qψ1,2(p)d3p

this equation becomes

((σ · p)2 − 2mE)

(ψ1(p)

ψ2(p)

)= (|p|2 − 2mE)

(ψ1(p)

ψ2(p)

)= 0 (31.2)

and as in chapter 17 our solution space is given by functions ψE,1,2(p) on the

sphere of radius√

2mE = |p| in momentum space (although now, two suchfunctions).

Another way to find solutions to this equation is to look for solutions toa pair of first-order equations involving the three-dimensional Dirac operator.Solutions to

σ · p

(ψ1(p)

ψ2(p)

)= ±√

2mE

(ψ1(p)

ψ2(p)

)will give solutions to 31.2, for either sign. One can rewrite this as

σ · p|p|

(ψ1(p)

ψ2(p)

)= ±

(ψ1(p)

ψ2(p)

)

and we will write solutions to this equation with the + sign as ψE,+(p), those for

the − sign as ψE,−(p). Note that ψE,+(p) and ψE,+(p) are each two-component

complex functions of the momentum, supported on the sphere√

2mE = |p|.In chapter 16 we saw that R ∈ SO(3) acts on single-component momentum

space solutions of the Schrodinger equation by

ψE(p)→ u(0, R)ψE(p) ≡ ψE(R−1p)

This takes solutions to solutions since the operator u(0, R) commutes with theCasimir operator |P|2

u(0, R)|P|2 = |P|2u(0, R) ⇐⇒ u(0, R)|P|2u(0, R)−1 = |P|2

This is true since

u(0, R)|P|2u(0, R)−1ψ(p) =u(0, R)|P|2ψ(Rp)

=|R−1P|2ψ(R−1Rp) = |P|2ψ(p)

For two-component wavefunctions, we could try to just take the representa-tion to be

u(0, R)

(ψ1(p)

ψ2(p)

)=

(ψ1(R−1p)

ψ2(R−1p)

)

330

(or, equivalently, thinking of two component wavefunctions as the tensor productof the space of single component wavefunction with C2, just acting on thefirst factor). If we do this, the operator σ · P does not commute with therepresentation u(0, R) because

u(0, R)(σ ·P)u(0, R)−1 = (σ ·R−1P) 6= σ ·P

Then rotations do not act separately on the spaces ψE,+(p) and ψE,−(p).If we want rotations to act separately on these spaces, we need to change

the action of rotations to

ψE,±(p)→ uS(0, R)ψE,±(p) = ΩψE,±(R−1p)

where Ω is one of the two elements of SU(2) corresponding to R ∈ SO(3) (or,in terms of tensor products, action by SU(2) on the C2 factor). Such an Ω canbe constructed using equation 6.3

Ω = Ω(φ,w) = e−iφ2 w·σ

Equation 6.5 shows that Ω is the SU(2) matrix corresponding to a rotation Rby an angle φ about the axis given by a unit vector w.

With this action on solutions we have

uS(0, R)(σ ·P)uS(0, R)−1ψE,±(p) =uS(0, R)(σ ·P)Ω−1ψE,±(Rp)

=Ω(σ ·R−1P)Ω−1ψE,±(R−1Rp)

=(σ ·P)ψE,±(p)

where we have used equation 6.5 to show

Ω(σ ·R−1P)Ω−1 = (σ ·RR−1P) = (σ ·P)

Note that the two representations we get this way are representations notof the rotation group SO(3) but of its double cover Spin(3) = SU(2) (becauseotherwise there is a sign ambiguity since we don’t know whether to choose Ωor −Ω). The translation part of the spatial symmetry group is easily seen tocommute with σ ·P, so we have constructed representations of E(3), or rather,of its double cover

E(3) = R3 o SU(2)

on the two spaces of solutions ψE,±(p). We will see that these two representa-tions are the E(3) representations described in section 17.3, the ones labeled bythe helicity ± 1

2 representations of the stabilizer group SO(2).The translation part of the group acts as in the one-component case, by the

multiplication operator

uS(a,1)ψE,±(p) = e−i(a·p)ψE,±(p)

anduS(a,1) = e−ia·P

so the Lie algebra representation is given by the usual P operator. The SU(2)part of the group acts by a product of two commuting different actions

331

1. The same action on the momentum coordinates as in the one-componentcase, just using R = Φ(Ω), the SO(3) rotation corresponding to the SU(2)group element Ω. For example, for a rotation about the x-axis by angle φwe have

ψE,±(p)→ ψE,±(R(φ, e1)−1p)

Recall that the operator that does this is e−iφL1 where

−iL1 = −i(Q2P3 −Q3P2) = −(q2∂

∂q3− q3

∂

∂q2)

and in general we have operators

−iL = −iQ×P

that provide the Lie algebra version of the representation (recall that atthe Lie algebra level, SO(3) and Spin(3) are isomorphic).

2. The action of the matrix Ω ∈ SU(2) on the two-component wavefunctionby

ψE,±(p)→ ΩψE,±(p)

For a rotation by angle φ about the x-axis we have

Ω = e−iφσ12

and the operators that provide the Lie algebra version of the representationare the

−iS = −i12σ

The Lie algebra representation corresponding to the action of both of thesetransformations is given by the operator

−iJ = −i(L + S)

and the standard terminology is to call L the “orbital” angular momentum, Sthe “spin” angular momentum, and J the “total” angular momentum.

The second Casimir operator for this case is

J ·P

and as in the one-component case the L · P part of this acts trivially on oursolutions ψE,±(p). The spin component acts non-trivially and we have

(J ·P)ψE,±(p) = (1

2σ · p)ψE,±(p) = ±1

2|p|ψE,±(p)

so we see that our solutions have helicity (eigenvalue of J · P divided by thesquare root of the eigenvalue of |P|2) values ± 1

2 , as opposed to the integralhelicity values discussed in chapter 17, where E(3) appeared and not its doublecover.

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31.2 The Dirac operator

One can generalize the above construction to the case of any dimension d asfollows. Recall from chapter 26 that associated to Rd with a standard innerproduct, but of a general signature (r, s) (where r + s = d, r is the numberof + signs, s the number of − signs) we have a Clifford algebra Cliff(r, s) withgenerators γj satisfying

γjγk = −γkγj , j 6= k

γ2j = +1 for j = 1, · · · , r γ2

j = −1, for j = r + 1, · · · , d

To any vector v ∈ Rd with components vj recall that we can associate a corre-sponding element /v in the Clifford algebra by

v ∈ Rd → /v =

d∑j=1

γjvj ∈ Cliff(r, s)

Multiplying this Clifford algebra element by itself and using the relations above,we get a scalar, the length-squared of the vector

/v2 = v2

1 + v22 · · ·+ v2

r − v2r+1 − · · · − v2

d = |v|2

This shows that by introducing a Clifford algebra, we can find an interestingnew sort of square-root for expressions like |v|2. We can define

Definition (Dirac operator). The Dirac operator is the operator

/∂ =

d∑j=1

γj∂

∂qj

This will be a first-order differential operator with the property that itssquare is the Laplacian

/∂2

=∂2

∂q21

+ · · ·+ ∂2

∂q2r

− ∂2

∂q2r+1

− · · · − ∂2

∂q2d

The Dirac operator /∂ acts not on functions but on functions taking valuesin the spinor vector space S that the Clifford algebra acts on. Picking a matrixrepresentation of the γj , the Dirac operator will be a constant coefficient firstorder differential operator acting on wavefunctions with dim S components. Inchapter 45 we will study in detail what happens for the case of r = 3, s = 1 andsee how the Dirac operator there provides an appropriate wave-equation withthe symmetries of special relativistic space-time.


The point of view here in terms of representations of E(3) is not very conven-tional, but the material here about spin and the Pauli equation can be found

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in any quantum mechanics book, see for example chapter 14 of [63]. For moredetails about supersymmetric quantum mechanics and the appearance of theDirac operator as the generator of a supersymmetry in the quantization of apseudo-classical system, see [71] and [1].

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Chapter 32

Lagrangian Methods andthe Path Integral

In this chapter we’ll give a rapid survey of a different starting point for devel-oping quantum mechanics, based on the Lagrangian rather than Hamiltonianclassical formalism. Lagrangian methods have quite different strengths andweaknesses than those of the Hamiltonian formalism, and we’ll try and pointthese out, while referring to standard physics texts for more detail about thesemethods.

The Lagrangian formalism leads naturally to an apparently very differentnotion of quantization, one based upon formulating quantum theory in terms ofinfinite-dimensional integrals known as path integrals. A serious investigationof these would require another and very different volume, so again we’ll haveto restrict ourselves to just outlining how path integrals work, describing theirstrengths and weaknesses, and giving references to standard texts for the details.

32.1 Lagrangian mechanics

In the Lagrangian formalism, instead of a phase space R2d of positions qj andmomenta pj , one considers just the position space Rd. Instead of a Hamiltonianfunction h(q,p), one has

Definition (Lagrangian). The Lagrangian L for a classical mechanical systemwith configuration space Rd is a function

L : (q,v) ∈ Rd ×Rd → L(q,v) ∈ R

Given differentiable paths in the configuration space defined by functions

γ : t ∈ [t1, t2]→ Rd

which we will write in terms of their position and velocity vectors as

γ(t) = (q(t), q(t))

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one can define a functional on the space of such paths

Definition. ActionThe action S for a path γ is

S[γ] =

∫ t2

t1

L(q(t), q(t))dt

The fundamental principle of classical mechanics in the Lagrangian formal-ism is that classical trajectories are given by critical points of the action func-tional. These may correspond to minima of the action (so this is sometimescalled the “principle of least action”), but one gets classical trajectories also forcritical points that are not minima of the action. One can define the appropriatenotion of critical point as follows

Definition. Critical point for S ) A path γ is a critical point of the functionalS[γ] if

δS(γ) ≡ d

dsS(γs)|s=0 = 0

whenγs : [t1, t2]→ Rd

is a smooth family of paths parametrized by an interval s ∈ (−ε, ε), with γ0 = γ.

We’ll now ignore analytical details and adopt the physicist’s interpretationof this as the first-order change in S due to an infinitesimal change δγ =(δq(t), δq(t)) in the path.

When (q(t), q(t)) satisfy a certain differential equation, the path γ will be acritical point and thus a classical trajectory:

Theorem. Euler-Lagrange equationsOne has

δS[γ] = 0

for all variations of γ with endpoints γ(t1) and γ(t2) fixed if

∂L

∂qj(q(t), q(t))− d

dt(∂L

∂qj(q(t), q(t))) = 0

for j = 1, · · · , d. These are called the Euler-Lagrange equations.

Proof. Ignoring analytical details, the Euler-Lagrange equations follow from thefollowing calculations, which we’ll just do for d = 1, with the generalization tohigher d straightfoward. We are calculating the first-order change in S due toan infinitesimal change δγ = (δq(t), δq(t))

δS[γ] =

∫ t2

t1

δL(q(t), q(t))dt

=

∫ t2

t1

(∂L

∂q(q(t), q(t))δq(t) +

∂L

∂q(q(t), q(t))δq(t))dt

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But

δq(t) =d

dtδq(t)

and, using integration by parts

∂L

∂qδq(t) =

d

dt(∂L

∂qδq)− (

d

dt

∂L

∂q)δq

so

δS[γ] =

∫ t2

t1

((∂L

∂q− d

dt

∂L

∂q)δq − d

dt(∂L

∂qδq))dt

=

∫ t2

t1

(∂L

∂q− d

dt

∂L

∂q)δqdt− (

∂L

∂qδq)(t2) + (

∂L

∂qδq)(t1)

If we keep the endpoints fixed so δq(t1) = δq(t2) = 0, then for solutions to

∂L

∂q(q(t), q(t))− d

dt(∂L

∂q(q(t), q(t))) = 0

the integral will be zero for arbitrary variations δq.

As an example, a particle moving in a potential V (q) will be described by aLagrangian

L(q, q) =1

2mq2 − V (q)

for which the Euler-Lagrange equations will be

−∂V∂qj

=d

dt(mqj)

This is just Newton’s second law which says that the force coming from thepotential is equal to the mass times the acceleration of the particle.

The derivation of the Euler-Lagrange equations can also be used to studythe implications of Lie group symmetries of a Lagrangian system. When a Liegroup G acts on the space of paths, preserving the action S, it will take classicaltrajectories to classical trajectories, so we have a Lie group action on the spaceof solutions to the equations of motion (the Euler-Lagrange equations). Ingood cases, this space of solutions is just the phase space of the Hamiltonianformalism. On this space of solutions, we have, from the calculation above

δS[γ] = (∂L

∂qδq(X))(t1)− (

∂L

∂qδq(X))(t2)

where now δq(X) is the infinitesimal change in a classical trajectory comingfrom the infinitesimal group action by an element X in the Lie algebra of G.From invariance of the action S under G we must have δS=0, so

(∂L

∂qδq(X))(t2) = (

∂L

∂qδq(X))(t1)

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This is an example of a more general result known as “Noether’s theorem”.In this context it says that given a Lie group action on a Lagrangian systemthat leaves the action invariant, for each element X of the Lie algebra we willhave a conserved quantity

∂L

∂qδq(X)

which is independent of time along the trajectory. A basic example is whenthe Lagrangian is independent of the position variable q, depending only on thevelocities q, for example in the case of a free particle, when V (q) = 0. In sucha case one has invariance under the Lie group R of space-translations.

An infinitesimal translation is given by

δq(t) = ε

and the conserved quantity is∂L

∂q

For the case of the free particle, this will be

∂L

∂q= mq

and the conservation law is conservation of momentum.Given a Lagrangian classical mechanical system, one would like to be able

to find a corresponding Hamiltonian system that will give the same equations ofmotion. To do this, we proceed by defining (for each configuration coordinateqj) the momenta pj as above, as the conserved quantities corresponding tospace-translations, so

pj =∂L

∂qj

Then, instead of working with trajectories characterized at time t by

(q(t), q(t)) ∈ R2d

we would like to instead use

(q(t),p(t)) ∈ R2d

where pj = ∂L∂qj

and this R2d is the Hamiltonian phase space with the conven-

tional Poisson bracket.This transformation between position-velocity and phase space is known as

the Legendre transform, and in good cases (for instance when L is quadraticin all the velocities) it is an isomorphism. In general though, this is not anisomorphism, with the Legendre transform often taking position-velocity spaceto a lower-dimensional subspace of phase space. Such cases are not unusualand require a much more complicated formalism, even as classical mechanicalsystems (this subject is known as “constrained Hamiltonian dynamics”). One

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important example we will study in chapter 44 is that of the free electromag-netic field, with equations of motion the Maxwell equations. In that case theconfiguration space coordinates are the components (A0, A1, A2, A3) of the vec-tor potential, with the problem arising because the Lagrangian does not dependon A0.

Besides a phase space, for a Hamiltonian system one needs a Hamiltonianfunction. Choosing

h =

d∑j=1

pj qj − L(q, q)

will work, provided one can use the relation

pj =∂L

∂qj

to solve for the velocities qj and express them in terms of the momentum vari-ables. In that case, computing the differential of h one finds (for d = 1, thegeneralization to higher d is straightforward)

dh =pdq + qdp− ∂L

∂qdq − ∂L

∂qdq

=qdp− ∂L

∂qdq

So one has∂h

∂p= q,

∂h

∂q= −∂L

∂q

but these are precisely Hamilton’s equations since the Euler-Lagrange equationsimply

∂L

∂q=

d

dt

∂L

∂q= p

The Lagrangian formalism has the advantage of depending concisely on thechoice of action functional on the space of possible classical trajectories. Thiscan be straightforwardly generalized to theories where the configuration spaceis a space of classical fields and is quite useful in the case of relativistic quan-tum field theories. There, unlike the conventional Hamiltonian formalism, theLagrangian allows one to treat space and time symmetrically, allowing one toexploit the full set of space-time symmetries, which can mix space and time di-rections. Noether’s theorem provides a powerful tool for finding the conservedquantities corresponding to symmetries of the system that are due to invarianceof the action under some group of transformations.

On the other hand, in the Lagrangian formalism one loses the infinite dimen-sional group which acts on phase space preserving the Poisson bracket, includingthe finite dimensional Heisenberg and symplectic subgroups. In the Hamiltonianformalism it is the moment map that provides functions corresponding to groupactions preserving the Poisson bracket, and these may or may not be symme-tries (i.e. Poisson-commute with the Hamiltonian function). Our study of the

339

harmonic oscillator exploited several techniques (use of a complex structure onphase space, and of the U(1) symmetry of rotations in the q− p plane) that areunavailable in the Lagrangian formalism, which just uses configuration space,not phase space.

While the Legendre transform method given above works in some situa-tions, more generally and more abstractly, one can pass from the Lagrangianto the Hamiltonian formalism by taking as phase space the space of solutionsof the Euler-Lagrange equations. This is sometimes called the “covariant phasespace”, and it can often more concretely be realized by fixing a time t = 0 andparametrizing solutions by their initial conditions at such a t = 0. One can alsogo directly from the action to a sort of Poisson bracket on this covariant phasespace (this is called the “Peierls bracket”). For a general Lagrangian, one cango to a version of the Hamiltonian formalism either by this method or by themethod of Hamiltonian mechanics with constraints. Only for a special class ofLagrangians though will one get a non-degenerate Poisson bracket on a linearphase space and recover all the good properties of the standard Hamiltonianformalism.

32.2 Quantization and path integrals

To be added to this section: explicit calculations for the free particle caseAfter use of the Legendre transform to pass to a Hamiltonian system, one

then faces the question of how to construct a corresponding quantum theory.The method of “canonical quantization” is the one we have studied, takingthe position coordinates qj to operators Qj and momentum coordinates pj tooperators Pj , with Qj and Pj satisfying the Heisenberg commutation relations.By the Stone von-Neumann theorem, up to unitary equivalence there is only onewas to do this and realize these operators on a state spaceH. Recall though thatthe Groenewold-van Hove no-go theorem says that there is an inherent operator-ordering ambiguity for operators of higher order than quadratic, thus for suchoperators providing many different possible quantizations of the same classicalsystem (different though only by terms proportional to ~). In cases where theLegendre transform is not an isomorphism, the constrained Hamiltonian systemcomes with a new set of problems that appear when one tries to pass to aquantum system, and new methods are needed.

There is however a very different approach to quantization, which completelybypasses the Hamiltonian formalism. This is the path integral formalism, whichis based upon a method for calculating matrix elements of the time-evolutionoperator

〈qT |e−i~HT |q0〉

in the position eigenstate basis in terms of an integral over the space of pathsthat go from position q0 to position qT in time T (we will here only treat thed = 1 case). Here |q0〉 is an eigenstate of Q with eigenvalue q0 (a delta-functionat q0 in the position space representation), and |qT 〉 has Q eigenvalue qT . Thismatrix element has a physical interpretation as the amplitude for a particle

340

starting at q0 at t = 0 to have position qT at time T , with its norm-squaredgiving the probability density for observing the particle at position qT . It isalso the kernel function that allows one to determine the wave function ψ(q, t)at any time t in terms of its initial value at t = 0, by calculating

ψ(q, t) =

∫ ∞−∞〈qt|e−

i~HT |q0〉ψ(q, 0)dq0

To try and derive a path-integral expression for this, one breaks up theinterval [0, T ] into N equal-sized sub-intervals and calculates

〈qT |(e−iN~HT )N |q0〉

If the Hamiltonian is a sum H = K + V , the Trotter product formula showsthat

〈qT |e−i~HT |q0〉 = lim

N→∞〈qT |(e−

iN~KT e−

iN~V T )N |q0〉

If K(P ) can be chosen to depend only on the momentum operator P and V (Q)depends only on the operator Q, then one can insert alternate copies of theidentity operator in the forms∫ ∞

−∞|q〉〈q|dq = 1,

∫ ∞−∞|p〉〈p|dp = 1

This gives a product of terms of the form

〈qtj |e−iN~K(P )T |ptj 〉〈ptj |e−

iN~V (Q)T |qtj−1

〉

where the index j goes from 0 to N , tj = jT/N and the variable qtj and ptjwill be integrated over.

Such a term can be evaluated as

〈qtj |ptj 〉〈ptj |qtj−1〉e−

iN~K(ptj )T e−

iN~V (qtj−1

)T

=1√2π~

ei~ qtj ptj

1√2π~

e−i~ qtj−1

ptj e−iN~K(ptj )T e−

iN~V (qtj−1

)T

=1

2π~ei~ptj (qtj−qtj−1

)e−iN~ (K(ptj )+V (qtj−1

))T

The N factors of this kind give an overall factor of ( 12π~ )N times something

which is a discretized approximation to

ei~∫ T0

(pq−h(q(t),p(t)))dt

where the phase in the exponential is just the action. Taking into account theintegrations over qtj and ptj one should have something like

〈qT |e−i~HT |q0〉 = lim

N→∞(

1

2π~)N

N∏j=1

∫ ∞−∞

∫ ∞−∞

dptjdqtjei~∫ T0

(pq−h(q(t),p(t)))dt

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although one should not do the first and last integrals over q but fix the firstvalue of q to q0 and the last one to qT . One can try and interpret this sort ofintegration in the limit as an integral over the space of paths in phase space,thus a “phase space path integral”.

This is an extremely simple and seductive expression, apparently saying that,once the action S is specified, a quantum system is defined just by consideringintegrals ∫

Dγ ei~S[γ]

over paths γ in phase space, where Dγ is some sort of measure on this space ofpaths. Since the integration just involves factors of dpdq and the exponential justpdq and h(p, q), this formalism seems to share the same sort of invariance underthe infinite-dimensional group of canonical transformations (transformations ofthe phase space preserving the Poisson bracket) as the classical Hamiltonianformalism. It also appears to solve our problem with operator ordering ambigu-ities, since introducing products of P s and Q’s at various times will correspondto a path integral with various p and q factors in the integrand, but these com-mute, giving just one way of producting products at equal times of any numberof P and Q operators.

Unfortunately, we know from the Groenewold-van Hove theorem that thisis too good to be true. This expression cannot give a unitary representation ofthe full group of canonical transformations, at least not one that is irreducibleand restricts to what we want on transformations generated by linear functionsq and p. Another way to see the problem is that a simple argument showsthat by canonical transformations one can transform any Hamiltonian into afree-particle Hamiltonian, so all quantum systems would just be free particlesin some choice of variables. For the details of these arguments and a carefulexamination of what goes wrong, see chapter 31 of [60]. One aspect of theproblem is that for successive values of j the coordinates qtj or ptj have noreason to be close together. This is an integral over “paths” that do not acquireany expected continuity property as N → ∞, so the answer one gets alwaysdepends on the details of the discretization chosen, reintroducing the operator-ordering ambiguity problem.

One can intuitively see that there is something disturbing about such paths,since one is alternately at each time interval switching back and forth betweena q-space representation where q has a fixed value and nothing is known aboutp, and a p space representation where p has a fixed value but nothing is knownabout q. The “paths” of the limit are objects with little relation to continuouspaths in phase space, so while one may be able to define the limit above, itwill not necessarily have any of the properties one expects of an integral overcontinuous paths.

When the Hamiltonian h is quadratic in the momentum p, the ptj integralswill be Gaussian integrals that can be performed exactly. Equivalently, thekinetic energy part K of the Hamiltonian operator will have a kernel in positionspace that can be computed exactly. As a result, the ptj integrals can be

342

eliminated, along with the problematic use of alternating q-space and p-spacerepresentations. The remaining integrals over the qtj one might hope to interpretas a path integral over paths not in phase space, but in position space. One

finds, if K = P 2

2m

〈qT |e−i~HT |q0〉 =

limN→∞

(i2π~TNm

)N2

√m

i2π~T

N∏j=1

∫ ∞−∞

dqtjei~∑Nj=1(

m(qtj−qtj−1

)2

2T/N−V (qtj ) TN )

In the limit N →∞ the phase of the exponential becomes

S(γ) =

∫ T

0

dt(1

2m(q2)− V (q(t)))

One can try and properly normalize things so that this limit becomes an integral∫Dγ e

i~S[γ]

where now the paths γ(t) are paths in the position space.Perhaps should add a figure of a one-dimensional path integral here.An especially attractive aspect of this expression is that it provides a simple

understanding of how classical behavior emerges in the classical limit as ~→ 0.The stationary phase approximation method for oscillatory integrals says that,for a function f with a single critical point at x = xc (i.e. f ′(xc) = 0) and for asmall parameter ε, one has

1√i2πε

∫ +∞

−∞dx eif/ε =

1√f ′′(c)

eif(xc)/ε(1 +O(ε))

Using the same principle for the infinite-dimensional path integral, with f = Sthe action functional on paths, and ε = ~, one finds that for ~ → 0 the pathintegral will simplify to something that just depends on the classical trajectory,since by the principle of least action, this is the critical point of S.

Such position-space path integrals do not have the problems of principleof phase space path integrals coming from the Groenewold-van Hove theorem,but they still have serious analytical problems since they involve an attemptto integrate a wildly oscillating phase over an infinite-dimensional space. Awayfrom the limit ~ → 0, it is not clear that whatever results one gets will beindependent of the details of how one takes the limit to define the infinite-dimensional integral, or that one will naturally get a unitary result for the timeevolution operator.

Path integrals for anticommuting variables can also be defined by analogywith the bosonic case, using the notion of fermionic integration discussed ear-lier. Such fermionic path integrals will usually be analogs of the phase spacepath integral, but in the fermionic case there are no points and no problem ofcontinuity of paths. In this case the “integral” is not really an integral, butrather an algebraic operation with some of the same properties, and it will notobviously suffer from the same problems as the phase space path integral.

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32.3 Euclidean path integrals, Brownian motionand statistical mechanics

This section to be significantly expanded. Begin with the free particle and Brow-nian motion and work that out explicitly. Add in a short discussion of statisticalmechanics and the partition function

Such path integrals though are closely related to integrals that are known tohave a rigorous definition with good properties, ones that occur in the theoryof random walks. There, a well-defined measure on paths does exist, Wienermeasure. In some sense Wiener measure is what one gets in the case of the pathintegral for a free particle, but taking the time variable t to be complex andanalytically continuing

t→ it

So, one can use Wiener measure techniques to define the path integral, gettingresults that need to be analytically continued back to the physical time variable.

32.4 Advantages and disadvantages of the pathintegral

In summary, the path integral method has the following advantages:

• An intuitive picture of the classical limit and a calculational method for“semi-classical” effects (quantum effects at small ~).

• Calculations for free particles or potentials V at most quadratic in q can bedone just using Gaussian integrals, and these are relatively easy to eval-uate and make sense of, despite the infinite-dimensionality of the spaceof paths. For higher order terms in V (q), one can get a series expan-sion by expanding out the exponential, giving terms that are moments ofGaussians so can be evaluated exactly.

• After analytical continuation, path integrals can be rigorously defined us-ing Wiener measure techniques, and often evaluated numerically even incases where no exact solution is known.

On the other hand, there are disadvantages:

• Some path integrals such as phase space path integrals do not at all havethe properties one might expect, so great care is required in any use ofthem.

• How to get unitary results can be quite unclear. The analytic continua-tion necessary to make path integrals well-defined can make their physicalinterpretation obscure.

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• Symmetries with their origin in symmetries of phase space that aren’tjust symmetries of configuration space are difficult to see using the con-figuration space path integral, with the harmonic oscillator providing agood example. One can see such symmetries using the phase-space pathintegral, but this is not reliable.

Path integrals for anticommuting variables can also be defined by analogywith the bosonic case, using the notion of fermionic integration discussed ear-lier. Such fermionic path integrals will usually be analogs of the phase spacepath integral, but in the fermionic case there are no points and no problem ofcontinuity of paths. In this case the “integral” is not really an integral, butrather an algebraic operation with some of the same properties, and it will notobviously suffer from the same problems as the phase space path integral. )


For much more about Lagrangian mechanics and its relation to the Hamiltonianformalism, see [2]. More details along the lines of the discussion here can befound in most quantum mechanics and quantum field theory textbooks. Anextensive discussion at an introductory level of the Lagrangian formalism andthe use of Noether’s theorem to find conserved quantities when it is invariantunder a group action can be found in [48]. For the formalism of constrainedHamiltonian dynamics, see [70], and for a review article about the covariantphase space and the Peierls bracket, see [37].

For the path integral, Feynman’s original paper [15] or his book [16] are quitereadable. A typical textbook discussion is the one in chapter 8 of Shankar [63].The book by Schulman [60] has quite a bit more detail, both about applicationsand about the problems of phase-space path integrals. Yet another fairly com-prehensive treatment, including the fermionic case, is the book by Zinn-Justin[86].

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Chapter 33

Correlation Functions

To be writtenStart with path integral, adding a source term. Give expression for cor-

relation functions of the qs. Interpretation in terms of the Heisenberg groupaction.

Then, do for complex variables, Bargmann-Fock quantization. Vacuum tovacuum amplitudes. The propagator in this case. General correlation functionsand their reduction to two-particle correlations.


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Chapter 34

Multi-particle Systems

In chapter 9 we saw how to use symmetric or antisymmetric tensor productsto describe a fixed number of identical quantum systems (for instance, freeparticles). From very early on in the history of quantum mechanics, it becameclear that at least certain kinds of particles, photons, required a formalism thatcould describe arbitrary numbers of particles, as well as phenomena involvingtheir creation and annihilation. This could be accomplished by thinking ofphotons as quantized excitations of a classical system with an infinite numberof harmonic oscillator degrees of freedom: the electromagnetic field. In ourmodern understanding of fundamental physics all elementary particles, not justphotons, are best described in this way. The same sort of formalism is also justwhat is needed to describe quantum phenomena in condensed matter physics,where one must deal with arbitrarily large numbers of particles, as well as thequantum behavior of materials that can approximately be described by somesort of field.

Conventional textbooks on quantum field theory often begin with relativis-tic systems, but we’ll start instead with the non-relativistic case. We’ll study asimple quantum field theory that extends the conventional single-particle quan-tum systems we have dealt with so far to deal with multi-particle systems. Thisversion of quantum field theory is what gets used in condensed matter physics,and is in many ways simpler than the relativistic case, which we’ll take up in alater chapter. It applies equally well to bosonic and fermionic particles.

Quantum field theory is a large and complicated subject, suitable for a full-year course at an advanced level. We’ll be giving only a very basic introduc-tion, mostly just considering free fields, which correspond to systems of non-interacting particles. Most of the complexity of the subject only appears whenone tries to construct quantum field theories of interacting particles. The theoryof free quantum fields in many ways it is little more than something we havealready discussed in great detail, the quantum harmonic oscillator problem. Amajor difference is that the classical harmonic oscillator phase space that is get-ting quantized in this case is an infinite dimensional one, the space of solutionsto the free particle Schrodinger equation.

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As usual, we’ll set ~ = 1, and for simplicity we’ll start with the case of a singlespatial dimension. We’ll also begin using x to denote a spatial variable insteadof the q conventional when this is the coordinate variable in a finite-dimensionalphase space.

34.1 Multi-particle quantum systems as quantaof a harmonic oscillator

It turns out that quantum systems of identical particles are best understood bythinking of such particles as quanta of a harmonic oscillator system. We willbegin with the bosonic case, then later consider the fermionic case, which usesthe fermionic oscillator system.

34.1.1 Bosons and the quantum harmonic oscillator

A fundamental postulate of quantum mechanics (see chapter 9) is that given aspace of states H1 describing a bosonic single particle, a collection of N identicalsuch particles is described by

SN (H1) = (H1 ⊗ · · · ⊗ H1︸︷︷︸N−times

)S

where the superscript S means we take elements of the tensor product invariantunder the action of the group SN by permutation of the factors. To describe anarbitrary number of identical particles, we should take as state space the sumof these

S∗(H1) = C⊕H1 ⊕ (H1 ⊗H1)S ⊕ · · · (34.1)

This same symmetric part of the tensor product occurs in the Bargmann-Fock construction of the state space of the harmonic oscillator, where the statespace Fd is called the Fock space and can be described in three different butisomorphic ways:

• A basis

|n1, n2, · · · , nd〉

is labeled by the eigenvalues of the number operators Nj for j = 1, · · · , dwhere nj ∈ 0, 1, 2, · · · . This is called the “occupation number” basis ofFd. In this basis, the annihilation and creation operators are

aj |n1, n2, · · · , nd〉 =√nj |n1, n2, · · · , nj − 1, · · · , nd〉

a†j |n1, n2, · · · , nd〉 =√nj + 1|n1, n2, · · · , nj + 1, · · · , nd〉

• Fd is the completion (with respect to the Bargmann-Fock inner product)of the space of polynomials C[w1, w2, · · · , wd] in d complex variables, with

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basis elements corresponding to the occupation number basis the mono-mials

wn11 wn2

2 · · ·wndd

Here the annihilation and creation operators are

aj =∂

∂wj, a†j = wj

• Fd is a completion of the algebra

S∗(W ∗) = C⊕W ∗ ⊕ (W ∗ ⊗W ∗)S ⊕ · · ·

of symmetric multilinear forms on W discussed in section 9.6, which isisomorphic to the algebra of polynomial functions on W (see the discussionin section 9.6). Here we take W to be the vector space Cd, with thecoordinate functions wj on W a basis of W ∗.

For each of these descriptions of Fd, we have an inner product on Fd andthe basis elements given above can be normalized to be an orthonormal basis.We also have a set of d annihilation and creation operators aj , a

†j that are each

other’s adjoints, and satisfy the canonical commutation relations

[aj , a†k] = δjk

We will describe the basis state |n1, n2, · · · , nd〉 as one containing n1 quanta oftype 1, n2 quanta of type 2, etc., and a total number of quanta

N =

d∑j=1

nj

We see from the description of the Fock space Fd as S∗(Cd) that it providesthe same state space as that of an arbitrary number of indistinguishable quan-tum systems, for the case H1 = Cd. So we can think of the d-dimensional quan-tum harmonic oscillator state space as that of an arbitrary number of quanta,where each quantum is a state in Cd. To describe arbitrary numbers of parti-cles, we just need to generalize from H1 = Cd to H1 the space of solutions ofthe theory describing a single particle. Note that this formalism implies indis-tinguishability of quanta and symmetry under interchange of quanta, since onlythe numbers of quanta appear in the description of the state. This will automat-ically implement the separate symmetry postulate needed in the conventionalquantum mechanical description of multiple particles by tensor products.

34.1.2 Fermions and the fermionic oscillator

For the case of fermionic particles, if H1 is the state space for a single particle,an arbitrary number of particles will be described by the state space

Λ∗(H1) = C⊕H1 ⊕ (H1 ⊗H1)A ⊕ · · ·

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where (unlike the bosonic case) this is a finite sum if H1 is finite-dimensional.One can proceed as in the bosonic case, instead of Fd using the fermionic oscil-lator state space HF = F+

d , which again has three isomorphic descriptions:

• A basis|n1, n2, · · · , nd〉

is labeled by the eigenvalues of the number operators Nj for j = 1, · · · , dwhere nj ∈ 0, 1. This is called the “occupation number” basis of F+

d .

• F+d is the space of polynomials C[η1, η2, · · · , ηd] in d anticommuting com-

plex variables, with basis elements corresponding to the occupation num-ber basis the monomials

ηn11 ηn2

2 · · · ηndd

• F+d is the Grassmann algebra

Λ∗(W ∗) = C⊕W ∗ ⊕ (W ∗ ⊗W ∗)A ⊕ · · ·

discussed in section 9.6, with product the wedge-product. Here we takeW to be the vector space Cd, with the coordinate functions ηj on W abasis of W ∗.

For each of these descriptions of F+d we have an inner product on F+

d andthe basis elements given above can be normalized to be an orthonormal basis.We also have a set of d annihilation and creation operators aF j , aF

†j that are

each other’s adjoints, and satisfy the canonical anticommutation relations

[aF j , aF†k]+ = δjk

We will describe the basis state |n1, n2, · · · , nd〉 as one containing n1 quanta oftype 1, n2 quanta of type 2, etc., and a total number of quanta

N =

d∑j=1

nj

We see from the description of the Fock space F+d as Λ∗(Cd) that it provides the

same state space as that of an arbitrary number of fermionic indistinguishablequantum systems, for the case H1 = Cd. The d-dimensional fermionic oscillatorstate space is that of an arbitrary number of fermionic quanta, where eachquantum is a state in Cd. As for bosons, our plan is to generalize this to H1

the space of solutions of the single-particle theory.The Fock space F+

d is finite-dimensional, of dimension 2d, so there are nostates with indefinitely large occupation numbers, and no need for a completionlike that of the bosonic case. The formalism automatically implies Pauli princi-ple (no more than one quantum per state) and the antisymmetry property forstates of multiple fermionic quanta that is a separate postulate in our earlierdescription of multiple particle states as tensor products.

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34.2 Multi-particle quantum systems of free par-ticles

To describe multi-particle quantum systems we will take as our single-particlestate space H1 the space of solutions of the free-particle Schrodinger equation,as already studied in chapter 10. As we saw in that chapter, one can use a “finitebox” normalization, which gives a discrete, countable basis for this space andthen try and take the limit of infinite box size. To fully exploit the symmetries ofphase space though, we will need the “continuum normalization”, which requiresconsidering not just functions but distributions.

34.2.1 Box normalization

Recall that for a free particle in one dimension the state space H consists ofcomplex-valued functions on R, with observables the self-adjoint operators formomentum

P = −i ddx

and energy (the Hamiltonian)

H =P 2

2m= − 1

2m

d2

dx2

Eigenfunctions for both P and H are the functions of the form

ψp(x) ∝ eipx

for p ∈ R, with eigenvalues p for P and p2

2m for H.

Note that these eigenfunctions are not normalizable, and thus not in theconventional choice of state space as L2(R). One way to deal with this issue isto do what physicists sometimes refer to as “putting the system in a box”, byimposing periodic boundary conditions

ψ(x+ L) = ψ(x)

for some number L, effectively restricting the relevant values of x to be consid-ered to those on an interval of length L. For our eigenfunctions, this conditionis

eip(x+L) = eipx

so we must have

eipL = 1

which implies that

p =2π

Lj ≡ pj

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for j an integer. Then the momentum will take on a countable number ofdiscrete values corresponding to the j ∈ Z, and

|j〉 = ψj(x) =1√Leipjx =

1√Lei

2πjL x

will be orthornormal eigenfunctions satisfying

〈j′|j〉 = δjj′

This “box” normalization is one form of what physicists call an “infrared cutoff”,a way of removing degrees of freedom that correspond to arbitrarily large sizes,in order to make a problem well-defined. To get a well-defined problem onestarts with a fixed value of L, then one studies the limit L→∞.

The number of degrees of freedom is now countable, but still infinite. Inorder to get a completely well-defined problem, one approach is to first makethe number of degrees of freedom finite. This can be done with an additionalcutoff, an “ultraviolet cutoff”, which means restricting attention to |p| ≤ Λfor some finite Λ, or equivalently |j| < ΛL

2π . This makes the state space finitedimensional and one then studies the Λ→∞ limit.

For finite L and Λ our single-particle state space H1 is finite dimensional,with orthonormal basis elements ψj(x). An arbitrary solution to the Schrodingerequation is then given by

ψ(x, t) =1√L

+ ΛL2π∑

j=−ΛL2π

αjeipjxe−i

p2j

2m t =1√L

+ ΛL2π∑

j=−ΛL2π

αjei 2πjL xe−i

4π2j2

2mL2 t

for arbitrary complex coefficients αj and can be completely characterized by itsinitial value at t = 0

ψ(x, 0) =1√L

+ ΛL2π∑

j=−ΛL2π

αjei 2πjL x

Vectors in H1 have coordinates αj ∈ C with respect to our chosen orthonormalbasis ψj(x), and these coordinates are in the dual space H∗1.

Multi-particle states are now described by the Fock spaces Fd, where d isthe number of values of j, which is the dimension of our cut-off single-particlestate space H1. In the occupation number representation of the Fock space,orthonormal basis elements are

| · · · , npj−1 , npj , npj+1 , · · · 〉

where the subscript j indexes the possible values of the momentum p (which arediscretized in units of 2π

L , and in the interval [−Λ,Λ]). The occupation numbernpj is the number of particles in the state with momentum pj , taking values0, 1, 2, · · · ,∞. The state with all occupation numbers equal to zero is denoted

| · · · , 0, 0, 0, · · · 〉 = |0〉

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and called the “vacuum” state.For each pj we can define annihilation and creation operators apj and a†pj .

These satisfy the commutation relations

[apj , a†pk

] = δjk

and act on states in the occupation number representation as

apj | · · · , npj−1 , npj , npj+1 , · · · 〉 =√npj | · · · , npj−1

, npj − 1, npj+1, · · · 〉

a†pj | · · · , npj−1, npj , npj+1

, · · · 〉 =√npj + 1| · · · , npj−1

, npj + 1, npj+1, · · · 〉

Observables one can build out of these operators include

• The number operator

N =∑k

a†pkapk

which will have as eigenvalues the total number of particles

N | · · · , npj−1, npj , npj+1

, · · · 〉 = (∑k

npk)| · · · , npj−1, npj , npj+1

, · · · 〉

• The momentum operator

P =∑k

pka†pkapk

with eigenvalues the total momentum of the multiparticle system.

P | · · · , npj−1, npj , npj+1

, · · · 〉 = (∑k

nkpk)| · · · , npj−1, npj , npj+1

〉

• The Hamiltonian

H =∑k

p2k

2ma†pkapk

which has eigenvalues the total energy

H| · · · , npj−1 , npj , npj+1 , · · · 〉 = (∑k

nkp2k

2m)| · · · , npj−1 , npj , npj+1 , · · · 〉

With ultraviolet and infrared cutoffs in place, the possible values of pj arefinite in number, H1 is finite dimensional and this is nothing but the standardquantized harmonic oscillator (with a Hamiltonian that has different frequencies

ωj =p2j

2m

for different values of j). Note that we are using normal-ordered operatorshere, which is necessary since in the limit as one or both cutoffs are removed,H1 becomes infinite dimensional, and only the normal-ordered version of theHamiltonian used here is well-defined (the non-normal-ordered version will differby an infinite sum of 1

2 s). For the case of the momentum operator, one does nottechnically need the normal-ordering since in that case the constant 1

2 term forpj would cancel the one for −pj .

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34.2.2 Continuum normalization

A significant problem introduced by using cutoffs such as the box normalizationis that these ruin some of the space-time symmetries of the system. The one-particle space with an infrared cutoff is a space of functions on a discrete set ofpoints, and this set of points will not have the same symmetries as the usualcontinuous momentum space (for instance in three dimensions it will not carryan action of the rotation group SO(3)). In our study of quantum field theorywe would like to exploit the action of space-time symmetry groups on the statespace of the theory, so need a formalism that preserves such symmetries.

In our earlier discussion of the free particle, we saw that physicists oftenwork with a continuum normalization formalism for H1, such that

|p〉 = ψp(x) =1√2πeipx, 〈p′|p〉 = δ(p− p′)

where formulas such as the second one need to be interpreted in terms of dis-tributions. In quantum field theory we want to be able to think of each valueof p as corresponding to a classical degree of freedom that gets quantized, andthis “continuum” normalization will then correspond to an uncountable numberof degrees of freedom, requiring great care when working in such a formalism.This will however allow us to readily see the action of space-time symmetrieson the states of the quantum field theory and to exploit the duality betweenposition and momentum space embodied in the Fourier transform.

In the continuum normalization, an arbitrary solution to the free-particleSchrodinger equation is given by

ψ(x, t) =1√2π

∫ ∞−∞

α(p)eipxe−ip2

2m tdp

for some complex-valued function α(p) on momentum space. Such solutions arein one-to-one correspondence with initial data

ψ(x, 0) =1√2π

∫ ∞−∞

α(p)eipxdp

This is exactly the Fourier inversion formula, expressing a function ψ(x, 0) in

terms of its Fourier transform ψ(x, 0)(p) = α(p). Note that we want to con-sider not just square integrable functions α(p), but non-integrable functionslike α(p) = 1 (which corresponds to ψ(x, 0) = δ(x)), and distributions such asα(p) = δ(p), which corresponds to ψ(x, 0) = 1.

We will generally work with this continuum normalization, taking as oursingle-particle spaceH1 the space of complex valued functions ψ(x, 0) on R. Onecan think of the |p〉 as an orthornomal basis of H1, with α(p) the coordinatefunction for the |p〉 basis vector. α(p) is then an element of H∗1, the linearfunction on H1 given by taking the coefficient of the |p〉 basis vector.

Quantization should take α(p) ∈ H∗1 to a corresponding annihilation opera-tor a(p), with a(p) and its adjoint satisfying

[a(p), a†(p′)] = δ(p− p′)

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We will follow the the standard physicist’s practice of writing formulas whichare straight-forward generalizations of the finite-dimensional case, with sumsbecoming integrals. Some examples are

N =

∫ +∞

−∞a(p)†a(p)dp

for the number operator,

P =

∫ +∞

−∞pa(p)†a(p)dp

for the momentum operator, and

H =

∫ +∞

−∞

p2

2ma(p)†a(p)dp

for the Hamiltonian operator. To properly manipulate these formulas one canalways return to the box normalization with cutoffs, where one has well-definedsums, and then examine the limit as cutoffs are removed.

Digression. While we’ll often work with the occupation number representa-tion, for N -particle states one can use the identification with the symmetrictensor description of states, as well as the isomorphism between tensor prod-ucts of function spaces and functions on product spaces (see equation 9.1) to getconventional multi-particle wave-functions ψ(x1, x2, · · · , xN ), symmetric underinterchange of the xj.

Much like the bosonic case, one can identify the antisymmetric multilinearform description of N -particle states as ΛN (H∗1) with a description in terms ofantisymmetric functions of N variables (this construction is known to physicistsas the Slater determinant).

Everything in this section has a straightforward analog describing a multi-particle system of fermionic particles with energy-momentum relation given bythe Schrodinger equation. The annihilation and creation operators will be thefermionic ones, satisfying the canonical anticommutation relations

[aF pj , aF†pk

]+ = δjk

implying that states will have occupation numbers npj = 0, 1, automaticallyimplementing the Pauli principle.

34.3 Multi-particle systems and Bargmann-Fockquantization

Recall that, to quantize the d-dimensional quantum harmonic oscillator system,the Bargmann-Fock method starts with the dual classical phase spaceM, a real

357

vector space of dimension 2d, then complexifies and chooses a complex structure,with standard choice J0. Then one has


with coordinates zj giving a basis of M−J0, and the zj a basis of M+

J0. Quanti-

zation takes the zj , zj to annihilation and creation operators, with the Poissonbracket relations becoming commutator relations. This is just a representationof the (complexified) Heisenberg Lie algebra, on the state space H = S∗(M+

J0).

Our construction of multi-particle quantum systems is just an infinite-dimensionalexample of this same construction, with

M+J0

= H∗1

Note that here our phase space (the space of solutions of the free-particleSchrodinger equation) is already a complex vector space since the wave-functionsare complex valued. We will use this as our standard complex structure (J0 isjust multiplication by i).

Also note that the Schrodinger equation is first-order in time, so solutionsare determined by the initial values ψ(x) = ψ(x, 0) of the wavefunction at t = 0.Elements of the space H1 of solutions can be identified with their initial-valuedata, the wavefunction at t = 0. This is unlike typical finite-dimensional classi-cal mechanical systems, where the equation of motion is second-order in time,with solutions determined by two pieces of initial-value data, the coordinatesand momenta (since one needs initial velocities as well as positions). Taking H1

as a classical phase space, it has the property that there is no natural splittingof coordinates into position-like variables and momentum-like variables. Whilethe Bargmann-Fock construction works well, there no natural way of setting uphere an infinite-dimensional version of the Schrodinger representation, wherestates would be functionals of position-like variables.

Instead of an index j, basis vectors |p〉 of H1 are labeled by a continuousvariable p, and coordinates with respect to this basis are the α(p). These co-ordinates live in H∗1 which is our M+

J0. We will also need complex conjugated

coordinates α(p). These live in M−J0, which we can take to be H∗1 with the

opposite complex structure, denoted H∗1. Treating each pair α(p), α(p) as coor-dinates of an oscillator, generalizing what one does for the coordinates zj , zj inthe finite-dimensional case, the Poisson bracket on

M+J0⊕M−J0

will be determined by

α(p), α(p′) = α(p), α(p′) = 0, α(p), α(p′) = −iδ(p− p′)

Quantization takes

α(p)→ a(p), α(p)→ a†(p), 1→ 1

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Multiplying these operators by −i gives a representation Γ′ of the HeisenbergLie algebra determined by the Poisson bracket relations. We have

Γ′(α(p)) = −ia(p), Γ′(α(p)) = −ia†(p)

and the commutator relations

[a(p), a(p′)] = [a†(p), a†(p′)] = 0, [a(p), a†(p′)] = δ(p− p′)1

The appearance of the delta-function in these relations shows that theseare actually relations for distributions, with the Poisson bracket actually welldefined on elements of H∗1 and H∗1 given by

α(f) =

∫ ∞−∞

f(p)α(p)dp, α(g) =

∫ ∞−∞

g(p)α(p)dp

for f, g sufficiently well-behaved functions of p. A pair (f, g) of functions of pgives an element of

M+J0⊕M−J0

= H∗1 ⊕H∗1which we can write

α(f) + α(g) =

∫ ∞−∞

(f(p)α(p) + g(p)α(p))dp

On these the Poisson bracket relations are

α(f1) + α(g1), α(f2) + α(g2) = −i∫ ∞−∞

(f1(p)g2(p)− f2(p)g1(p))dp

which is the symplectic form

Ω((f1, g1), (f2, g2)) = −i∫ ∞−∞

(f1(p)g2(p)− f2(p)g1(p))dp

on M+J0⊕M−J0

.

Similarly, the operators a(p), a†(p) are “operator-valued distributions”. Toget well-defined operators one needs to choose sufficiently well-behaved functionsf of the momenta, and then one will have operators a(f), a†(f) which one canthink of as related to the a(p), a†(p) by integration

a(f) =

∫ ∞−∞

f(p)a(p)dp, a†(f) =

∫ ∞−∞

f(p)a†(p)dp

These operators satisfy the commutation relations

[a(f1) + a†(g1), a(f2) + a†(g2)] =

∫ ∞−∞

(f1(p)g2(p)− f2(p)g1(p))dp

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which is the analog of equation 21.3 from the finite-dimensional case. Thesecould also be written

[a(f), a(g)] = [a†(f), a†(g)] = 0, [a(f), a†(g)] =

∫ ∞−∞

f(p)g(p)dp

The fomalism for many-particle fermionic systems is much the same, with thesameH1 and the same choice of bases. The only difference is that the coordinatefunctions are now taken to be anticommuting, satisfying the fermionic Poissonbracket relations of a Lie superalgebra rather than a Lie algebra. The finitedimensional construction of section 28.3 now becomes

α(p), α(p′)+ = α(p), α(p′)+ = 0, α(p), α(p′)+ = δ(p− p′)

and one has

α(f1) + α(g1), α(f2) + α(g2)+ =

∫ ∞−∞

(f1(p)g2(p) + f2(p)g1(p))dp

After quantization the annihilation and creation operators a(p), a†(p) are operator-valued distributions satisfying anticommutation relations

[a(p), a(p′)]+ = [a†(p), a†(p′)]+ = 0, [a(p), a†(p′)]+ = δ(p− p′)1

and operators a(f), a†(f) satisfying

[a(f), a(g)]+ = [a†(f), a†(g)]+ = 0, [a(f), a†(g)]+ =

∫ ∞−∞

f(p)g(p)dp

and generating an infinite-dimensional complex Clifford algebra.

34.4 Dynamics

To describe the time evolution of a quantum field theory system, it is generallyeasier to work with the Heisenberg picture (in which the time dependence is inthe operators) than the Schrodinger picture (in which the time dependence is inthe states). This is especially true in relativistic systems where one wants to asmuch as possible treat space and time on the same footing. It is however alsotrue in non-relativistic cases due to the complexity of the description of the states(inherent since one is trying to describe arbitrary numbers of particles) versusthe description of the operators, which are built simply out of the annihilationand creation operators.

Recall that in classical Hamiltonian mechanics, the Hamiltonian function hdetermines how an observable f evolves in time by the differential equation

d

dtf = f, h

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Quantization takes f to an operator f , and h to a self-adjoint operator H.Multiplying this by −i gives a skew-adjoint operator that exponentiates (we’llassume here H time-independent) to the unitary operator

U(t) = e−iHt

that determines how states (in the Schrodinger picture) evolve under time trans-lation. In the Heisenberg picture states stay the same and operators evolve, withtheir time evolution given by

f(t) = eiHtf(0)e−iHt

Such operators satisfyd

dtf = [f ,−iH]

which is the quantization of the classical dynamical equation.In our case the classical Hamiltonian is

h =

∫ ∞−∞

p2

2mα(p)α(p)dp

and the classical equations of motion are

d

dtα(p, t) = α(p, t), h = −i p

2

2mα(p, t)

with solutions

α(p, t) = e−ip2

2m tα(p, 0)

For operators, the equations are

d

dta(p, t) = [a(p, t),−iH] = −i p

2

2ma(p, t)

with solutions

a(p, t) = e−ip2

2m ta(p, 0) (34.2)


Conventional textbooks on quantum field theory often begin with relativisticsystems, but we’ll start instead with the non-relativistic case. We’ll study asimple quantum field theory that extends the conventional single-particle quan-tum systems we have dealt with so far to deal with multi-particle systems. Thisversion of quantum field theory is what gets used in condensed matter physics,and is in many ways simpler than the relativistic case, which we’ll take up in alater chapter.

This material is discussed in essentially any quantum field theory textbook.Many do not explicitly discuss the non-relativistic case, two that do are [24]

361

and [34]. Two books aimed at mathematicians that cover the subject muchmore carefully than those for physicists are [21] and [12]. In particular section4.5 of [21] gives a detailed description of the bosonic and fermionic Fock spaceconstructions. For a rigorous version of the material in this chapter (althoughgiven in the relativistic case), see chapter X.7 of [56].

Some good sources for learning about quantum field theory from the pointof view of non-relativistic many-body theory are Feynman’s lecture notes onstatistical mechanics [19] as well as [40] and [67].

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Chapter 35

Non-relativistic QuantumFields

Introduction to come

35.1 Quantum field operators

The multi-particle formalism developed in chapter 34 works well to describestates of multiple free particles, but does so purely in terms of states with well-defined momenta, with no information at all about their position. Instead ofstarting with a basis |p〉 of the single-particle space H1, one could have insteadstarted with the basis

|x〉 = ψx(x′) = δ(x′ − x)

which are eigenfunctions of the position operator Q with well-defined positionx, then considered an oscillator for each value of x. The corresponding statespace would be an infinite-dimensional Fock space, with an occupation numberfor each value of x, something one could make well-defined by introducing aspatial cutoff and discretizing space, so that x only takes on a finite number ofvalues. This however would be a much less useful formalism than the one usingmomentum space, because such states in the occupation number basis would notbe free-particle energy eigenstates. While a state with a well-defined momentumevolves as a state with the same momentum, a state with well-defined positionat some time does not evolve into states with well-defined positions (its wavefunction spreads out).

One does however want to be able to discuss states with well-defined posi-tions, and introduce local interactions between particles. One can define opera-tors that correspond to creation or annihilation of a particle at a fixed position,doing so by taking a Fourier transform of the annihilation and creation operatorsfor momentum eigenstates:

363

Definition (Quantum field operators). The quantum field operators for the freeparticle system are

ψ(x) =1√2π

∫ ∞−∞

eipxa(p)dx (35.1)

and its adjoint

ψ†(x) =1√2π

∫ ∞−∞

e−ipxa†(p)dx

Note that, just like a(p) and a†(p), these are not self-adjoint operators, andthus not themselves observables, but physical observables can be constructed bytaking simple (typically quadratic) combinations of them. As noted earlier, formany-particle systems the state space H is complicated to describe and workwith, it is the operators that behave simply, and these will generally be builtout of the field operators.

In the continuum normalization the annihilation and creation operators sat-isfy the distributional equation

[a(p), a†(p′)] = δ(p− p′)

and one can compute the commutators

[ψ(x), ψ(x′)] = [ψ†(x), ψ†(x′)] = 0

[ψ(x), ψ†(x′)] =1

2π

∫ ∞−∞

∫ ∞−∞

eipxe−ip′x′ [a(p), a†(p′)]dpdp′

=1

2π

∫ ∞−∞

∫ ∞−∞

eipxe−ip′x′δ(p− p′)dpdp′

=1

2π

∫ ∞−∞

eip(x−x′)dp

=δ(x− x′)

To get some idea of the behavior of quantum field operators, one can calcu-late what they do to the vacuum state. One has

ψ(x)|0〉 = 0

ψ†(x)|0〉 =1√2π

∫ ∞−∞

e−ipx| · · · , 0, np = 1, 0, · · · 〉dp

With cutoffs in place this state would be a well-defined sum over discrete valuesof p, and an eigenstate of the number operator N with total particle numberof one. For the subspace of H with fixed particle number N , instead of usingthe occupation number description, one can use the symmetric tensor product

364

description (see 34.1). The subspace with N = 1 is just H1 itself, so given by awavefunction (of a coordinate we’ll call x′), with

| · · · , 0, np = 1, 0, · · · 〉 ↔ 1√2πeipx

′

In this description of the one-particle states we have

ψ†(x)|0〉 =1√2π

∫ ∞−∞

e−ipx1√2πeipx

′dp = δ(x− x′)

and we see that the field operator ψ†(x) is thus the operator which when appliedto the vacuum creates a single particle localized at x′ = x.

Rewrite the next two paragraphs, better explanation of the relation to con-ventional multi-particle wave-functions

The field operators allow one to recover conventional wavefunctions, for sin-gle and multiple-particle states. One sees by orthonormality of the occupationnumber basis states that

〈· · · , 0, np = 1, 0, · · · |ψ†(x)|0〉 =1√2πe−ipx = ψp(x)

the complex conjugate wavefunction of the single-particle state of momentump. An arbitrary one particle state |Ψ1〉 with wavefunction ψ(x) is a linearcombination of such states, and taking complex conjugates one finds

〈0|ψ(x)|Ψ1〉 = ψ(x)

In the quantum field theory formalism, the conventional single particle wavefunction thus reappears as the matrix element of the field operator between asingle particle state and the vacuum.

Similarly, for a two-particle state of identical particles with momenta p1 andp2 one finds

〈0|ψ(x1)ψ(x2)| · · · , 0, np1 = 1, 0, · · · , 0, np2 = 1, 0, · · · 〉 = ψp1,p2(x1, x2)

where

ψp1,p2(x1, x2) =

1√2

1

2π(eip1x1eip2x2 + eip1x2eip2x1)

is the wavefunction (symmetric under interchange of x1 and x2 for bosons) forthis two particle state. For a general two-particle state |Ψ2〉 with wavefunctionψ(x1, x2) one has

〈0|ψ(x1)ψ(x2)|Ψ2〉 = ψ(x1, x2)

and one can easily generalize this to see how field operators are related towavefunctions for an arbitrary number of particles.

There are observables that one can define simply using field operators. Theseinclude:

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• The number operator N . One can define a number density operator

n(x) = ψ†(x)ψ(x)

and integrate it to get an operator with eigenvalues the total number ofparticles in a state

N =

∫ ∞−∞

n(x)dx

=

∫ ∞−∞

∫ ∞−∞

∫ ∞−∞

1√2πe−ip

′xa†(p′)1√2πeipxa(p)dpdp′dx

=

∫ ∞−∞

∫ ∞−∞

δ(p− p′)a†(p′)a(p)dpdp′

=

∫ ∞−∞

a†(p)a(p)dp

• The total momentum operator P . This can be defined in terms of fieldoperators as

P =

∫ ∞−∞

ψ†(x)(−i ddx

)ψ(x)dx

=

∫ ∞−∞

∫ ∞−∞

∫ ∞−∞

1√2πe−ip

′xa†(p′)(−i)(ip) 1√2πeipxa(p)dpdp′dx

=

∫ ∞−∞

∫ ∞−∞

δ(p− p′)pa†(p′)a(p)dpdp′

=

∫ ∞−∞

pa†(p)a(p)dp

• The Hamiltonian H. This can be defined much like the momentum, justchanging

−i ddx→ − 1

2m

d2

dx2

to find

H =

∫ ∞−∞

ψ†(x)(− 1

2m

d2

dx2)ψ(x)dx =

∫ ∞−∞

p2

2ma†(p)a(p)dp

One can easily extend the above to three spatial dimensions, getting field op-erators ψ(x) and ψ†(x), defined by integrals over three-dimensional momentumspace. For instance, in the continuum normalization

ψ(x) = (1

2π)

32

∫R3

eip·xa(p)d3p

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and the Hamiltonian for the free field is

H =

∫R3

ψ†(x)(− 1

2m∇2)ψ(x)d3x

One can also very easily write down quantum field describing fermionicparticles, just by changing commutators to anticommutators for the creation-annihilation operators and using fermionic instead of bosonic oscillators. Onegets fermionic fields that satisfy anticommutation relations

[ψ(x), ψ†(x′)]+ = δ(x− x′)

and states that in the occupation number representation have np = 0, 1.

35.2 Field quantization

One could equally well motivate the definition of quantum field operators bystarting with the single-particle space H1 and quantizing, taking as a basis ofcoordinates on H1 not the Fourier components α(p), but the elements of H∗1given by

ψ(x) : ψ ∈ H1 → ψ(x, 0)

One also has complex antilinear maps

ψ(x) : ψ ∈ H1 → ψ(x, 0)

which one can think of as a basis of H∗1, meaning H∗1 with a complex conjugatedaction of i. There will be an ambiguity of notation here, with ψ(x) sometimesmeaning the complex number ψ(x) for a function ψ, but more often the linearfunction on the space of solutions given by evaluating the solution ψ at (x, 0).

This basis for H1 is related by Fourier transformation to the basis usingmomentum eigenstates, and all formulas for ψ(x) and ψ(x) given by Fouriertransforming the α(p) and a(p). In particular, one has Poisson bracket relations

ψ(x), ψ(x′) = −iδ(x− x′)

As with annihilation and creation operators, these distributional relations indi-cate that one is really dealing with elements of H∗1 and H∗1 of the form

ψ(f) =

∫ ∞−∞

f(x)ψ(x)dx, ψ(g) =

∫ ∞−∞

g(x)ψ(x)dx

for f, g sufficiently well-behaved functions of x.A pair (f, g) of functions of x gives an element of H∗1 ⊕ H∗1 which we can

write

ψ(f) + ψ(g) =

∫ ∞−∞

(f(x)α(x) + g(x)α(x))dx

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On these the Poisson bracket relations are

ψ(f1) + ψ(g1), ψ(f2) + ψ(g2) = −i∫ ∞−∞

(f1(x)g2(x)− f2(x)g1(x))dx

which is the symplectic form

Ω((f1, g1), (f2, g2)) = −i∫ ∞−∞

(f1(x)g2(x)− f2(x)g1(x))dx

The operators ψ(x), ψ†(x) are operator-valued distributions, which for func-tions f give operators

ψ(f) =

∫ ∞−∞

f(x)ψ(x)dx, a†(f) =

∫ ∞−∞

f(x)ψ†(x)dx

These satisfy the commutation relations

[ψ(f1) + ψ†(g1), ψ(f2) + ψ†(g2)] =

∫ ∞−∞

(f1(x)g2(x)− f2(x)g1(x))dx

which could also be written

[ψ(f), ψ(g)] = [ψ†(f), ψ†(g)] = 0, [ψ(f), ψ†(g)] =

∫ ∞−∞

f(x)g(x)dx

35.3 Dynamics of the free quantum field

One usually describes the dynamics of the theory in the Heisenberg picture,and one can treat the evolution of the field operators just as the annihilationand creation operators were treated in section 35.1. The calculations are morecomplicated here since, unlike in momentum space, the Hamiltonian operatoris not a diagonal operator in position space.

The fields satisfy the Schrodinger equation

i∂

∂tψ(x, t) = − 1

2m

∂2

∂x2ψ(x, t)

which isd

dtψ(x, t) = ψ(x, t), h

if we take our Hamiltonian function on H1 to be

h =

∫ +∞

−∞ψ(x)

−1

2m

∂2

∂x2ψ(x)dx

368

since

∂

∂tψ(x, t) =ψ(x, t), h

=ψ(x, t),

∫ +∞

−∞ψ(x′, t)

−1

2m

∂2

∂x′2ψ(x′, t)dx′

=−1

2m

∫ +∞

−∞(ψ(x, t), ψ(x′, t) ∂

2

∂x′2ψ(x′, t)+

ψ(x′, t)ψ(x, t),∂2

∂x′2ψ(x′, t))dx′)

=−1

2m

∫ +∞

−∞(−iδ(x− x′) ∂2

∂x′2ψ(x′, t) + ψ(x′, t)

∂2

∂x′2ψ(x, t), ψ(x′, t))dx′)

=i

2m

∂2

∂x2ψ(x, t)

Here we have used the derivation property of the Poisson bracket and the lin-

earity of the operator ∂2

∂x′2.

Note that there are other forms of the same Hamiltonian function, relatedto the one we chose by integration by parts. One has

ψ(x)d2

dx2ψ(x) =

d

dx(ψ(x)

d

dxψ(x))− | d

dxψ(x)|2

=d

dx(ψ(x)

d

dxψ(x)− (

d

dxψ(x))ψ(x)) + (

d2

dx2ψ(x))ψ(x)

so neglecting integrals of derivatives (assuming boundary terms go to zero atinfinity), one could have used

h =1

2m

∫ +∞

−∞| ddxψ(x)|2dx or h = − 1

2m

∫ +∞

−∞(d2

dx2ψ(x))ψ(x)dx

In position space the expression for the Hamiltonian operator (again normal-ordered) will be:

H =

∫ +∞

−∞ψ†(x)

−1

2m

d2

dx2ψ(x)dx

Using this quantized form, essentially the same calculation as before (now withoperators and commutators instead of functions and Poisson brackets) showsthat the quantum field dynamical equation

d

dtψ(x, t) = −i[ψ(x, t), H]

becomes∂

∂tψ(x, t) =

i

2m

∂2

∂x2ψ(x, t)

The field operator ψ(x, t) satisfies the Schrodinger equation which now ap-pears as a differential equation for operators rather than for wavefunctions.

369

One can explicitly solve such a differential equation just as for wavefunctions,by Fourier transforming and turning differentiation into multiplication. Theoperator ψ(x, t) is related to the operator a(p, t) by

ψ(x, t) =1√2π

∫ ∞−∞

eipxa(p, t)dp

so using the solution 34.2 we find

ψ(x, t) =1√2π

∫ ∞−∞

eipxe−ip2

2m ta(p)dp

where the operators a(p) ≡ a(p, 0) are the initial values.

35.4 Correlation functions and the propagator

We will not enter into the important topic of how to compute observables inquantum field theory that can be connected to experimentally important quan-tities such as scattering cross-sections. A crucial role in such calculations isplayed by the following observables:

Definition (Green’s function or propagator). The Green’s function or propa-gator for a quantum field theory is the amplitude, for t > t′

G(x, t, x′, t′) = 〈0|ψ(x, t)ψ†(x′, t′)|0〉

The physical interpretation of these functions is that they describe the am-plitude for a process in which a one-particle state localized at x is created at timet′, propagates for a time t − t′, and then its wavefunction is compared to thatof a one-particle state localized at x. Using the solution for the time-dependentfield operator given earlier we find

G(x, t, x′, t′) =1

2π

∫ +∞

−∞

∫ +∞

−∞〈0|eipxe−i

p2

2m ta(p)e−ip′xei

p′22m t

′a†(p′)|0〉dpdp′

=

∫ +∞

−∞

∫ +∞

−∞eipxe−i

p2

2m te−ip′x′ei

p′22m t

′δ(p− p′)dpdp′

=

∫ +∞

−∞eip(x−x

′)e−ip2

2m (t−t′)dp

One can evaluate this integral, finding

G(x′, t′, x, t) = (−im

2π(t′ − t))

12 e

−im2π(t′−t) (x′−x)2

and thatlimt→t′

G(x′, t′, x, t) = δ(x′ − x)

To be added: do in momentum space first. Comparison to the path integralpropagator, and to the correlation function calculations of previous chapter.

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35.5 Interacting quantum fields

To describe an arbitrary number of particles moving in an external potentialV (x), one can take the Hamiltonian to be

H =

∫ ∞−∞

ψ†(x)(− 1

2m

d2

dx2+ V (x))ψ(x)dx

If one can solve the one-particle Schrodinger equation for a complete set oforthonormal wavefunctions ψn(x), one can describe this quantum system usingsimilar techniques to those for the free particle, taking as basis for H1 the ψn(x)instead of plane waves of momentum p. A creation-annihilation operator pairan, a

†n is associated to each eigenfunction, and quantum fields are defined by

ψ(x) =∑n

ψn(x)an, ψ†(x) =∑n

ψn(x)a†n

For Hamiltonians just quadratic in the quantum fields, quantum field the-ories are quite tractable objects. They are in some sense just free quantumoscillator systems, with all of their symmetry structure intact, but taking thenumber of degrees of freedom to infinity. Higher order terms though makequantum field theory a difficult and complicated subject, one that requires ayear-long graduate level course to master basic computational techniques, andone that to this day resists mathematician’s attempts to prove that many ex-amples of such theories have even the basic expected properties. In the theoryof charged particles interacting with an electromagnetic field, when the electro-magnetic field is treated classically one still has a Hamiltonian quadratic in thefield operators for the particles. But if the electromagnetic field is treated as aquantum system, it acquires its own field operators, and the Hamiltonian is nolonger quadratic in the fields, a vastly more complicated situation described asan“interacting quantum field theory”.

Even if one restricts attention to the quantum fields describing one kind ofparticles, there may be interactions between particles that add terms to theHamiltonian, and these will be higher order than quadratic. For instance, ifthere is an interaction between such particles described by an interaction energyv(y − x), this can be described by adding the following quartic term to theHamiltonian

1

2

∫ ∞−∞

∫ ∞−∞

ψ†(x)ψ†(y)v(y − x)ψ(y)ψ(x)dxdy

The study of “many-body” quantum systems with interactions of this kind is amajor topic in condensed matter physics.

To be added: Work out the expressions in momentum space for the Hamilto-nian with a potential, non-diagonal. Give expression for the quartic interactionin momentum space, with the delta-function potential.

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35.6 The Lagrangian density and the path inte-gral

While we have worked purely in the Hamiltonian formalism, one could insteadstart with an action for this system. An action that will give the Schrodingerequation as an Euler-Lagrange equation is

S =

∫ ∞−∞

∫ ∞−∞

(iψ∂

∂tψ − h)dxdt

=

∫ ∞−∞

∫ ∞−∞

(iψ∂

∂tψ + ψ

1

2m

∂2

∂x2ψ)dxdt

=

∫ ∞−∞

∫ ∞−∞

(iψ∂

∂tψ − 1

2m| ∂∂xψ|2)dxdt

where the last form comes by using integration by parts to get an alternate formof h as mentioned earlier. In the Lagrangian approach to field theory, the actionis an integral over space and time of a Lagrangian density, which in this case is

L(x, t) = iψ∂

∂tψ − 1

2m| ∂∂xψ|2

If one tries to define a canonical momentum for ψ as ∂L∂ψ

one gets iψ. This

justifies the Poisson bracket relation

ψ(x), iψ(x′) = δ(x− x′)

but, as expected for a case where the equation of motion is first-order in time,such a canonical momentum is not independent of ψ. The space H1 of wave-functions ψ is already a phase space. One could try and quantize this systemby path integral methods, for instance computing the propagator by doing theintegral ∫

Dψ(x, t)ei~S[ψ]

over paths in H1 parametrized by t = 0 to t = T . This is a highly infinite-dimensional integral, over paths in an infinite-dimensional space. In additon,one needs to keep in mind the warnings given earlier about path integrals overpaths in a phase space, since that is what one has here.


As with the last chapter, the material here is discussed in essentially any quan-tum field theory textbook, with two that explicitly discuss the non-relativisticcase [24] and [34]. For a serious mathematical treatment of quantum fields asdistribution-valued operators, a standard reference is [68].

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Chapter 36

Symmetries andNon-relativistic QuantumFields

In our study of the harmonic oscillator (chapters 22 and 23) we found thatthe symmetries of the system could be studied using quadratic functions on thephase space. Classically these gave a Lie algebra under the Poisson bracket, andquantization by the Bargmann-Fock method provided a unitary representationΓ′ of the Lie algebra, with quadratic functions becoming quadratic operators.Since we treat quantum fields as infinite dimensional harmonic oscillators, theirsymmetries can be studied in the same way, with the phase space now the singleparticle Hilbert space H1. Certain specific quadratic expressions in the fieldswill provide a Lie algebra under the Poisson bracket, with quantization thengiving a unitary representation of the Lie algebra in terms of quadratic fieldoperators.

In chapter 35 we saw how this works for time translation symmetry, whichdetermines the dynamics of the theory. For the case of a free particle, thefield theory Hamiltonian is a quadratic function of the fields, providing a basicexample of how such functions generate a unitary representation on the statesof the quantum theory by use of a quadratic combination of the quantum fieldoperators. In this chapter we will see how other group actions on the spaceH1 also lead to quadratic operators and unitary transformations on the fullquantum field theory. We would like to find a formula to these, somethingthat will be simplest to do in the case that the group acts on phase space asunitary transformations, preserving the complex structure used in Bargmann-Fock quantization.

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36.1 Internal symmetries

Since the phase space H1 is a space of complex functions, there is a group thatacts unitarily on this space: the group U(1) of phase transformations of thecomplex values of the function. Such a group action that acts trivially on thespatial coordinates but non-trivially on the values of ψ(x) is called an “internalsymmetry”. If the fields ψ have multiple components, taking values in Cm,there will be a unitary action of the larger group U(m).

36.1.1 U(1) symmetry

In chapter 2 we saw that the fact that irreducible representations of U(1) arelabeled by integers is what is responsible for the term “quantization”: sincequantum states are representations of this group, they break up into statescharacterized by integers, with these integers counting the number of “quanta”.In the non-relativistic quantum field theory, this integer will just be the totalparticle number. Such a theory can be thought of as an harmonic oscillatorwith an infinite number of degrees of freedom, and the total particle number isjust the total occupation number, summed over all degrees of freedom.

The U(1) action on the fields ψ(x) which provide coordinates on H1 is givenby

ψ(x)→ e−iθψ(x), ψ(x)→ eiθψ(x) (36.1)

(recall that the fields are in the dual space to H1, so the action is the inverse tothe action of U(1) on H1 itself by multiplication by eiθ).

To understand the infinitesimal generator of this symmetry, first recall thesimple case of a harmonic oscillator in one variable, identifying the phase spaceR2 with C so the coordinates are z, z, with a U(1) action

z → e−iθz, z → eiθz

The Poisson bracket isz, z = −i

which implieszz, z = iz, zz, z = −iz

Quantizing takes z → −ia, z → −ia† and we saw in chapter 22 that we havetwo choices for the unitary operator that will be the quantization of zz:

•zz → − i

2(a†a+ aa†)

This will have eigenvalues −i(n+ 12 ), n = 0, 1, 2 . . . .

•zz → −ia†a

This is the normal-ordered form, with eigenvalues −in.

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With either choice, we get a number operator

N =1

2(a†a+ aa†), or N =

1

2:(a†a+ aa†): = a†a

In both cases we have[N, a] = −a, [N, a†] = a†

soeiθNae−iθN = e−iθa, eiθNa†e−iθN = eiθa†

Either choice of N will give the same action on operators. Hoever, on statesonly the normal-ordered one will have the desirable feature that

N |0〉 = 0, eiNθ|0〉 = |0〉

Since we now want to treat fields, adding together an infinite number of suchoscillator degrees of freedom, we will need the normal-ordered version in orderto not get ∞ · 1

2 as the number eigenvalue for the vacuum state.In momentum space, we simply do the above for each value of p and sum,

getting

N =

∫ +∞

−∞a†(p)a(p)dp (36.2)

where one needs to keep in mind that this is really an operator valued distribu-tion, which must be integrated against some weighting function on momentumspace to get a well-defined operator. What really makes sense is

N(f) =

∫ +∞

−∞a†(p)a(p)f(p)dp

for a suitable class of functions f .Instead of working with a(p), the quantization of the Fourier transform of

ψ(x), one could work with ψ(x) itself, and write

N =

∫ +∞

−∞ψ†(x)ψ(x)dx (36.3)

with the Fourier transform relating equations 36.2 and 36.3 for N . ψ†(x)ψ(x)is also an operator valued distribution, with the interpretation of measuring thenumber density at x.

On field operators, N satisfies

[N , ψ] = −ψ, [N , ψ†] = ψ†

so ψ acts on states by reducing the eigenvalue of N by one, while ψ† acts onstates by increasing the eigenvalue of N by one. Exponentiating, one has

eiθN ψe−iθN = e−iθψ, eiθN ψ†e−iθN = eiθψ†

375

which are the quantized versions of the U(1) action on the phase space coordi-nates (see equations 36.1) that we began our discussion with.

An important property of N that can be straightforwardly checked is that

[N , H] = [N ,

∫ +∞

−∞ψ†(x)

−1

2m

∂2

∂x2ψ(x)dx] = 0

This implies that particle number is a conserved quantity: if we start out witha state with a definite particle number, this will remain constant. Note thatthe origin of this conservation law comes from the fact that N is the quantizedgenerator of the U(1) symmetry of phase transformations on complex-valuedfields ψ. If we start with any Hamiltonian function h on H1 that is invariantunder the U(1) (i.e. built out of terms with an equal number of ψs and ψs),

then for such a theory N will commute with H and particle number will beconserved.

36.1.2 U(m) symmetry

By taking fields with values in Cm, or, equivalently, m different species ofcomplex-valued field ψj , j = 1, 2, . . . ,m, one can easily construct quantum fieldtheories with larger internal symmetry groups than U(1). Taking as Hamilto-nian function

h =

∫ +∞

−∞

m∑j=1

ψj(x)−1

2m

∂2

∂x2ψj(x)dx

one can see that this will be invariant not just under U(1) phase transformations,but also under transformations

ψ1

ψ2

...ψm

→ U

ψ1

ψ2

...ψm

where U is an m by m unitary matrix. The Poisson brackets will be

ψj(x), ψk(x′) = −iδ(x− x′)δjk

and are also invariant under such transformations by U ∈ U(m).As in the U(1) case, one can begin by considering the case of one particular

value of p or of x, for which the phase space is Cm, with coordinates zj , zj . As wesaw in section 23.1, the m2 quadratic combinations zjzk for j = 1, . . . ,m, k =1, . . . ,m will generalize the role played by zz in the m = 1 case, with theirPoisson bracket relations exactly the Lie bracket relations of the Lie algebrau(m) (or, considering all complex linear combinations, gl(m,C)).

After quantization, these quadratic combinations become quadratic combi-nations in annihilation and creation operators aj , a

†j satisfying

[aj , a†k] = δjk

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Recall (theorem 23.2) that for m by m matrices X and Y one will have

[

m∑j,k=1

a†jXjkak,

m∑j,k=1

a†jYjkak] =

m∑j,k=1

a†j [X,Y ]jkak

So, for each X in the Lie algebra gl(m,C), quantization will give us a represen-tation of gl(m,C) where X acts as the operator

m∑j,k=1

a†jXjkak

When the matrices X are chosen to be skew-adjoint (Xjk = −Xkj) this con-struction will give us a unitary representation of u(m).

As in the U(1) case, one gets an operator in the quantum field theory just bysumming over either the a(p) in momentum space, or the fields in configurationspace, finding for each X ∈ u(m) an operator

X =

∫ +∞

−∞

m∑j,k=1

ψ†j (x)Xjkψk(x)dx

that provides a representation of u(m) and U(m) on the quantum field theorystate space. This representation takes

eX ∈ U(m)→ U(eX) = eX = e∫ +∞−∞

∑mb,c=1 ψ

†j (x)Xjkψk(x)dx

When, as for the free-particle h we chose, the Hamiltonian is invariant underU(m) transformations of the fields ψj , then we will have

[X, H] = 0

Energy eigenstates of the quantum field theory will break up into irreduciblerepresentations of U(m) and can be labeled accordingly.

36.2 Spatial symmetries

We saw in chapter 17 that the action of the group E(3) on physical spaceR3 induces a unitary action on the space H1 of solutions to the free-particleSchrodinger equation. Quantization of this phase space with this group actionproduces a quantum field theory state space carrying a unitary representationof the group E(3). There are three different actions of the group E(3) that oneneeds to keep straight here. Given an element (a, R) ∈ E(3) one has:

1. An action on R3, preserving the inner product on R3

x→ Rx + a

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2. A unitary action on H1 induced by the action in 1. on R3, given by

ψ(x)→ u(a, R)ψ(x) = ψ(R−1(x− a))

on wavefunctions, or, on Fourier transforms by

ψ(p)→ u(a, R)ψ(p) = e−ia·R−1pψ(R−1p)

Recall from chapter 17 that this is not an irreducible representation ofE(3), but one can get an irreducible representation by taking the space ofsolutions ψE that are energy eigenfunctions with eigenvalue E.

3. The action u(a, R) in 2. of E(3) preserves the symplectic structure onthe classical phase space H1, and the Bargmann-Fock construction gives arepresentation of the group of such symplectic transformations, of whichE(3) is a finite-dimensional subgroup. We thus have a representation ofE(3) on the quantum field theory state spaceH, given by unitary operatorsU(a, R).

It is the last of these that we want to examine here, and as usual for quantumfield theory, we don’t want to try and explicitly construct the state space H andsee the E(3) action on that construction, but instead want to use the analog ofthe Heisenberg picture in the time-translation case, taking the group to act onoperators. For each (a, R) ∈ E(3) we want to find operators U(a, R) that willbe built out of the field operators, and act on the field operators as

ψ(x)→ U(a, R)ψ(x)U(a, R)−1 = ψ(Rx + a) (36.4)

Note that here the way the group acts on the argument of the operator-valueddistribution is inverse to the way that it acts on the argument of a solution inH1. This is because ψ(x) is an operator associated not to an element of H1, butto a distribution on this space, in particular the distribution ψ(x), here meaning“evaluation of the solution ψ at x.” The group will act with an inverse on suchlinear functions on H1, compared to its action on elements of H1. For a moregeneral distribution of the form

ψ(f) =

∫R3

f(x)ψ(x)d3x

E(3) will act on f by

f → (a, R) · f(x) = f(R−1(x− a))

and on ψ(f) by

ψ(f)→ (a, R) · ψ(f) =

∫R3

f(R−1(x− a))ψ(x)d3x

The distribution ψ(x) corresponds to taking f as the delta-function, and it willtransform as

ψ(x)→ (a, R) · ψ(x) = ψ(Rx + a)

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For spatial translations, we want to construct momentum operators −iPthat give a Lie algebra representation of the translation group. Exponentiationwill then give the unitary represenation

U(a,1) = e−ia·P

after exponentiation. Note that these are not the momentum operators P thatact on H1, but are operators in the quantum field theory that will be built outof quadratic combinations of the field operators. By equation 36.4 we want

e−ia·Pψ(x)eia·P = ψ(x + a)

Such an operator P can be constructed in terms of quadratic operators in thefields in the same way as the Hamiltonian H was in section 35.1, although thereis an opposite choice of sign for time versus space translations (−iH = ∂

∂t , and

−iP = − ∂∂x ) for reasons that appear when we combine space and time later in

special relativity. The calculation proceeds by just replacing the single-particleHamiltonian operator by the single-particle momentum operator P = −i∇. Soone has

P =

∫R3

ψ†(x)(−i∇)ψ(x)d3x

In the last chapter we saw that, in terms of annihilation and creation operators,this operator is just

P =

∫R3

p a†(p)a(p)d3p

which is just the integral over momentum space of the momentum times thenumber-density operator in momentum space.

For spatial rotations, we found in chapter 17 that these had generators theangular momentum operators

L = X×P = X× (−i∇)

acting on H1. Just as for energy and momentum, we can construct angularmomentum operators in the quantum field theory as quadratic field operatorsby

L =

∫R3

ψ†(x)(x× (−i∇))ψ(x)d3x

These will generate the action of rotations on the field operators. For instance,if R(θ) is a rotation about the x3 axis by angle θ, we will have

ψ(R(θ)x) = e−iθL3 ψ(x)eiθL3

Note that these constructions are infinite-dimensional examples of theorem23.2 which showed how to take an action of the unitary group on phase space(preserving Ω) and produce a representation of this group on the state spaceof the quantum theory. In our study of quantum field theory, we will be con-tinually exploiting this construction, for groups acting unitarily on the infinite-dimensional phase space H1 of solutions of some linear field equations.

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36.3 Fermion and spin 1/2 particle systems

Everything that was done in this chapter carries over straightforwardly to thecase of a fermionic non-relativistic quantum field theory of free particles. Fieldoperators will in this case generate an infinite-dimensional Clifford algebra andthe quantum state space will be an infinite-dimensional version of the spinorrepresentation. All the symmetries considered in this chpter also appear in thefermionic case, and have Lie algebra representations constructed using quadraticcombinations of the field operators in just the same way as in the bosonic case.In section 28.3 we saw in finite dimensions how unitary group actions on thefermionic phase space gave a unitary representation on the fermionic oscillatorspace, by the same method of annihilation and creation operators as in thebosonic case. The construction of the Lie algebra representation operators inthe fermionic case is an infinite-dimensional example of that method.

For the case of two-component wavefunctions ψ =

(ψ1

ψ2

)satisfying the Pauli

equation (see chapter 31), one has to use the double cover of E(3), with elements(a,Ω),Ω ∈ SU(2) and on these the action is

ψ(x)→ u(a,Ω)ψ(x) = Ωψ(R−1(x− a))

andψ(p)→ u(a,Ω)ψ(p) = e−ia·R

−1pΩψ(R−1p)

where R = Φ(Ω) is the SO(3) group element corresponding to Ω.The infinitesimal version of this unitary action is given by the operators

−iP and −iL (in the two-component case, instead of −iL, one needs −iJ =−i(L + S)).


The material of this chapter is often developed in conventional quantum fieldtheory texts in the context of relativistic rather than non-relativistic quantumfield theory. Symmetry generators are also more often derived via Lagrangianmethods (Noether’s theorem) rather than the Hamiltonian methods used here.For an example of a detailed discussion relatively close to this one, see [24].

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Chapter 37

Quantization ofInfinite-dimensional PhaseSpaces

Rewrite this introduction, adding reference to QFT material in now earlier chap-ters. Explain different character of this section, providing mathematical andphysical context concerning what happens in infinite dimensions.

Up until this point we have been dealing with finite-dimensional phase spacesand their quantization in terms of Weyl and Clifford algebras. We will now turnto the study of quantum systems (both bosonic and fermionic) correspondingto infinite-dimensional phase spaces. The phase spaces of interest are spacesof solutions of some partial differential equation, so these solutions are classi-cal fields. The corresponding quantum theory is thus called a “quantum fieldtheory”. In this chapter we’ll just make some general comments about the newphenomena that appear when one deals with such infinite-dimensional exam-ples, without going into any detail at all. Formulating quantum field theories ina mathematically rigorous way is a major and ongoing project in mathematicalphysics research, one far beyond the scope of this text. We will treat this subjectat a physicist’s level of rigor, while trying in places to give some hint of how onemight proceed with precise mathematical constructions when they exist.

While finite-dimensional Lie groups and their representations are rather well-understood mathematical objects, this is not at all true for infinite-dimensionalLie groups, where mathematical results are rather fragmentary. For the caseof infinite-dimensional phase spaces, bosonic or fermionic, the symplectic ororthogonal groups acting on these spaces will be infinite-dimensional. One topicof this chapter will be some of the new phenomena that arise when looks forinfinite-dimensional analogs of the role these groups and their representationsplay in quantum theory in the finite-dimensional case.

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37.1 Inequivalent irreducible representations

In our discussion of the Weyl and Clifford algebras in finite dimensions, an im-portant part of this story was the Stone-von Neumann theorem and its fermionicanalog, which say that these algebras each have only one interesting irreduciblerepresentation (the Schrodinger representation in the bosonic case, the spinorrepresentation in the fermionic case). Once we go to infinite dimensions, this isno longer true: there will be an infinite number of inequivalent irreducible rep-resentations, with no known complete classification of the possibilities. Beforeone can even begin to compute things like expectation values of observables,one needs to find an appropriate choice of representation, adding a new layer ofdifficulty to the problem that goes beyond that of just increasing the number ofdegrees of freedom.

To get some idea of how the Stone-von Neumann theorem can fail, onecan consider the Bargmann-Fock quantization of the harmonic oscillator with ddegrees of freedom, and recall that it necessarily depends upon making a choiceof an appropriate complex structure J (see chapter 22), with the conventionalchoice denoted J0. Changing from J0 to a different J corresponds to a changein the lowest-energy or vacuum state:

|0〉J0→ |0〉J

But |0〉J is still an element of the same state space H as |0〉J0, and one gets

the same H by acting with annihilation and creation operators on |0〉J0or on

|0〉J . The two constructions of the same H correspond to unitarily equivalentrepresentations of the Heisenberg group H2d+1.

For a phase space with d =∞, what can happen is that there can be choicesof J such that

J〈0|0〉J0= 0

in which case acting with annihilation and creation operators on |0〉J0and |0〉J

gives two orthogonal state spaces HJ0and HJ , providing two inequivalent rep-

resentations of the Heisenberg group. For quantum systems with an infinitenumber of degrees of freedom, one can thus have a unique algebra of operators,but many inequivalent representations of the operators. Different choices of aHamiltonian operator, or different choices of the J distinguishing annihilationand creation operators can give both a different vacuum state and a differentstate space H on which the operators act. This same phenomenon occurs bothin the bosonic and fermionic cases, for infinite dimensional Weyl or Cliffordalgebras.

To get examples of this phenomenon, one can consider the following casesof group actions on H studied earlier for d finite and take the limit as d→∞:

• In chapter 21 we saw that coherent states could be defined by acting on|0〉 by the Heisenberg group, getting, for the case of one degree of freedom

|α〉 = eαa†−αa|0〉

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Here α ∈ C and one can show

|〈α|0〉|2 = e−|α|2

For arbitrary d one gets states parametrized by a vector α ∈ Cd, and

|〈α|0〉|2 = e−∑dj=1 |αj |

2

In the infinite dimensional case, for any sequence of αj with divergent∑∞j=1 |αj |2 one will have

〈α|0〉 = 0

For each such sequence, one gets a state |α〉 that will be in an inequivalentrepresentation of the d =∞ Heisenberg Lie algebra.

Examples of this kind of phenomenon can occur in quantum field theories,in cases where it is energetically favorable for many quanta of the field to“condense” into the lowest energy state. This can then be a state like |α〉,with ε having a physical interpretation as the density of the condensate.

• In both the standard oscillator case with Sp(2d,R) acting, and the fermionicoscillator case with SO(2d,R) acting, we found that there were “Bogoli-ubov transformations”, with elements of the group not in the U(d) sub-group distinguished by the choice of J taking |0〉 to a different state. As inthe case of the Heisenberg group action above, such action by Bogoliubovtransformations can, in the limit of d → ∞, take |0〉 to an orthogonalstate, introducing the possibility of inequivalent representations of thecommutation relations. The physical interpretation again is that suchstates correspond to condensates of quanta. For the usual oscillator case,this phenomenon occurs in the theory of superfluidity, for fermionic os-cillators it occurs in the theory of superconductivity. It was in the studyof such systems that Bogoliubov discovered the transformations that nowbear his name.

If one restricts the class of complex structures J to ones not that differentfrom J0, then one can recover a version of the Stone-von Neumann theorem andhave much the same behavior as in the finite-dimensional case. Note that foreach invertible linear map g on phase space, g acts on the complex structure,taking

J0 → Jg = g · J0

One can define subgroups of the infinite-dimensional symplectic or orthogonalgroups as follows:

Definition (Restricted symplectic and orthogonal groups). The group of lineartransformations g of an infinite-dimensional symplectic vector space preservingthe symplectic structure and also satisfying the condition

tr(A†A) <∞

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on the operatorA = [Jg, J0]

is called the restricted symplectic group and denoted Spres. The group of lineartransformations g of an infinite-dimensional inner-product space preserving theinner-product and satisfying the same condition as above on [Jg, J0] is called therestricted orthogonal group and denoted SOres.

An operator A satisfying tr(A†A) < ∞ is said to be a Hilbert-Schmidtoperator.

One then has the following replacement for the Stone-von Neumann theorem:

Theorem. Given two complex structures J1, J2 on a Hilbert space such that[J1, J2] is Hilbert-Schmidt, acting on the states

|0〉J1, |0〉J2

by annihilation and creation operators will give unitarily equivalent representa-tions of the Weyl algebra (in the bosonic case), or the Clifford algebra (in thefermionic case).

The standard reference for the proof of this statement is the original papersof Shale [61] and Shale-Stinespring [62]. A detailed discussion of the theoremcan be found in [49].

For some motivation for this theorem, consider the finite-dimensional casestudied in section 23.3 (this is for the symplectic group case, a similar calculationholds in the orthgonal group case). Elements of sp(2d,R) corresponding toBogoliubov transformations (i.e. with non-zero commutator with J0) were ofthe form

1

2

∑jk

(Bjkzjzk +Bjkzjzk)

for symmetric complex matrices B. These acted on the metaplectic representa-tion by

− i

2

∑jk

(Bjka†ja†k +Bjkajak) (37.1)

and commuting two of them gave a result (equation 23.6) corresponding toquantization of an element of the u(d) subgroup, differing from its normal-ordered version by a term

−1

2tr(BC − CB)1 = −1

2tr(BC† − CB†)1

For d = ∞, this trace in general will be infinite and undefined. An alter-nate characterization of Hilbert-Schmidt operators is that for B and C Hilbert-Schmidt operators, the traces

tr(BC†) and tr(CB†)

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will be finite and well-defined. So, at least to the extent normal-ordered op-erators quadratic in annihilation and creation operators are well-defined, theHilbert-Schmidt condtion on operators not commuting with the complex struc-ture implies that they will have well-defined commutation relations with eachother.

37.2 The anomaly and the Schwinger term

The argument above gives some motivation for the existence as d goes to ∞ ofwell-defined commutators of operators of the form 37.1 and thus for the existenceof an analog of the metaplectic representation for the infinite-dimensional Liealgebra spres of Spres. There is one obvious problem though with this argument,in that while it tells us that normal-ordered operators will have well-definedcommutation relations, they are not quite the right commutation relations, dueto the occurrence of the extra scalar term

−1

2tr(BC† − CB†)1

This term is sometimes called the “Schwinger term”.The Schwinger term causes a problem with the standard expectation that

given some group G acting on the phase space preserving the Poisson bracket,one should get a unitary representation of G on the quantum state space H.This problem is sometimes called the “anomaly”, meaning that the expectedunitary Lie algebra representation does not exist (due to extra scalar termsin the commutation relations). Recall from section 13.3 that this potentialproblem was already visible at the classical level, in the fact that given L ∈ g,the corresponding moment map µL is only well-defined up to a constant. Whilefor the finite-dimensional cases we studied, the constants could be chosen so asto make the map

L→ µL

a Lie algebra homomorphism, that turns out to no longer be true for the caseg = spres (or sores) acting on an infinite-dimensional phase space. The potentialproblem of the anomaly is thus already visible classically, but it is only whenone constructs the quantum theory and thus a representation on the state spacethat one can see whether the problem cannot be removed by a constant shiftin the representation operators. This situation, despite its classical origin, issometimes characterized as a form of symmetry-breaking due to the quantizationprocedure.

Note that this problem will not occur for G that commute with the complexstructure, since for these the normal-ordered Lie algebra representation opera-tors will be a true representation of u(∞) ⊂ spres. We will call U(∞) ⊂ Spresthe subgroup of elements that commute with J0 exactly, not just up to a Hilbert-Schmidt operator. It turns out that G ⊂ U(∞) for most of the cases we areinterested in, allowing construction of the Lie algebra representation by normal-ordered quadratic combinations of the annihilation and creation operators (as in

385

23.3). Also note that since normal-ordering just shifts operators by somethingproportional to a constant, when this constant is finite there will be no anomalysince one can get operators with correct commutators by such a finite shift of thenormal-ordered ones. The anomaly is an inherently infinite-dimensional prob-lem since it is only then that infinite shifts are necessary. When the anomalydoes appear, it will appear as a phase-ambiguity in the group representationoperators (not just a sign ambiguity as in finite-dimensional case of Sp(2d,R)),and H will be a projective representation of the group.

Such an undetermined phase factor only creates a problem for the action onstates, not for the action on operators. Note that in the finite-dimensional casethe action of Sp(2d,R) on operators (see 18.4) is independent of any constantshift in the Lie algebra represenation operators. Equivalently, if one has aunitary projective representation on states, the phase ambiguity cancels out inthe action on operators, which is by conjugation.

37.3 Spontaneous symmetry breaking

Generalities about multiple vacua and spontaneous symmetry breaking, work outan example, perhaps including some of the text below about a U(1) symmetry.

In quantum field theories, due to the infinite number of degrees of freedom,the Stone-von Neumann theorem does not apply, and one can have unitarilyinequivalent representations of the algebra generated by the field operators,leading to new kinds of behavior not seen in finite dimensional quantum systems.In particular, one can have a space of states where the lowest energy state |0〉does not have the property

N |0〉 = 0, e−iθN |0〉 = |0〉

but instead gets taken by e−iθN to some other state, with

N |0〉 6= 0, e−iθN |0〉 ≡ |θ〉 6= |0〉 (for θ 6= 0)

In this case, the vacuum state is not an eigenstate of N so does not have awell-defined particle number. If [N , H] = 0, the states |θ〉 will all have the sameenergy as |0〉 and there will be a multiplicity of different vacuum states, labeledby θ. In such a case the U(1) symmetry is said to be “spontaneously broken”.This phenomenon occurs when non-relativistic quantum field theory is used todescribe a superconductor. There the lowest energy state will be a state withouta definite particle number, with electrons pairing up in a way that allows themto lower their energy, “condensing” in the lowest energy state.

When, as for the free-particle h we chose, the Hamiltonian is invariant underU(m) transformations of the fields ψj , then we will have

[X, H] = 0

In this case, if |0〉 is invariant under the U(m) symmetry, then energy eigenstatesof the quantum field theory will break up into irreducible representations of

386

U(m) and can be labeled accordingly. As in the U(1) case, the U(m) symmetrymay be spontaneously broken, with

X|0〉 6= 0

for some directions X in u(m). When this happens, just as in the U(1) casestates did not have well-defined particle number, now they will not carry well-defined irreducible U(m) representation labels.

37.4 Higher order operators and renormaliza-tion

We have generally restricted ourselves to considering only products of the fun-damental position and momentum operators of degree less than or equal to two,since it is these that have an interpretation as the operators of a Lie algebrarepresentation. By the Groenewold-van Hove theorem, higher-order products ofposition and momentum variables have no unique quantization (the operator-ordering problem). In the finite dimensional case one can of course considerhigher-order products of operators, for instance systems with Hamiltonian op-erators of higher order than quadratic. Unlike the quadratic case, typicallyno exact solution for eigenvectors and eigenvalues will exist, but various ap-proximation methods may be available. In particular, for Hamiltonians that arequadratic plus a term with a small parameter, perturbation theory methods canbe used to compute a power-series approximation in the small parameter. Thisis an important topic in physics, covered in detail in the standard textbooks.

The standard approach to quantization of infinite-dimensional systems isto begin with “regularization”, somehow modifying the system to only havea finite-dimensional phase space. One quantizes this theory, taking advantageof the uniqueness of its representation, then tries to take a limit that recoversthe infinite-dimensional system. Such a limit will generally be quite singular,leading to an infinite result, and the process of manipulating these potentialinfinities is called “renormalization”. Techniques for taking limits of this kindin a manner that leads to a consistent and physically sensible result typicallytake up a large part of standard quantum field theory textbooks. For manytheories, no appropriate such techniques are known, and conjecturally none arepossible. For others there is good evidence that such a limit can be successfullytaken, but the details of how to do this remain unknown (with for instance a$1 million Millenium Prize offered for showing rigorously this is possible in thecase of Yang-Mills gauge theory).

In succeeding chapters we will go on to study a range of quantum fieldtheories, but due to the great complexity of the issues involved, will not addresswhat happens for non-quadratic Hamiltonians. Physically this means that wewill just be able to study theories of free particles, although with methods thatgeneralize to particles moving in non-trivial background classical fields. Fora treatment of the subject that includes interacting quantized multi-particle

387

systems, one of the conventional textbooks will be needed to supplement whatis here.


Berezin’s The Method of Second Quantization [5] develops in detail the infinite-dimensional version of the Bargmann-Fock construction, both in the bosonicand fermionic cases. Infinite-dimensional versions of the metaplectic and spinorrepresentations are given there in terms of operators defined by integral kernels.For a discussion of the infinite-dimensional Weyl and Clifford algebras, togetherwith a realization of their automorphism groups Spres and Ores (and the corre-sponding Lie algebras) in terms of annihilation and creation operators acting onthe infinite-dimensional metaplectic and spinor representations, see [49]. Thebook [52] contains an extensive discussion of the groups Spres and Ores and theinfinite-dimensional version of their metaplectic and spinor representations. Itemphasizes the origin of novel infinite-dimensional phenomena in the nature ofthe complex structures used in infinite dimensional examples.

The use of Bogoliubov transformations in the theories of superfluidity andsuperconductivity is a standard topic in quantum field theory textbooks thatemphasize condensed matter applications, see for example [40]. The book [9]discusses in detail the occurrence of inequivalent representations of the commu-tation relations in various physical systems.

388

Chapter 38

Minkowski Space and theLorentz Group

For the case of non-relativistic quantum mechanics, we saw that systems withan arbitrary number of particles, bosons or fermions, could be described bytaking as the Hamiltonian phase space the state space H1 of the single-particlequantum theory (e.g. the space of complex-valued wavefunctions on R3 in thebosonic case). This phase space is infinite-dimensional, but it is linear and itcan be quantized using the same techniques that work for the finite-dimensionalharmonic oscillator. This is an example of a quantum field theory since it is aspace of functions (fields, to physicists) that is being quantized.

We would like to find some similar way to proceed for the case of rela-tivistic systems, finding relativistic quantum field theories capable of describ-ing arbitrary numbers of particles, with the energy-momentum relationshipE2 = |p|2c2 + m2c4 characteristic of special relativity, not the non-relativistic

limit |p| mc where E = |p|22m . In general, a phase space can be thought of as

the space of initial conditions for an equation of motion, or equivalently, as thespace of solutions of the equation of motion. In the non-relativistic field theory,the equation of motion is the first-order in time Schrodinger equation, and thephase space is the space of fields (wavefunctions) at a specified initial time, sayt = 0. This space carries a representation of the time-translation group R, thespace-translation group R3 and the rotation group SO(3). To construct a rela-tivistic quantum field theory, we want to find an analog of this space. It will besome sort of linear space of functions satisfying an equation of motion, and wewill then quantize by applying harmonic oscillator methods.

Just as in the non-relativistic case, the space of solutions to the equationof motion provides a representation of the group of space-time symmetries ofthe theory. This group will now be the Poincare group, a ten-dimensionalgroup which includes a four-dimensional subgroup of translations in space-time,and a six-dimensional subgroup (the Lorentz group), which combines spatialrotations and “boosts” (transformations mixing spatial and time coordinates).

389

The representation of the Poincare group on the solutions to the relativisticwave equation will in general be reducible. Irreducible such representations willbe the objects corresponding to elementary particles. Our first goal will be tounderstand the Lorentz group, in later sections we will find representations ofthis group, then move on to the Poincare group and its representations.

38.1 Minkowski space

Special relativity is based on the principle that one should consider space andtime together, and take them to be a four-dimensional space R4 with an indef-inite inner product:

Definition (Minkowski space). Minkowski space M4 is the vector space R4

with an indefinite inner product given by

(x, y) ≡ x · y = −x0y0 + x1y1 + x2y2 + x3y3

where (x0, x1, x2, x3) are the coordinates of x ∈ R4, (y0, y1, y2, y3) the coordi-nates of y ∈ R4.

Digression. We have chosen to use the −+ ++ instead of the more common+−−− sign convention for the following reasons:

• Analytically continuing the time variable x0 to ix0 gives a positive definiteinner product.

• Restricting to spatial components, there is no change from our previousformulas for the symmetries of Euclidean space E(3).

• Only for this choice will we have a purely real spinor representation (sinceCliff(3, 1) = M(22,R) 6= Cliff(1, 3)).

• Weinberg’s quantum field theory textbook [78] uses this convention (al-though, unlike him, we’ll put the 0 component first).

This inner product will also sometimes be written using the matrix

ηµν =

−1 0 0 00 1 0 00 0 1 00 0 0 1

as

x · y =

3∑µ,ν=0

ηµνxµxν

390

Digression (Upper and lower indices). In many physics texts it is conventionalin discussions of special relativity to write formulas using both upper and lowerindices, related by

xµ =

3∑ν=0

ηµνxν = ηµνx

ν

with the last form of this using the Einstein summation convention.

One motivation for introducing both upper and lower indices is that specialrelativity is a limiting case of general relativity, which is a fully geometricaltheory based on taking space-time to be a manifold M with a metric g thatvaries from point to point. In such a theory it is important to distinguish betweenelements of the tangent space Tx(M) at a point x ∈M and elements of its dual,the co-tangent space T ∗x (M), while using the fact that the metric g provides aninner product on Tx(M) and thus an isomorphism Tx(M) ' T ∗x (M). In thespecial relativity case, this distinction between Tx(M) and T ∗x (M) just comesdown to an issue of signs, but the upper and lower index notation is useful forkeeping track of those.

A second motivation is that position and momenta naturally live in dualvector spaces, so one would like to distinguish between the vector space M4 ofpositions and the dual vector space of momenta. In the case though of a vectorspace like M4 which comes with a fixed inner product ηµν , this inner productgives a fixed identification of M4 and its dual, an identification that is also anidentification as representations of the Lorentz group. For simplicity, we willnot here try and distinguish by notation whether a vector is in M4 or its dual,so will just use lower indices, not both upper and lower indices.

The coordinates x1, x2, x3 are interpreted as spatial coordinates, and thecoordinate x0 is a time coordinate, related to the conventional time coordinatet with respect to chosen units of time and distance by x0 = ct where c is thespeed of light. Mostly we will assume units of time and distance have beenchosen so that c = 1.

Vectors v ∈M4 such that |v|2 = v · v > 0 are called “spacelike”, those with|v|2 < 0 “time-like” and those with |v|2 = 0 are said to lie on the “light-cone”.Suppressing one space dimension, the picture to keep in mind of Minkowskispace looks like this:

391

We can take Fourier transforms with respect to the four space-time variables,which will take functions of x0, x1, x2, x3 to functions of the Fourier transformvariables p0, p1, p2, p3. The definition we will use for this Fourier transform willbe

f(p) =1

(2π)2

∫M4

e−ip·xf(x)d4x

=1

(2π)2

∫M4

e−i(−p0x0+p1x1+p2x2+p3x3)f(x)dx0d3x

and the Fourier inversion formula is

f(x) =1

(2π)2

∫M4

eip·xf(p)d4p

Note that our definition puts one factor of 1√2π

with each Fourier (or inverse

Fourier) transform with respect to a single variable. A common alternate con-vention among physicists is to put all factors of 2π with the p integrals (and

thus in the inverse Fourier transform), none in the definition of f(p), the Fouriertransform itself.

The reason why one conventionally defines the Hamiltonian operator as i ∂∂tbut the momentum operator with components −i ∂

∂xjis due to the sign change

between the time and space variables that occurs in this Fourier transform inthe exponent of the exponential.

Discuss the sign conventions and the Fourier transform in more detail here.

392

38.2 The Lorentz group and its Lie algebra

Recall that in 3 dimensions the group of linear transformations of R3 pre-serving the standard inner product was the group O(3) of 3 by 3 orthogonalmatrices. This group has two disconnected components: SO(3), the subgroupof orientation preserving (determinant +1) transformations, and a componentof orientation reversing (determinant −1) transformations. In Minkowksi space,one has

Definition (Lorentz group). The Lorentz group O(3, 1) is the group of lineartransformations preserving the Minkowski space inner product on R4.

In terms of matrices, the condition for a 4 by 4 matrix Λ to be in O(3, 1)will be

ΛT

−1 0 0 00 1 0 00 0 1 00 0 0 1

Λ =

−1 0 0 00 1 0 00 0 1 00 0 0 1

The Lorentz group has four components, with the component of the iden-

tity a subgroup called SO(3, 1) (which some call SO+(3, 1)). The other threecomponents arise by multiplication of elements in SO(3, 1) by P, T, PT where

P =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

is called the “parity” transformation, reversing the orientation of the spatialvariables, and

T =

−1 0 0 00 1 0 00 0 1 00 0 0 1

reverses the time orientation.

The Lorentz group has a subgroup SO(3) of transformations that just acton the spatial components, given by matrices of the form

Λ =

1 0 0 000 R0

where R is in SO(3). For each pair j, k of spatial directions one has the usualSO(2) subgroup of rotations in the j−k plane, but now in addition for each pair0, j of the time direction with a spatial direction, one has SO(1, 1) subgroups

393

of matrices of transformations called “boosts” in the j direction. For example,for j = 1, one has the subgroup of SO(3, 1) of matrices of the form

Λ =

coshφ sinhφ 0 0sinhφ coshφ 0 0

0 0 1 00 0 0 1

for φ ∈ R.

The Lorentz group is six-dimensional. For a basis of its Lie algebra one cantake six matrices Mµν for µ, ν ∈ 0, 1, 2, 3 and j < k. For the spatial indices,these are

M12 =

0 0 0 00 0 −1 00 1 0 00 0 0 0

, M13 =

0 0 0 00 0 0 10 0 0 00 −1 0 0

, M23 =

0 0 0 00 0 0 00 0 0 −10 0 1 0

which correspond to the basis elements of the Lie algebra of SO(3) that we sawin an earlier chapter. One can rename these using the same names as earlier

l1 = M23, l2 = M13, l3 = M12

and recall that these satisfy the so(3) commutation relations

[l1, l2] = l3, [l2, l3] = l1, [l3, l1] = l2

and correspond to infinitesimal rotations about the three spatial axes.Taking the first index 0, one gets three elements corresponding to infinitesi-

mal boosts in the three spatial directions

M01 =

0 1 0 01 0 0 00 0 0 00 0 0 0

, M02 =

0 0 1 00 0 0 01 0 0 00 0 0 0

, M03 =

0 0 0 10 0 0 00 0 0 01 0 0 0

These can be renamed as

k1 = M01, k2 = M02, k3 = M03

One can easily calculate the commutation relations between the kj and lj , whichshow that the kj transform as a vector under infinitesimal rotations. For in-stance, for infinitesimal rotations about the x1 axis, one finds

[l1, k1] = 0, [l1, k2] = k3, [l1, k3] = −k2

Commuting infinitesimal boosts, one gets infinitesimal spatial rotations

[k1, k2] = −l3, [k3, k1] = −l2, [k2, k3] = −l1

394

Digression. A more conventional notation in physics is to use Jj = ilj forinfinitesimal rotations, and Kj = ikj for infinitesimal boosts. The intention ofthe different notation used here is to start with basis elements of the real Liealgebra so(3, 1), (the lj and kj) which are purely real objects, before complexifyingand considering representations of the Lie algebra.

Taking the following complex linear combinations of the lj and kj

Aj =1

2(lj + ikj), Bj =

1

2(lj − ikj)

one finds[A1, A2] = A3, [A3, A1] = A2, [A2, A3] = A1

and[B1, B2] = B3, [B3, B1] = B2, [B2, B3] = B1

This construction of the Aj , Bj requires that we complexify (allow complexlinear combinations of basis elements) the Lie algebra so(3, 1) of SO(3, 1) andwork with the complex Lie algebra so(3, 1) ⊗ C. It shows that this Lie alge-bra splits into a product of two sub-Lie algebras, which are each copies of the(complexified) Lie algebra of SO(3), so(3)⊗C. Since

so(3)⊗C = su(2)⊗C = sl(2,C)

we haveso(3, 1)⊗C = sl(2,C)× sl(2,C)

In the next section we’ll see the origin of this phenomenon at the group level.

38.3 Spin and the Lorentz group

Just as the groups SO(n) have double covers Spin(n), the group SO(3, 1) has adouble cover, which we will show can be identified with the group SL(2,C) of2 by 2 complex matrices with unit determinant. This group will have the sameLie algebra as the SO(3, 1), and we will sometimes refer to either group as the“Lorentz group”.

Recall that for SO(3) the spin double cover Spin(3) can be identified witheither Sp(1) (the unit quaternions) or SU(2), and then the action of Spin(3)as SO(3) rotations of R3 was given by conjugation of imaginary quaternions orcertain 2 by 2 complex matrices respectively. In the SU(2) case this was doneexplicitly by identifying

(x1, x2, x3)↔(

x3 x1 − ix2

x1 + ix2 −x3

)and then showing that conjugating this matrix by an element of SU(2) was alinear map leaving invariant

det

(x3 x1 − ix2

x1 + ix2 −x3

)= −(x2

1 + x22 + x2

3)

395

and thus a rotation in SO(3).The same sort of thing works for the Lorentz group case. Now we identify

R4 with the space of 2 by 2 complex self-adjoint matrices by

(x0, x1, x2, x3)↔(x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)and observe that

det

(x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)= x2

0 − x21 − x2

2 − x23

This provides a very useful way to think of Minkowski space: as complex self-adjoint 2 by 2 matrices, with norm-squared minus the determinant of the matrix.

The linear transformation(x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)→ Ω

(x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)Ω†

for Ω ∈ SL(2,C) preserves the determinant and thus the inner-product, since

det(Ω

(x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)Ω†) =(det Ω) det

(x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)(det Ω†)

=x20 − x2

1 − x22 − x2

3

It also takes self-adjoint matrices to self-adjoints, and thus R4 to R4, since

(Ω

(x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)Ω†)† =(Ω†)†

(x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)†Ω†

=Ω

(x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)Ω†

Note that both Ω and −Ω give the same linear transformation when they actby conjugation like this. One can show that all elements of SO(3, 1) arise assuch conjugation maps, by finding appropriate Ω that give rotations or boostsin the µ− ν planes, since these generate the group.

Recall that the double covering map

Φ : SU(2)→ SO(3)

was given for Ω ∈ SU(2) by taking Φ(Ω) to be the linear transformation inSO(3) (

x3 x1 − ix2

x1 + ix2 −x3

)→ Ω

(x3 x1 − ix2

x1 + ix2 −x3

)Ω−1

We have found an extension of this map to a double covering map from SL(2,C)to SO(3, 1). This restricts to Φ on the subgroup SU(2) of SL(2,C) matricessatisfying Ω† = Ω−1.

396

Digression (The complex group Spin(4,C) and its real forms). Recall thatwe found that Spin(4) = Sp(1) × Sp(1), with the corresponding SO(4) trans-formation given by identifying R4 with the quaternions H and taking not justconjugations by unit quaternions, but both left and right multiplication by dis-tinct unit quaternions. Rewriting this in terms of complex matrices instead ofquaternions, we have Spin(4) = SU(2) × SU(2), and a pair Ω1,Ω2 of SU(2)matrices acts as an SO(4) rotation by(

x0 − ix3 −x2 − ix1

x2 − ix1 x0 + ix3

)→ Ω1

(x0 − ix3 −x2 − ix1

x2 − ix1 x0 + ix3

)Ω2

preserving the determinant x20 + x2

1 + x22 + x2

3.For another example, consider the identification of R4 with 2 by 2 real ma-

trices given by

(x0, x1, x2, x3)↔(x0 + x3 x2 + x1

x2 − x1 x0 − x3

)Given a pair of matrices Ω1,Ω2 in SL(2,R), the linear transformation(

x0 + x3 x2 + x1

x2 − x1 x0 − x3

)→ Ω1

(x0 + x3 x2 + x1

x2 − x1 x0 − x3

)Ω2

preserves the reality condition on the matrix, and preserves

det

(x0 + x3 x2 + x1

x2 − x1 x0 − x3

)= x2

0 + x21 − x2

2 − x23

so gives an element of SO(2, 2) and we see that Spin(2, 2) = SL(2,R) ×SL(2,R).

The three different examples

Spin(4) = SU(2)× SU(2), Spin(3, 1) = SL(2,C)

andSpin(2, 2) = SL(2,R)× SL(2,R)

that we have seen are all so-called “real forms” of a fact about complex groupsthat one can get by complexifying any of the examples, i.e. considering elements(x0, x1, x2, x3) ∈ C4, not just in R4. For instance, in the Spin(4) case, takingthe x0, x1, x2, x3 in the matrix(

x0 − ix3 −x2 − ix1

x2 − ix1 x0 + ix3

)to have arbitrary complex values z0, z1, z2, z3 one gets arbitrary 2 by 2 complexmatrices, and the transformation(

z0 − iz3 −z2 − iz1

z2 − iz1 z0 + iz3

)→ Ω1

(z0 − iz3 −z2 − iz1

z2 − iz1 z0 + iz3

)Ω2

397

preserves this space as well as the determinant (z20 +z2

1 +z22 +z2

3) for Ω1 and Ω2

not just in SU(2), but in the larger group SL(2,C). So we find that the groupSO(4,C) of complex orthogonal transformations of C4 has spin double cover

Spin(4,C) = SL(2,C)× SL(2,C)

Since spin(4,C) = so(3, 1)⊗C, this relation between complex Lie groups corre-sponds to the Lie algebra relation

so(3, 1)⊗C = sl(2,C)× sl(2,C)

we found explicitly earlier when we showed that by taking complex coefficientsof generators lj and kj of so(3, 1) we could find generators Aj and Bj of twodifferent sl(2,C) sub-algebras.


Those not familiar with special relativity should consult a textbook on thesubject for the physics background necessary to appreciate the significance ofMinkowski space and its Lorentz group of invariances. An example of a suitablesuch book aimed at mathematics students is Woodhouse’s Special Relativity [83].

Most quantum field theory textbooks have some sort of discussion of theLorentz group and its Lie algebra, although the issue of its complexification isoften not treated. A typical example is Peskin-Schroeder [51], see the beginningof Chapter 3. Another example is Quantum Field Theory in a Nutshell byTony Zee, see Chapter II.3 [84] (and test your understanding by interpretingproperly some of the statements included there such as “The mathematicallysophisticated say the algebra SO(3, 1) is isomorphic to SU(2)⊗ SU(2)”).

398

Chapter 39

Representations of theLorentz Group

Having seen the importance in quantum mechanics of understanding the repre-sentations of the rotation group SO(3) and its double cover Spin(3) = SU(2)one would like to also understand the representations of the Lorentz groupSO(3, 1) and its double cover Spin(3, 1) = SL(2,C). One difference from theSO(3) case is that all non-trivial finite-dimensional irreducible representationsof the Lorentz group are non-unitary (there are infinite-dimensional unitary ir-reducible representations, of no known physical significance, which we will notdiscuss). While these finite-dimensional representations themselves only pro-vide a unitary action of the subgroup Spin(3) ⊂ Spin(3, 1), they will later beused in the construction of quantum field theories whose state spaces will havea unitary action of the Lorentz group.

add more about this point later

39.1 Representations of the Lorentz group

In the SU(2) case we found irreducible unitary representations (πn, Vn) of di-

mension n+1 for n = 0, 1, 2, . . .. These could also be labeled by s = n2 , called the

“spin” of the representation, and we will do that from now on. These represen-tations can be realized explicitly as homogeneous polynomials of degree n = 2sin two complex variables z1, z2. For the case of Spin(4) = SU(2)× SU(2), theirreducible representations will just be tensor products

V s1 ⊗ V s2

of SU(2) irreducibles, with the first SU(2) acting on the first factor, the secondon the second factor. The case s1 = s2 = 0 is the trivial representation, s1 =12 , s2 = 0 is one of the half-spinor representations of Spin(4) on C2, s1 = 0, s2 =12 is the other, and s1 = s2 = 1

2 is the representation on four-dimensional(complexified) vectors.

399

Turning now to Spin(3, 1) = SL(2,C), one can use the same constructionusing homogeneous polynomials as in the SU(2) case to get irreducible repre-sentations of dimension 2s+ 1 for s = 0, 1

2 , 1, . . .. Instead of acting by SU(2) onz1, z2, one acts by SL(2,C), and then as before uses the induced action on poly-nomials of z1 and z2. This gives representations (πs, V

s) of SL(2,C). Amongthe things that are different though about these representations:

• They are not unitary (except in the case of the trivial representation). For

example, for the defining representation V12 on C2, the Hermitian inner

product

〈(ψφ

),

(ψ′

φ′

)〉 =

(ψ φ

)·(ψ′

φ′

)= ψψ′ + φφ′

is invariant under SU(2) transformations Ω since

〈Ω(ψφ

),Ω

(ψ′

φ′

)〉 =

(ψ φ

)Ω† · Ω

(ψ′

φ′

)and Ω†Ω = 1 by unitarity. This is no longer true for Ω ∈ SL(2,C).

The representation V12 of SL(2,C) does have a non-degenerate bilinear

form, which we’ll denote by ε,

ε(

(ψφ

),

(ψ′

φ′

)) =

(ψ φ

)( 0 1−1 0

)(ψ′

φ′

)= ψφ′ − φψ′

that is invariant under the SL(2,C) action on V12 and can be used to

identify the representation and its dual. This is just the complexificationof the symplectic form on R2 studied in section 14.1.1, and the samecalculation there which showed that it was SL(2,R) invariant here showsthat the complex version is SL(2,C) invariant.

• In the case of SU(2) representations, the complex conjugate representa-tion one gets by taking as representation matrices π(g) instead of π(g) isequivalent to the original representation (the same representation, with adifferent basis choice, so matrices changed by a conjugation). To see thisfor the spin- 1

2 representation, note that SU(2) matrices are of the form

Ω =

(α β

−β α

)and one has (

0 1−1 0

)(α β

−β α

)(0 1−1 0

)−1

=

(α β−β α

)so the matrix (

0 1−1 0

)400

is the change of basis matrix relating the representation and its complexconjugate.

This is no longer true for SL(2,C). Conjugation by a fixed matrix will notchange the eigenvalues of the matrix, and these can be complex (unlikeSU(2) matrices, which have real eigenvalues). So such a (matrix) conju-gation cannot change all SL(2,C) matrices to their complex conjugates,since in general (complex) conjugation will change their eigenvalues.

The classification of irreducible finite dimensional SU(2) representation wasdone earlier in this course by considering its Lie algebra su(2), complexified togive us raising and lowering operators, and this complexification is sl(2,C). Ifone examines that argument, one finds that it mostly also applies to irreduciblefinite-dimensional sl(2,C) representations. There is a difference though: nowflipping positive to negative weights (which corresponds to change of sign ofthe Lie algebra representation matrices, or conjugation of the Lie group rep-resentation matrices) no longer takes one to an equivalent representation. Itturns out that to get all irreducibles, one must take both the representationswe already know about and their complex conjugates. One can show (we won’tprove this here) that the tensor product of one of each type of irreducible isstill an irreducible, and that the complete list of finite-dimensional irreduciblerepresentations of sl(2,C) is given by:

Theorem (Classification of finite dimensional sl(2,C) representations). Theirreducible representations of sl(2,C) are labeled by (s1, s2) for sj = 0, 1

2 , 1, . . ..These representations are given by the tensor product representations

(πs1 ⊗ πs2 , V s1 ⊗ V s2)

where (πs, Vs) is the irreducible representation represenation of dimension 2s+1

and (πs, Vs) its complex conjugate. Such representations have dimension (2s1 +

1)(2s2 + 1).

All these representations are also representations of the group SL(2,C) andone has the same classification theorem for the group, although we will not tryand prove this. We will also not try and study these representations in general,but will restrict attention to the cases of most physical interest, which are

• (0, 0): The trivial representation on C, also called the “spin 0” or scalarrepresentation.

• ( 12 , 0): These are called left-handed (for reasons we will see later on) “Weyl

spinors”. We will often denote the representation space C2 in this case asSL, and write an element of it as ψL.

• (0, 12 ): These are called right-handed Weyl spinors. We will often denote

the representation space C2 in this case as SR, and write an element of itas ψR.

401

• ( 12 ,

12 ): This is called the “vector” representation since it is the complexifi-

cation of the action of SL(2,C) as SO(3, 1) transformations of space-timevectors that we saw earlier. It is a representations of SO(3, 1) as well asSL(2,C).

• ( 12 , 0) ⊕ (0, 1

2 ): This reducible 4 complex dimensional representation isknown as the representation on “Dirac spinors”.

One can manipulate these Weyl spinor representations ( 12 , 0) and (0, 1

2 ) ina similar way to the treatment of tangent vectors and their duals in tensoranalysis. Just like in that formalism, one can distinguish between a represen-tation space and its dual by upper and lower indices, in this case using notthe metric but the SL(2,C) invariant bilinear form ε to raise and lower indices.With complex conjugates and duals, there are four kinds of irreducible SL(2,C)representations on C2 to keep track of:

• SL: This is the standard defining representation of SL(2,C) on C2, withΩ ∈ SL(2,C) acting on ψL ∈ SL by

ψL → ΩψL

A standard index notation for such things is called the “van der Waer-den notation”. It uses a lower index A taking values 1, 2 to label thecomponents with respect to a basis of SL as

ψL =

(ψ1

ψ2

)= ψA

and in this notation Ω acts by

ψA → ΩBAψB

For instance, the element

Ω = e−iθ2σ3

corresponding to an SO(3) rotation by an angle θ around the z-axis actson SL by (

ψ1

ψ2

)→ e−i

θ2σ3

(ψ1

ψ2

)• S∗L: This is the dual of the defining representation, with Ω ∈ SL(2,C)

acting on ψ∗L ∈ S∗L byψ∗L → (Ω−1)Tψ∗L

This is a general property of representations: given any finite-dimensionalrepresentation (π(g), V ), the pairing between V and its dual V ∗ is pre-served by acting on V ∗ by matrices (π(g)−1)T , and these provide a repre-sentation ((π(g)−1)T , V ∗). In van der Waerden notation, one uses upperindices and writes

ψA → ((Ω−1)T )ABψB

402

Writing elements of the dual as row vectors, our example above of a par-ticular Ω acts by (

ψ1 ψ2)→(ψ1 ψ2

)eiθ2σ3

Note that the bilinear form ε gives an isomorphism of representationsbetween SL and S∗L, written in index notation as

ψA = εABψB

where

εAB =

(0 1−1 0

)• SR: This is the complex conjugate representation to SL, with Ω ∈ SL(2,C)

acting on ψR ∈ SR byψR → ΩψR

The van der Waerden notation uses a separate set of dotted indices forthese, writing this as

ψA → ΩB

AψB

Another common notation among physicists puts a bar over the ψ todenote that the vector is in this representation, but we’ll reserve thatnotation for complex conjugation. The Ω corresponding to a rotationabout the z-axis acts as (

ψ1

ψ2

)→ ei

θ2σ3

(ψ1

ψ2

)• S∗R: This is the dual representation to SR, with Ω ∈ SL(2,C) acting onψ∗R ∈ S∗R by

ψ∗R → (Ω−1

)Tψ∗R

and the index notation uses raised dotted indices

ψA → ((Ω−1

)T )ABψB

Our standard example of a Ω acts by(ψ1 ψ2

)→(ψ1 ψ2

)e−i

θ2σ3

Another copy of ε

εAB =

(0 1−1 0

)gives the isomorphism of SR and S∗R as representations, by

ψA = εABψB

403

Restricting to the SU(2) subgroup of SL(2,C), all these representationsare unitary, and equivalent. As SL(2,C) representations, they are not unitary,and while the representations are equivalent to their duals, SL and SR areinequivalent (since as we have seen, one cannot complex conjugate SL(2,C)matrices by a matrix conjugation).

For the case of the ( 12 ,

12 ) representation, to see explicitly the isomorphism

between SL⊗SR and vectors, recall that we can identify Minkowski space with2 by 2 self-adjoint matrices. Ω ∈ SL(2,C) acts by(

x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)→ Ω

(x0 + x3 x1 − ix2

x1 + ix2 x0 − x3

)Ω†

We can identify such matrices as linear maps from S∗R to SL (and thus isomor-phic to the tensor product SL ⊗ (S∗R)∗ = SL ⊗ SR, see chapter 9).

39.2 Dirac γ matrices and Cliff(3, 1)

In our discussion of the fermionic version of the harmonic oscillator, we definedthe Clifford algebra Cliff(r, s) and found that elements quadratic in its gener-ators gave a basis for the Lie algebra of so(r, s) = spin(r, s). Exponentiatingthese gave an explicit construction of the group Spin(r, s). We can apply thatgeneral theory to the case of Cliff(3, 1) and this will give us the representations( 1

2 , 0) and (0, 12 ).

If we complexify our R4, then its Clifford algebra becomes just the algebraof 4 by 4 complex matrices

Cliff(3, 1)⊗C = Cliff(4,C) = M(4,C)

We will represent elements of Cliff(3, 1) as such 4 by 4 matrices, but shouldkeep in mind that we are working in the complexification of the Clifford algebrathat corresponds to the Lorentz group, so there is some sort of condition on thematrices that needs to be kept track of to identify Cliff(3, 1) ⊂M(4,C) . Thereare several different choices of how to explicitly represent these matrices, andfor different purposes, different ones are most convenient. The one we will beginwith and mostly use is sometimes called the chiral or Weyl representation, andis the most convenient for discussing massless charged particles. We will tryand follow the conventions used for this representation in [78].

Digression. Note that the Aj and Bj we constructed using the lj and kj werealso complex 4 by 4 matrices, but they were acting on complex vectors (( 1

2 ,12 ),

the complexification of the vector representation). Now we want 4 by 4 matriceswhich act on a different C4, a sum of two Weyl spinor representations.

Writing 4 by 4 matrices in 2 by 2 block form and using the Pauli matricesσj we assign the following matrices to Clifford algebra generators

γ0 = −i(

0 11 0

), γ1 = −i

(0 σ1

−σ1 0

), γ2 = −i

(0 σ2

−σ2 0

), γ3 = −i

(0 σ3

−σ3 0

)404

One can easily check that these satisfy the Clifford algebra relations for gener-ators of Cliff(1, 3): they anticommute with each other and

γ20 = −1, γ2

1 = γ22 = γ2

3 = 1

The quadratic Clifford algebra elements − 12γjγk for j < k satisfy the com-

mutation relations of so(3, 1). These are explicitly

−1

2γ1γ2 = − i

2

(σ3 00 σ3

), −1

2γ1γ3 = − i

2

(σ2 00 σ2

), −1

2γ2γ3 = − i

2

(σ1 00 σ1

)and

−1

2γ0γ1 =

1

2

(−σ1 0

0 σ1

), −1

2γ0γ2 =

1

2

(−σ2 0

0 σ2

), −1

2γ0γ3 =

1

2

(−σ3 0

0 σ3

)They provide a representation (π′,C4) of the Lie algebra so(3, 1) with

π′(l1) = −1

2γ2γ3, π

′(l2) = −1

2γ1γ3, π

′(l3) = −1

2γ1γ2

and

π′(k1) = −1

2γ0γ1, π

′(k2) = −1

2γ0γ2, π

′(k3) = −1

2γ0γ3

Note that the π′(lj) are skew-adjoint, since this representation of the so(3) ⊂so(3, 1) sub-algebra is unitary. The π′(kj) are self-adjoint and this representa-tion π′ of so(3, 1) is not unitary.

On the two commuting sl(2,C) subalgebras of so(3, 1)⊗C with bases

Aj =1

2(lj + ikj), Bj =

1

2(lj − ikj)

this representation is

π′(A1) = − i2

(σ1 00 0

), π′(A2) = − i

2

(σ2 00 0

), π′(A3) = − i

2

(σ3 00 0

)and

π′(B1) = − i2

(0 00 σ1

), π′(B2) = − i

2

(0 00 σ2

), π′(B3) = − i

2

(0 00 σ3

)We see explicitly that the action of the quadratic elements of the Clifford

algebra on the spinor representation C4 is reducible, decomposing as the directsum SL ⊕ S∗R of two inequivalent representations on C2

Ψ =

(ψLψ∗R

)with complex conjugation (interchange of Aj and Bj) relating the sl(2,C) ac-tions on the components. The Aj act just on SL, the Bj just on S∗R. An

405

alternative standard notation to the two-component van der Waerden notationis to use the four components of C4 with the action of the γ matrices. Therelation between the two notations is given by

ΨA ↔(ψBφB

)where the index A on the left takes values 1, 2, 3, 4 and the indices B, B on theright each take values 1, 2.

Note that identifying Minkowski space with elements of the Clifford algebraby

(x0, x1, x2, x3)→ /x = x0γ0 + x1γ1 + x2γ2 + x3γ3

identifies Minkowski space with certain 4 by 4 matrices. This again gives theidentification used earlier of Minkowski space with linear maps from S∗R to SL,since the upper right two by two block of the matrix will be given by

−i(x0 + x3 x1 − ix2

x1 + ix2 −x0 − x3

)and takes S∗R to SL.

An important element of the Clifford algebra is constructed by multiplyingall of the basis elements together. Physicists traditionally multiply this by i tomake it self-adjoint and define

γ5 = iγ0γ1γ2γ3 =

(−1 00 1

)This can be used to produce projection operators from the Dirac spinors ontothe left and right-handed Weyl spinors

1

2(1− γ5)Ψ = ψL,

1

2(1 + γ5)Ψ = ψ∗R

There are two other commonly used representations of the Clifford algebrarelations, related to the one above by a change of basis. The Dirac representationis useful to describe massive charged particles, especially in the non-relativisticlimit. Generators are given by

γD0 = −i(

1 00 −1

), γD1 = −i

(0 σ1

−σ1 0

)

γD2 = −i(

0 σ2

−σ2 0

), γD3 = −i

(0 σ3

−σ3 0

)and the projection operators for Weyl spinors are no longer diagonal, since

γD5 =

(0 11 0

)406

A third representation, the Majorana representation, is given by (now nolonger writing in 2 by 2 block form, but as 4 by 4 matrices)

γM0 =

0 0 0 −10 0 1 00 −1 0 01 0 0 0

, γM1 =

1 0 0 00 −1 0 00 0 1 00 0 0 −1

γM2 =

0 0 0 10 0 −1 00 −1 0 01 0 0 0

, γM3 =

0 −1 0 0−1 0 0 00 0 0 −10 0 −1 0

with

γM5 = i

0 −1 0 01 0 0 00 0 0 10 0 −1 0

The importance of the Majorana representation is that it shows the interestingpossibility of having (in signature (3, 1)) a spinor representation on a real vectorspace R4, since one sees that the Clifford algebra matrices can be chosen to bereal. One has

γ0γ1γ2γ3 =

0 −1 0 01 0 0 00 0 0 10 0 −1 0

and

(γ0γ1γ2γ3)2 = −1

The Majorana spinor representation is on SM = R4, with γ0γ1γ2γ3 a realoperator on this space with square −1, so it provides a complex structure onSM . Recall that a complex structure on a real vector space gives a splitting ofthe complexification of the real vector space into a sum of two complex vectorspaces, related by complex conjugation. In this case this corresponds to

SM ⊗C = SL ⊕ S∗R

the fact that complexifying Majorana spinors gives the two kinds of Weylspinors.


Most quantum field theory textbook have extensive discussions of spinor rep-resentations of the Lorentz group and gamma matrices, although most use theopposite convention for the signature of the Minkowski metric. Typical exam-ples are Peskin-Schroeder [51] and Quantum Field Theory in a Nutshell by TonyZee, see Chapter II.3 and Appendix E [84].

407

Chapter 40

The Poincare Group and itsRepresentations

In the previous chapter we saw that one can take the semi-direct product ofspatial translations and rotations and that the resulting group has infinite-dimensional unitary representations on the state space of a quantum free parti-cle. The free particle Hamiltonian plays the role of a Casimir operator: to getirreducible representations one fixes the eigenvalue of the Hamiltonian (the en-ergy), and then the representation is on the space of solutions to the Schrodingerequation with this energy. This is a non-relativistic procedure, treating time andspace (and correspondingly the Hamiltonian and the momenta) differently. Fora relativistic analog, we will use instead the semi-direct product of space-timetranslations and Lorentz transformations. Irreducible representations of thisgroup will be labeled by a continuous parameter (the mass) and a discrete pa-rameter (the spin or helicity), and these will correspond to possible relativisticelementary particles.

In the non-relativistic case, the representation occurred as a space of solu-tions to a differential equation, the Schrodinger equation. There is an analogousdescription of the irreducible Poincare group representations as spaces of solu-tions of relativistic wave equations, but we will put off that story until succeedingchapters.

40.1 The Poincare group and its Lie algebra

Definition (Poincare group). The Poincare group is the semi-direct product

P = R4 o SO(3, 1)

with double-coverP = R4 o SL(2,C)

The action of SO(3, 1) or SL(2,C) on R4 is the action of the Lorentz group onMinkowski space.

409

We will refer to both of these groups as the “Poincare group”, meaning bythis the double-cover only when we need it because spinor representations of theLorentz group are involved. The two groups have the same Lie algebra, so thedistinction is not needed in discussions that only need the Lie algebra. Elementsof the group P will be written as pairs (a,Λ), with a ∈ R4 and Λ ∈ SO(3, 1).The group law is

(a1,Λ1)(a2,Λ2) = (a1 + Λ1a2,Λ1Λ2)

The Lie algebra LieP = LieP has dimension 10, with basis

t0, t1, t2, t3, l1, l2, l3, k1, k2, k3

where the first four elements are a basis of the Lie algebra of the translationgroup, and the next six are a basis of so(3, 1), with the lj giving the subgroup ofspatial rotations, the kj the boosts. We already know the commutation relationsfor the translation subgroup, which is commutative so

[tj , tk] = 0

We have seen that the commutation relations for so(3, 1) are

[l1, l2] = l3, [l2, l3] = l1, [l3, l1] = l2

[k1, k2] = −l3, [k3, k1] = −l2, [k2, k3] = −l1and that the commutation relations between the lj and kj correspond to thefact that the kj transform as a vector under spatial rotations, so for examplecommuting the kj with l1 gives an infinitesimal rotation about the 1-axis and

[l1, k1] = 0, [l1, k2] = k3, [l1, k3] = −k2

The Poincare group is a semi-direct product group of the sort discussed inchapter 16 and it can be represented as a group of 5 by 5 matrices in much thesame way as elements of the Euclidean group E(3) could be represented by 4by 4 matrices (see chapter 17). Writing out this isomorphism explicitly for abasis of the Lie algebra, we have

l1 ↔

0 0 0 0 00 0 0 0 00 0 0 −1 00 0 1 0 00 0 0 0 0

l2 ↔

0 0 0 0 00 0 0 1 00 0 0 0 00 −1 0 0 00 0 0 0 0

l3 ↔

0 0 0 0 00 0 −1 0 00 1 0 0 00 0 0 0 00 0 0 0 0

k1 ↔

0 1 0 0 01 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

k2 ↔

0 0 1 0 00 0 0 0 01 0 0 0 00 0 0 0 00 0 0 0 0

k3 ↔

0 0 0 1 00 0 0 0 00 0 0 0 01 0 0 0 00 0 0 0 0

410

t0 ↔

0 0 0 0 10 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

t1 ↔

0 0 0 0 00 0 0 0 10 0 0 0 00 0 0 0 00 0 0 0 0

t2 ↔

0 0 0 0 00 0 0 0 00 0 0 0 10 0 0 0 00 0 0 0 0

t3 ↔

0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 10 0 0 0 0

We can use this explicit matrix representation to compute the commutators

of the infinitesimal translations tj with the infinitesimal rotations and boosts(lj , kj). t0 commutes with the lj and t1, t2, t3 transform as a vector underrotations, For instance, for infinitesimal rotations about the 1-axis

[l1, t1] = 0, [l1, t2] = t3, [l1, t3] = −t2

with similar relations for the other axes.

For boosts one has

[kj , t0] = tj , [kj , tj ] = t0, [kj , tk] = 0 if j 6= k, k 6= 0

Note that infinitesimal boosts do not commute with infinitesimal time transla-tion, so after quantization boost will not commute with the Hamiltonian andthus are not the sort of symmetries which act on spaces of energy eigenstates,preserving the energy.

40.2 Representations of the Poincare group

We want to find unitary irreducible representations of the Poincare group. Thesewill be infinite dimensional, so given by operators π(g) on a Hilbert space H,which will have an interpretation as a single-particle relativistic quantum statespace. The standard physics notation for the operators giving the representationis U(a,Λ), with the U emphasizing their unitarity. To classify these representa-tions, we recall from chapter 18 that irreducible representations of semi-directproducts N oK are associated with pairs of a K-orbit Oα in the space N andan irreducible representation of the corresponding little group Kα.

For the Poincare group, N = R4 is the space of characters (one-dimensionalrepresentations) of the translation group of Minkowski space. These are labeledby an element p = (p0, p1, p2, p3) that has a physical interpretation as the energy-momentum vector of the state such that π(x) (for x in the translation groupN) acts as multiplication by

ei(−p0x0+p1x1+p2x2+p3x3)

411

Equivalently, the p0, p1, p2, p3 are the eigenvalues of the energy and momentumoperators

P0 = −iπ′(t0), P1 = iπ′(t1), P2 = iπ′(t2), P3 = iπ′(t3)

that give the representation of the translation part of the Poincare group Liealgebra on the states.

The Lorentz group acts on this R4 by

p→ Λp

and, restricting attention to the p0 − p3 plane, the picture of the orbits lookslike this

Unlike the Euclidean group case, here there are several different kinds oforbits Oα. We’ll examine them and the corresponding stabilizer groups Kα

each in turn, and see what can be said about the associated representations.One way to understand the equations describing these orbits is to note thatthe different orbits correspond to different eigenvalues of the Poincare groupCasimir operator

P 2 = −P 20 + P 2

1 + P 22 + P 2

3

412

This operator commutes with all the generators of the Lie algebra of the Poincaregroup, so by Schur’s lemma it must act as a scalar times the identity on anirreducible representation (recall that the same phenomenon occurs for SU(2)representations, which can be characterized by the eigenvalue j(j+1) of the Cas-mir operator J2 for SU(2)). At a point p = (p0, p1, p2, p3) in energy-momentumspace, the Pj operators are diagonalized and P 2 will act by the scalar

−p20 + p2

1 + p22 + p2

3

which can be positive, negative, or zero, so given by m2,−m2, 0 for variousm. The value of the scalar will be the same everywhere on the orbit, so inenergy-momentum space orbits will satisfy one of the three equations

−p20 + p2

1 + p22 + p2

3 =

−m2

m2

0

Note that in this chapter we are just classifying Poincare group representa-tions, not actually constructing them. It is possible to construct these represen-tations using the data we will find that classifies them, but this would requireintroducing some techniques (for so-called “induced representations”) that gobeyond the scope of this course. In later chapters we will explicitly constructthese representations in certain specific cases as solutions to certain relativisticwave equations.

40.2.1 Positive energy time-like orbits

One way to get negative values −m2 of the Casimir P 2 is to take the vectorp = (m, 0, 0, 0), m > 0 and generate an orbit Om,0,0,0 by acting on it withthe Lorentz group. This will be the upper, positive energy, hyperboloid of thehyperboloid of two sheets

−p20 + p2

1 + p22 + p2

3 = −m2

so

p0 =√p2

1 + p22 + p2

3 +m2

The stabilizer group of Km,0,0,0 is the subgroup of SO(3, 1) of elements ofthe form (

1 00 Ω

)where Ω ∈ SO(3), so Km,0,0,0 = SO(3). Irreducible representations of thisgroup are classified by the spin. For spin 0, points on the hyperboloid canbe identified with positive energy solutions to a wave equation called the Klein-Gordon equation and functions on the hyperboloid both correspond to the spaceof all solutions of this equation and carry an irreducible representation of thePoincare group. In the next chapter we will study the Klein-Gordon equation,

413

as well as the quantization of the space of its solutions by quantum field theorymethods.

We will later study the case of spin 12 , where one must use the double cover

SU(2) of SO(3). The Poincare group representation will be on functions onthe orbit that take values in two copies of the spinor representation of SU(2).These will correspond to solutions of a wave equation called the massive Diracequation.

For choices of higher spin representations of the stabilizer group, one canagain find appropriate wave equations and construct Poincare group represen-tations on their space of solutions, but we will not enter into this topic.

40.2.2 Negative energy time-like orbits

Starting instead with the energy-momentum vector p = (−m, 0, 0, 0), m > 0,the orbit O−m,0,0,0 one gets is the lower, negative energy component of thehyperboloid

−p20 + p2

1 + p22 + p2

3 = −m2

satisfying

p0 = −√p2

1 + p22 + p2

3 +m2

Again, one has the same stabilizer group K−m,0,0,0 = SO(3) and the same con-stuctions of wave equations of various spins and Poincare group representationson their solution spaces as in the positive energy case. Since negative energieslead to unstable, unphysical theories, we will see that these representations aretreated differently under quantization, corresponding physically not to particles,but to antiparticles.

40.2.3 Space-like orbits

One can get positive values m2 of the Casimir P 2 by considering the orbitO0,0,0,m of the vector p = (0, 0, 0,m). This is a hyperboloid of one sheet,satisfying the equation

−p20 + p2

1 + p22 + p2

3 = m2

It is not too difficult to see that the stabilizer group of the orbit is K0,0,0,m =SO(2, 1). This is isomorphic to the group SL(2,R), and it has no finite-dimensional unitary representations. These orbits correspond physically to“tachyons”, particles that move faster than the speed of light, and there isno known way to consistently incorporate them in a conventional theory.

40.2.4 The zero orbit

The simplest case where the Casimir P 2 is zero is the trivial case of a pointp = (0, 0, 0, 0). This is invariant under the full Lorentz group, so the orbitO0,0,0,0 is just a single point and the stabilizer group K0,0,0,0 is the entire Lorentz

414

group SO(3, 1). For each finite-dimensional representation of SO(3, 1), one getsa corresponding finite dimensional representation of the Poincare group, withtranslations acting trivially. These representations are not unitary, so not usablefor our purposes.

40.2.5 Positive energy null orbits

One has P 2 = 0 not only for the zero-vector in momentum space, but for athree-dimensional set of energy-momentum vectors, called the null-cone. Bythe term “cone” one means that if a vector is in the space, so are all productsof the vector times a positive number. Vectors p = (p0, p1, p2, p3) are called“light-like” or “null” when they satisfy

|p|2 = −p20 + p2

1 + p22 + p2

3 = 0

One such vector is p = (1, 0, 0, 1) and the orbit of the vector under the actionof the Lorentz group will be the upper half of the full null-cone, the half withenergy p0 > 0, satisfying

p0 =√p2

1 + p22 + p2

3

The stabilizer group K1,0,0,1 of p = (1, 0, 0, 1) includes rotations about thex3 axis, but also boosts in the other two directions. It is isomorphic to theEuclidean group E(2). Recall that this is a semi-direct product group, and ithas two sorts of irreducible representations

• Representations such that the two translations act trivially. These areirreducible representations of SO(2), so one-dimensional and characterizedby an integer n (half-integers when one uses the Poincare group doublecover).

• Infinite dimensional irreducible representations on a space of functions ona circle of radius r

The first of these two gives irreducible representations of the Poincare groupon certain functions on the positive energy null-cone, labeled by the integer n,which is called the “helicity” of the representation. We will in later chaptersconsider the cases n = 0 (massless scalars, wave-equation the Klein-Gordonequation), n = ± 1

2 (Weyl spinors, wave equation the Weyl equation), and n =±1 (photons, wave equation the Maxwell equations).

The second sort of representation of E(2) gives representations of the Poincaregroup known as “continuous spin” representations, but these seem not to cor-respond to any known physical phenomena.

40.2.6 Negative energy null orbits

Looking instead at the orbit of p = (−1, 0, 0, 1), one gets the negative energy partof the null-cone. As with the time-like hyperboloids of non-zero mass m, thesewill correspond to antiparticles instead of particles, with the same classificationas in the positive energy case.

415


The Poincare group and its Lie algebra is discussed in pretty much any quantumfield theory textbook. Weinberg [78] (Chapter 2) has some discussion of therepresentations of the Poincare group on single particle state spaces that we haveclassified here. Folland [21] (Chapter 4.4) and Berndt [7] (Chapter 7.5) discussthe actual construction of these representations using the induced representationmethods that we have chosen not to try and explain here.

416

Chapter 41

The Klein-Gordon Equationand Scalar Quantum Fields

In the non-relativistic case we found that it was possible to build a quantumtheory describing arbitrary numbers of particles by taking as classical phasespace the single-particle space of solutions to the free particle Schrodinger equa-tion. This phase space carries a unitary representation of the Euclidean groupE(3). Quantization gives multi-particle state space with momentum and angu-lar momentum operators that provide a Lie algebra representation of E(3) onthe multi-particle states.

To get the same sort of construction for relativistic systems, we want to startwith an irreducible unitary representation not of E(3), but of the Poincare groupP as our single-particle space H1. The simplest example of this is provided bythe space of solutions of a relativistic wave-equation known as the Klein-Gordonequation. Its solutions have no consistent interpretation as a physical theory ofa single particle (the energy is not positive), but they can be used to producea consistent quantum field theory describing an arbitrary number of particles.This is done in much the same way as in the non-relativistic case, with a crucialdifference in choice of complex structure. In the relativistic case we need touse annihilation and creation operators determined by the decomposition ofthe space of solutions into positive and negative energy. This will lead to thepossibility of a theory w ith both particles and antiparticles.

41.1 The Klein-Gordon equation and its solu-tions

Recall from chapter 40 that irreducible representations of the Poincare groupare characterized in part by the value that the Casimir operator P 2 takes onthe representation. When this is negative, we have the operator equation

P 2 = −P 20 + P 2

1 + P 22 + P 2

3 = −m2

417

Just as in the non-relativistic case, where we could represent momentum oper-ators as either multiplication operators on functions of the momenta, or differ-entiation operators on functions of the positions, here we can do the same, withfunctions now depending on the four space-time coordinates. Taking P0 = −i ∂∂tas well as the conventional momentum operators Pj = −i ∂

∂xj, the above equa-

tion for the Casimir operator becomes:

Definition (Klein-Gordon equation). The Klein-Gordon equation is the second-order partial differential equation

(− ∂2

∂t2+

∂2

∂x21

+∂2

∂x22

+∂2

∂x23

)φ = m2φ

or

(− ∂2

∂t2+ ∆−m2)φ = 0

for functions φ(x) on Minkowski space (which may be real or complex valued).)

This equation is the simplest Lorentz-invariant wave equation, and histori-cally was the one first tried by Schrodinger (he soon realized it could not accountfor atomic spectra and instead used the non-relativistic equation that bears hisname).

Taking Fourier transforms by

φ(p) =1

(2π)2

∫R4

e−i(−p0x0+p·x)φ(x)d4x

the momentum operators become multiplication operators, and the Klein-Gordonequation is now

(p20 − p2

1 − p22 − p2

3 −m2)φ = 0

Solutions to this will be functions φ of the energy-momenta that are non-zeroonly on the hyperboloid

p20 − p2

1 − p22 − p2

3 −m2 = 0

in energy-momentum space R4. This hyperboloid has two components, withpositive and negative energy

p0 = ±ωp

where

ωp =√p2

1 + p22 + p2

3 +m2

Ignoring one dimension these look like

418

and are the orbits of the energy-momentum vectors (m, 0, 0, 0) and (−m, 0, 0, 0)under the Poincare group action discussed in chapter 40.

In the non-relativistic case, a continuous basis of solutions of the free-particleSchrodinger equation labeled by p ∈ R3 was given by the functions

eip·xe−i|p|22m t

with a general solution a superposition of these with coefficients ψ(p) given bythe Fourier inversion formula

ψ(x, t) =1

(2π)3/2

∫R3

ψ(p)eip·xe−i|p|22m td3p

The complex valued ψ(p) could be thought of as coordinates on the non-relativistic single-particle space H1 (and thus in H∗1). Note that in the non-relativistic case the space of all solutions is a representation of the Euclideangroup E(3), but to get an irreducible representation one must restrict to solu-

tions of a fixed energy E (and thus ψ(p) supported on the sphere |p|2 = 2mE).In the relativistic case the space of solutions to the Klein-Gordon equation

for a fixed m will give an irreducible representation of the Poincare group.Coordinates on this space will be given by the Fourier components φ(p) on theenergy-momentum space R4, which are non-zero for p on the two-componenthyperboloid

p20 = ω2

p

419

(here p = (p0,p)). An arbitrary solution will be given by a function φ on thisspace, and the Fourier inversion formula giving the position space solution interms of φ will be

φ(x, t) =1

(2π)3/2

∫R4

δ(p20 − ω2

p)φ(p)ei(p·x−p0t)d4p

Here the integral is over a three-dimensional hyperboloid, but expressed as afour-dimensional integral over R4 with a delta-function on the hyperboloid inthe argument.

The delta function distribution with argument a function f(x) depends onlyon the zeros of f , and if f ′ 6= 0 at such zeros, one has

δ(f(x)) =∑

xj :f(xj)=0

δ(f ′(xj)(x− xj)) =1

|f ′(xj)|δ(x− xj)

For each p, one can apply this to the case of the function of p0 given by

f = p20 − ω2

p

on R4, and using

d

dp0(p2

0 − ω2p)|p0=±ωp

= 2p0|p0=±ωp= ±2ωp

one finds

φ(x, t) =1

(2π)3/2

∫R4

1

2ωp(δ(p0 − ωp) + δ(p0 + ωp))φ(p)ei(p·x−p0t)dp0d

3p

=1

(2π)3/2

∫R3

(φ+(p)e−iωpt + φ−(p)eiωpt)eip·xd3p

2ωp(41.1)

Hereφ+(p) = φ(ωp,p), φ−(p) = φ(−ωp,p)

are the values of φ on the positive and negative energy hyperboloids.We see that instead of thinking of the Fourier transforms of solutions as

taking values on energy-momentum hyperboloids, we can think of them as takingvalues on the space R3 of momenta, just as in the non-relativistic case. Onedifference is that we have to use both positive and negative energy Fouriercomponents. Another is the use of

d3p

2ωp

instead of d3p, which is necessary to get a Lorentz invariant integral.A general complex-valued solution to the Klein-Gordon equation will thus

be given by the two complex-valued functions φ+, φ−. We can impose the

420

condition that the solution be real-valued, in which case one can check that thepair of functions must satisfy the condition

φ−(p) = φ+(−p)

Real-valued solutions of the Klein-Gordon equation thus correspond to arbi-trary complex-valued functions defined on either the positive or negative energyhyperboloid (the value on the other hyperboloid is then fixed).

By analogy with the non-relativistic case, one is tempted to try and thinkof solutions of the Klein-Gordon equation as wavefunctions describing a sin-gle relativistic particle. To do this, one needs to consider complex solutions,not just the real-valued solutions. It turns out though that one cannot notconsistently make a physical interpretation of these solutions as single-particlewavefunctions.. Some reasons for this are:

• The occurrence of solutions with negative energy will cause an instabilityif the theory is coupled to other physics. Single-particle states could thentypically evolve to states with arbitrarily negative energy, transferringpositive energy to the other physical system they are coupled to.

• One could try and restrict to the space of solutions of positive energy, withthe inner product

〈φ, ψ〉 =1

(2π)3/2

∫R3

φ+(p)ψ+(p)d3p

2ωp

Such a theory though will not have a conventional position operator, since

x = i∇

will not be self-adjoint. One will also not have conventional localized delta-function states, since these will not have Fourier transforms will supportonly on the positive-energy hyperboloid.

• No positive conserved probability, this will be the charge.

• Propagator not Lorentz invariant, need to include anti-particles.

• A standard physical argument is that a relativistic single-particle theorydescribing localized particles is not possible, since once the position un-certainty of a particle is small enough, its momentum uncertainty will belarge enough to provide the energy needed to create new particles. If youtry and put a relativistic particle in a smaller and smaller box, at somepoint you will no longer have just one particle, and it is this situation thatimplies that only the many-particle theory will be consistent.

The last of these problems indicates that one should be looking for a many-particle theory. It will turn out that while the space of solutions to the Klein-Gordon equation cannot be used to get a consistent single-particle theory, it canbe used as a phase space to construct a multi-particle theory in much the sameway as the Schrodinger equation was used in the relativistic case.

421

41.2 The complex structure on the space of Klein-Gordon solutions

Recall from chapter 21 that if we intend to quantize a classical phase space Mby the Bargmann-Fock method, we need to choose a complex structure J onthe dual phase space M = M∗. Then

M⊗C =M+J ⊕M

−J

where M+J is the +i eigenspace of J , M−J the −i eigenspace. The quantum

state space will be the space of polynomials on the dual of M+J . The choice of

J corresponds to a choice of distinguished state |0〉J ∈ H, the Bargmann-Fockstate given by the constant polynomial function 1.

In the non-relativistic quantum field theory case we saw that since wave-functions were complex-valued, the space of solutions of the Schrodinger equa-tion came with a natural complex structure J given by multiplication by i. Thelinear functionals ψ(x) in position space or α(p) in momentum space could beinterpreted as a basis of the desired M+

J . We could get M−J by taking thecomplex conjugate space toM+

J , with basis the linear functionals ψ(x) or α(p).A surprising aspect of the relativistic theory is that we must make a very

different choice of J to get a consistent multi-particle quantum theory. Thesimplest situation is that of taking the single particle space H1 to be the spaceof real-valued solutions of the Klein-Gordon equation, and we will restrict tothis case in the rest of this chapter, dealing with the case of complex-valuedsolutions in chapter 42. Our phase space M = H1 to be quantized is thus a realvector space, the usual starting point for a Bargmann-Fock quantization.

When we complexify and look at the spaceM⊗C, it naturally decomposesinto two pieces: M+, the complex functions on the positive energy hyperboloidand M−, the complex functions on the negative energy hyperboloid. Moreexplicitly, we can decompose a complexified solution φ(x, t) of the Klein-Gordonequation (see 41.1) as φ = φ+ + φ−, where

φ+(x, t) =1

(2π)3/2

∫R3

φ+(p)e−iωpteip·xd3p

2ωp

and

φ−(x, t) =1

(2π)3/2

∫R3

φ−(−p)eiωpte−ip·xd3p

2ωp

We will take as complex structure the operator J that is +i on positive energywavefunctions and −i on negative energy wavefunctions. Since the Schrodingerequation says that

id

dt= H

is the operator with eigenvalues the energy, wavefunctions with time depen-dence e−iωpt will be said to have positive energy or, equivalently, positive fre-quency. Those with time dependence eiωpt have negative energy or frequency.

422

The Fourier components φ+ of positive energy solutions will be complexifiedcoordinates on the solution space, give a continuous basis of M+

J , and will be

quantized as annihilation operator. The Fourier components φ− of negativeenergy solutions will be basis elements forM−J and quantized as creation oper-ators.

There are many different possible choices of how to normalize basis elementsof M+

J . We will choose to set

α(p) =φ+(p)√

2ωp

, α(p) =φ−(−p)√

2ωp

and in terms of these real solutions of the Klein-Gordon equation are given by

φ(x, t) =1

(2π)3/2

∫R3

(α(p)e−iωpteip·x + α(p)eiωpte−ip·x)d3p√2ωp

(41.2)

With this normalization of the α(p), the Poisson bracket relations andHamiltonian will be

α(p), α(p′) = −iδ3(p− p′), α(p), α(p′)) = α(p), α(p′)) = 0 (41.3)

h =

∫R3

ωpα(p)α(p)d3p

The equations of motion are

d

dtα = α, h = ωp

∫R3

α(p′)α(p), α(p′)d3p′ = −iωpα

d

dtα = α, h = iωpα

with solutions

α(p, t) = e−iωptα(p, 0), α(p, t) = eiωptα(p, 0)

Add discussion of the symplectic structure and Hermitian form on the spaceof Klein-Gordon solutions in momentum space.

41.3 Quantization of the Klein-Gordon theoryin momentum space

Given the description we have found in momentum space of a real scalar fieldsatisfying the Klein-Gordon equation, we can proceed to quantize the theory inexactly the same way as was done with the non-relativistic Schrodinger equation,taking momentum components of fields to operators by replacing

α(p)→ a(p), α(p)→ a†(p)

423

where a(p), a†(p) are operator valued distributions satisfying the commutationrelations

[a(p), a†(p′)] = δ3(p− p′)

For the Hamiltonian we take the normal-ordered form

H =

∫R3

ωpa†(p)a(p)d3p

Starting with a vacuum state |0〉, by applying creation operators one can createarbitary positive energy multiparticle states of free relativistic particles withsingle-particle states having the energy momentum relation

E(p) = ωp =√|p|2 +m2

This description of the quantum system is essentially the same as that ofthe non-relativistic theory, which seems to differ only in the energy-momentum

dispersion relation, with E(p) = |p|22m . The change in complex structure J

though means that in the relativistic theory the vacuum state |0〉 and thatannihilation and creation operators carry a different meaning:

• Non-relativistic theory. M+J = H∗1 is the space of coordinates on solutions

of the free particle Schrodinger equation, which all have positive energy.M−J = H∗1 is the space of coordinates of complex conjugates of solutions,which again have positive energy (they satisfy the complex-conjugatedSchrodinger equation). For each basis element of H1 (a positive energysolution with momentum p) , quantization takes the coordinate in M+

J

to an annihilation operator a(p) and complex conjugate coordinate to acreation operator a†(p).

• Relativistic theory. M+J is the space of coordinates on positive energy

complexified solutions of the Klein-Gordon equation, while M−J is thespace of coordinates on negative energy complexified solutions. To a basiselement in the space of positive energy complexified solutions quantizationgives us an annihilation operator, whereas for negative energy complexifiedsolution basis elements we have a creation operator.

To make physical sense of the quanta in the relativistic theory, assigningall non-vacuum states a positive energy, we take such quanta as having twoequivalent descriptions:

• A positive energy particle moving forward in time with momentum p.

• A positive energy antiparticle moving backwards in time with momentum−p.

The operator a†(p) adds such quanta to a state, the operator a(p) destroysthem. Note that for a theory of quantized real-valued Klein-Gordon fields, anon-zero field has components in both M+ and M− so its quantization willboth create and destroy quanta.

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41.4 Classical relativistic scalar fields

Instead of describing the phase space H1 using momentum space coordinates,we can as in the non-relativistic case instead take as coordinates on H1 thevalues of the initial data at a fixed time t = 0. Since the Klein-Gordon equationis second order in time, solutions will be parametrized by initial data which,unlike the non-relativistic case, now requires the specification at t = 0 of notone, but two functions,

φ(x) = φ(x, 0), φ(x) =∂

∂tφ(x, t)|t=0

the values of the field and its first time derivative.We can take as our phase space H1 the space of pairs of functions (φ, π),

with coordinates φ(x), π(x) and Poisson brackets

φ(x), π(x′) = δ(x− x′), φ(x), φ(x′) = π(x), π(x′) = 0 (41.4)

The Klein-Gordon equation for φ(x, t) in Hamiltonian form is the following pairof first order equations

∂

∂tφ = π,

∂

∂tπ = (∆−m2)φ

which together imply∂2

∂t2φ = (∆−m2)φ

To get these as equations of motion, we just need to find a Hamiltonianfunction h on the phase space H1 such that

∂

∂tφ =φ, h = π

∂

∂tπ =π, h = (∆−m2)φ

One can check (by calculations similar to those of section 35.3) that twochoices of Hamiltonian function with this property are

h =

∫R3

H(x)d3x

where

H =1

2(π2 − φ∆φ+m2φ2) or H =

1

2(π2 + (∇φ)2 +m2φ2)

Here the two different integrands H(x) are related (as in the non-relativisticcase) by integration by parts so these just differ by boundary terms that areassumed to vanish.

425

One could instead have taken as starting point the Lagrangian formalism,with an action

S =

∫M4

L d4x

where

L =1

2((∂

∂tφ)2 − (∇φ)2 −m2φ2)

This action is a functional of fields on Minkowski space M4 and is Lorentzinvariant. The Euler-Lagrange equations give as equation of motion the Klein-Gordon equation

(2−m2)φ = 0

One recovers the Hamiltonian formalism by seeing that the canonical momentumfor φ is

π =∂L∂φ

= φ

and the Hamiltonian density is

H = πφ− L =1

2(π2 + (∇φ)2 +m2φ2)

Recalling equation 41.2 for the Klein-Gordon solutions in terms of momen-tum space coordinates

φ(x, t) =1

(2π)3/2

∫R3

(α(p)e−iωpteip·x + α(p)eiωpte−ip·x)d3p√2ωp

we can compute Poisson brackets and show that the field Poisson bracket rela-tions 41.4 are the same as the momentum space Poisson bracket relations 41.3.To see this, one can compute the Poisson brackets for the fields as follows. Wehave

π(x) =∂

∂tφ(x, t)|t=0 =

1

(2π)3/2

∫R3

(−iωp)(α(p)eip·x − α(p)e−ip·x)d3p√2ωp

and

φ(x) =1

(2π)3/2

∫R3

(α(p)eip·x + α(p)e−ip·x)d3p√2ωp

so

φ(x), π(x′) =1

2(2π)3

∫R3×R3

(α(p), iα(p′)ei(p·x−p′x′)

− iα(p), α(p′)ei(−p·x+p′x′))d3pd3p′

=1

2(2π)3

∫R3×R3

δ3(p− p′)(ei(p·x−p′x′) + ei(−p·x+p′x′))d3pd3p′

=1

2(2π)3

∫R3

(eip·(x−x′) + e−ip·(x−x

′))d3p

=δ3(x− x′)

426

As in the non-relativistic case, this distributional equation means that, forpairs (f, g) of functions from an appropriately chosen class, it is elements of H∗1of the form

φ(f) + π(g) =

∫R3

(f(x)φ(x) + g(x)π(x))d3x

that have well-defined Poisson bracket relations

φ(f1) + π(g1), φ(f2) + π(g2) =

∫R3

(f1(x)g2(x)− f2(x)g1(x))d3x

This is just the infinite-dimensional analog of 12.1, the Poisson bracket of twolinear combinations of position and momentum variables. Here the right-handside is a symplectic form Ω on H∗1.

Add discussion of the Hermitian form on the space of Klein-Gordon solutionsin position space

41.5 Quantization of the real scalar field

Just as in the non-relativistic case (see equation 35.1) one can define quantumfield operators using the momentum space decomposition and annihilation andcreation operators:

Definition (Real scalar quantum field). The real scalar quantum field operatorsare the operator-valued distributions defined by

φ(x) =1

(2π)3/2

∫R3

(a(p)eip·x + a†(p)e−ip·x)d3p√2ωp

(41.5)

π(x) =1

(2π)3/2

∫R3

(−iωp)(a(p)eip·x − a†(p)e−ip·x)d3p√2ωp

(41.6)

By essentially the same computation as for Poisson brackets, one can com-pute commutators, finding

[φ(x), π(x′)] = iδ3(x− x′), [φ(x), φ(x′)] = [π(x), π(x′)] = 0

These can be interpreted as the relations of a unitary representation of a Heisen-berg Lie algebra, the infinite dimensional Lie algebra corresponding to the dualM = H∗1 of the space of solutions of the Klein-Gordon equation.

The Hamiltonian operator will be quadratic in the field operators and canbe chosen to be

H =

∫R3

1

2:(π(x)2 + (∇φ(x))2 +m2φ(x)2):d3x

This operator is normal ordered, and a computation (see for instance chapter 5of [11]) shows that in terms of momentum space operators this is just

H =

∫R3

ωpa†(p)a(p)d3p

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the Hamiltonian operator discussed earlier.The dynamical equations of the quantum field theory are now

∂

∂tφ = [φ,−iH] = π

∂

∂tπ = [π,−iH] = (∆−m2)φ

which have as solution the following equation for the time-dependent field op-erator:

φ(x, t) =1

(2π)3/2

∫R3

(a(p)e−iωpteip·x + a†(p)eiωpte−ip·x)d3p√2ωp

(41.7)

As we saw in section 41.3 this is a theory of quanta that can be interpreted aseither particles moving forward in time or antiparticles of opposite momentummoving backwards in time, with the self-adjoint field operator having both anannihilation operator component and a creation operator component.

41.6 The propagator

As explained in the non-relativistic case, in quantum field theory explicitlydealing with states and their time-dependence is awkward, so we work in theHeisenberg picture, expressing everything in terms of a fixed, unchangeable state|0〉 and time-dependent operators. For the free scalar field theory, in equation41.7 we have explicitly solved for the time-dependence of the field operators. Abasic quantity needed for describing the propagation of quanta of a quantumfield theory is the propagator:

Definition (Green’s function or propagator, scalar field theory). The Green’sfunction or propagator for a scalar field theory is the amplitude, for t > t′

G(x, t,x′, t′) = 〈0|φ(x, t)φ(x′, t′)|0〉

By translation invariance, the propagator will only depend on t − t′ andx − x′, so we can just evaluate the case (x′, t′) = (0, 0), using the formula forthe time dependent field to get

G(x, t,0, 0) =1

(2π)3

∫R3×R3

〈0|(a(p)e−iωpteip·x + a†(p)eiωpte−ip·x)

(a(p′) + a†(p′))|0〉 d3p√2ωp

d3p′√2ωp′

=

∫R3×R3

δ3(p− p′)e−iωpteip·xd3p√2ωp

d3p′√2ωp′

=

∫R3

e−iωpteip·xd3p

2ωp

For t > 0, this gives the amplitude for propagation of a particle in time tfrom the origin to the point x.

428

• Describe properties of this function, show that space-like fall-off as expo-nential.

• Give Lorentz invariant version, the Feynman propagator.

• Compute the commutator, show that it vanishes at space-like separation.

• Compare to non-relativistic propagator.

41.7 Interacting scalar field theories

Our discussion so far has dealt purely with a theory of non-interacting quanta,so this theory is called a quantum theory of free fields. The field however canbe used to introduce interactions between these quanta, interactions which arelocal in space. The simplest such theory is the one given by adding a quarticterm to the Hamiltonian, taking

H =

∫R3

:1

2(π(x)2 + (∇φ(x))2 +m2φ(x)2) + λφ(x)4: d3x

This interacting theory is vastly more complicated and much harder to under-stand than the non-interacting theory. Among the difficult problems it raisesare those of how to make sense of the operator φ(x)4 since it is a product not ofoperators but of operator-valued distributions. One also needs to find an appro-priate state space on which such a Hamiltonian operator will be well-defined,with a well-defined ground state (the free-particle state space will not work).

One calculational technique one can try is a series expansion in powers of λabout the free-field value λ = 0, a calculation whose terms are given by Feynmandiagrams, and described in detail in standard quantum field theory textbooks.A basic physical calculation in this theory is that of the amplitude for scatteringof two particles, something which is not there in the free-field theory. One doessuch calculations by first introducing a cut-off to get finite results, and thentrying to get a sensible limit as the cut-off is removed (this is the theory of“renormalization”). Our best understanding currently is that one can get finiteresults for the terms in a series expansion, but the expansion is not convergent.It appears that removal of the cut-off will always in the limit give back the non-interacting theory, so there is no non-trivial continuum quantum field theory ofscalar fields that can be constructed in this way.

41.8 Fermionic scalars

Commutator at space-like separations does not vanish. Non-positivity of theHamiltonian

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Pretty much every quantum field theory textbook has a treatment of the rela-tivistic scalar field with more details than here, and significantly more physicalmotivation. A good example with some detailed versions of the calculationsdone here is chapter 5 of [11]. See Folland [21], chapter 5 for a mathematicallymore careful treatment of the distributional nature of the scalar field operators.

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Chapter 42

Symmetries and RelativisticScalar Quantum Fields

Just as for non-relativistic quantum fields, the theory of free relativistic scalarquantum fields starts by taking as phase space an infinite dimensional space ofsolutions of an equation of motion. Quantization of this phase space involvesconstructing field operators which provide a representation of the correspondingHeisenberg Lie algebra, by an infinite dimensional version of the Bargmann-Fockconstruction. The equation of motion has its own representation-theoreticalsignificance: it is an eigenvalue equation for a Casimir operator of a group ofspace-time symmetries, picking out an irreducible representation of that group.In this case the Casimir operator is the Klein-Gordon operator, and the space-time symmetry group is the Poincare group. The Poincare group acts on thephase space of solutions to the Klein-Gordon equation, preserving the Poissonbracket. One can thus use the same methods as in the finite-dimensional caseto get a representation of the Poincare group by intertwining operators forthe Heisenberg Lie algebra representation (that representation is given by thefield operators). These methods give a representation of the Lie algebra of thePoincare group in terms of quadratic combinations of the field operators.

We’ll begin with the case of an even simpler group action on the phase space,that coming from an “internal symmetry” one gets if one takes multi-componentscalar fields, with an orthogonal group or unitary group acting on the real orcomplex vector space in which the classical fields take their values.

42.1 Internal symmetries

The real scalar field theory of chapter 41 lacks one feature of the non-relativistictheory, which is an action of the group U(1) by phase changes on complex fields.This is needed to provide a notion of “charge” and allow the introduction ofelectromagnetism into the theory. In the real scalar field theory there is no dis-tinction between states describing particles and states describing antiparticles.

431

To get a theory with such a distinction we need to introduce fields with morecomponents. Two possibilities are to consider real fields with m components, inwhich case we will have a theory with SO(m) symmetry, and U(1) = SO(2) them = 2 special case, or to consider complex fields with m components, in whichcase we have theories with U(m) symmetry, and m = 1 the U(1) special case.

42.1.1 SO(m) symmetry and real scalar fields

Taking as single particle phase spaceH1 the space of pairs φ1, φ2 of real solutionsto the Klein-Gordon equation, elements g(θ) of the group SO(2) will act on thedual phase space H∗1 of coordinates on such solutions by(

φ1(x)φ2(x)

)→ g(θ) ·

(φ1(x)φ2(x)

)=


)(φ1(x)φ2(x)

)(π1(x)π2(x)

)→ g(θ) ·

(π1(x)π2(x)

)=


)(π1(x)π2(x)

)Here φ1(x), φ2(x), π1(x), π2(x) are the coordinates for initial values at t = 0 of aKlein-Gordon solution. The Fourier transforms of solutions behave in the samemanner.

This group action on H1 breaks up into a direct sum of an infinite number(one for each value of x) of identical cases of rotations in a configuration spaceplane, as discussed in section 18.2.2. We will use the calculation there, where wefound that for a basis element L of the Lie algebra of SO(2) the correspondingquadratic function on the phase space with coordinates q1, q2, p1, p2 was

µL = q1p2 − q2p1

For the case here, we just take

q1, q2, p1, p2 → φ1(x), φ2(x), π1(x), π2(x)

To get a quadratic functional on the fields that will have the desired Poissonbracket with the fields for each value of x, we need to just integrate the analogof µL over R3. We will denote the result by Q, since it is an observable thatwill have a physical interpretation as electric charge when this theory is coupledto the electromagnetic field (see chapter 43):

Q =

∫R3

(π2(x)φ1(x)− π1(x)φ2(x))d3x

One can use the field Poisson bracket relations

φj(x), πk(x′) = δjkδ(x− x′)

to check that

Q,(φ1(x)φ2(x)

) =

(−φ2(x)φ1(x)

), Q,

(π1(x)π2(x)

) =

(−π2(x)π1(x)

)432

Quantization of the classical field theory gives us a unitary representation Uof SO(2), with

U ′(L) = −iQ = −i∫R3

(π2(x)φ1(x)− π1(x)φ2(x))d3x

The operator

U(θ) = e−iθQ

will act by conjugation on the fields:

U(θ)

(φ1(x)

φ2(x)

)U(θ)−1 =


)(φ1(x)

φ2(x)

)

U(θ)

(π1(x)π2(x)

)U(θ)−1 =


)(π1(x)π2(x)

)It will also give a representation of SO(2) on states, with the state space de-

composing into sectors each labeled by the integer eigenvalue of the operator Q(which will be called the “charge” of the state).

Using the definitions of φ and π (41.5 and 41.6) one can compute Q in termsof annihilation and creation operators, with the result

Q = i

∫R3

(a†2(p)a1(p)− a†1(p)a2(p))d3p (42.1)

One expects that since the time evolution action on the classical field spacecommutes with the SO(2) action, the operator Q should commute with the

Hamiltonian operator H. This can readily be checked by computing [H, Q]using

H =

∫R3

ωp(a†1(p)a1(p) + a†2(p)a2(p))d3p

Note that the vacuum state |0〉 is an eigenvector for Q and H with eigenvalue

0: it has zero energy and zero charge. States a†1(p)|0〉 and a†2(p)|0〉 are eigen-

vectors of H with eigenvalue and thus energy ωp, but these are not eigenvectors

of Q, so do not have a well-defined charge.All of this can be generalized to the case of m > 2 real scalar fields, with a

larger group SO(m) now acting instead of the group SO(2). The Lie algebra isnow multi-dimensional, with a basis the elementary antisymmetric matrices εjk,with j, k = 1, 2, · · · ,m and j < k, which correspond to infinitesimal rotations inthe j − k planes. Group elements can be constructed by multiplying rotationseθεjk in different planes. Instead of a single operator Q, we get multiple operators

−iQjk = −i∫R3

(πk(x)φj(x)− πj(x)φk(x))d3x

and conjugation by

Ujk(θ) = e−iθQjk

433

rotates the field operators in the j − k plane. These also provide unitary oper-ators on the state space, and, taking appropriate products of them, a unitaryrepresentation of the full group SO(m) on the state space. The Qjk commutewith the Hamiltonian, so the energy eigenstates of the theory break up into ir-reducible representations of SO(m) (a subject we haven’t discussed for m > 3).

42.1.2 U(1) symmetry and complex scalar fields

Instead of describing a scalar field system with SO(2) symmetry using a pairφ1, φ2 of real fields, it is more covenient to identify the R2 that the fields takevalues in with C, and work with complex scalar fields and a U(1) symmetry.This will allow us to work with a set of annihilation and creation operatorsfor particle states with a definite value of the charge observable. Note that wewere already forced to introduce a complex structure (given by the splitting ofcomplexified solutions of the Klein-Gordon equation into positive and negativeenergy solutions) as part of the Bargmann-Fock quantization. This is a secondand independent source of complex numbers in the theory.

We express a pair of real-valued fields as complex-valued fields using

φ =1√2

(φ1 + iφ2), π =1√2

(π1 − iπ2)

These can be thought of as initial-value data parametrizing complex solutionsof the Klein-Gordon equation, giving a phase space that is infinite-dimensional,with four real dimensions for each value of x. Instead of

φ1(x), φ2(x), π1(x), π2(x)

we can think of the complex-valued fields and their complex conjugates

φ(x), φ(x), π(x), π(x)

as providing a real basis of the coordinates on phase space.The Poisson bracket relations on such complex fields will be

φ(x), φ(x′) = π(x), π(x′) = φ(x), π(x′) = φ(x), π(x′) = 0

φ(x), π(x′) = φ(x), π(x′) = δ(x− x′)

and the classical Hamiltonian is

h =

∫R3

(|π|2 + |∇φ|2 +m2|φ|2)d3x

Note that introducing complex fields in a theory like this with field equationsthat are second-order in time means that for each x we have a phase spacewith two complex dimensions (φ(x) and π(x)). Using Bargmann-Fock methodsrequires complexifying one’s phase space, which is a bit confusing here sincethe phase space is already is given in terms of complex fields. We can howeverproceed to find the operator that generates the U(1) symmetry as follows.

434

In terms of complex fields, the SO(2) transformations on the pair φ1, φ2 ofreal fields become U(1) phase transformations, with Q now given by

Q = −i∫R3

(π(x)φ(x)− π(x)φ(x))d3x

satisfying

Q,φ(x) = iφ(x), Q,φ(x) = −iφ(x)

Quantization of the classical field theory gives a representation of the infinitedimensional Heisenberg algebra with commutation relations

[φ(x), φ(x′)] = [π(x), π(x′)] = [φ(x), π†(x′)] = [φ†(x), π(x′)] = 0

[φ(x), π(x′)] = [φ†(x), π†(x′)] = iδ3(x− x′)

Quantization of the quadratic functional Q of the fields is done with the normal-ordering prescription, to get

Q = −i∫R3

: (π(x)φ(x)− π†(x)φ†(x)) : d3x

Taking L = i as a basis element for u(1), one gets a unitary representationU of U(1) using

U ′(L) = −iQ

and

U(θ) = e−iθQ

U acts by conjugation on the fields:

U(θ)φU(θ)−1 = e−iθφ, U(θ)φ†U(θ)−1 = eiθφ†

U(θ)πU(θ)−1 = eiθπ, U(θ)π†U(θ)−1 = e−iθπ†

It will also give a representation of U(1) on states, with the state space de-composing into sectors each labeled by the integer eigenvalue of the operatorQ.

In the Bargmann-Fock quantization of this theory, we can express quantumfields now in terms of a different set of two annihilation and creation operators

a(p) =1√2

(a1(p) + ia2(p)), a†(p) =1√2

(a†1(p)− ia†2(p))

b(p) =1√2

(a1(p)− ia2(p)), b†(p) =1√2

(a†1(p) + ia†2(p))

The only non-zero commutation relations between these operators will be

[a(p), a†(p′)] = δ(p− p′), [b(p), b†(p′)] = δ(p− p′)

435

so we see that we have, for each p, two independent sets of standard annihilationand creation operators, which will act on a tensor product of two standardharmonic oscillator state spaces. The states created and annihilated by thea†(p) and a(p) operators will have an interpretation as particles of momentump, whereas those created and annihilated by the b†(p) and b(p) operators willbe antiparticles of momentum p. The vacuum state will satisfy

a(p)|0〉 = b(p)|0〉 = 0

Using these creation and annihilation operators, the definition of the complexfield operators is

Definition (Complex scalar quantum field). The complex scalar quantum fieldoperators are the operator-valued distributions defined by

φ(x) =1

(2π)3/2

∫R3

(a(p)eip·x + b†(p)e−ip·x)d3p√2ωp

φ†(x) =1

(2π)3/2

∫R3

(b(p)eip·x + a†(p)e−ip·x)d3p√2ωp

π(x) =1

(2π)3/2

∫R3

(−iωp)(a(p)eip·x − b†(p)e−ip·x)d3p√2ωp

π†(x) =1

(2π)3/2

∫R3

(−iωp)(b(p)eip·x − a†(p)e−ip·x)d3p√2ωp

These operators provide a representation of the infinite-dimensional Heisen-berg algebra given by the linear functions on the phase space of solutions tothe complexified Klein-Gordon equation. This representation will be on a statespace describing both particles and antiparticles. The commutation relationsare

[φ(x), φ(x′)] = [π(x), π(x′)] = [φ†(x), φ†(x′)] = [π†(x), π†(x′)] = 0

[φ†(x), π(x′)] = [φ(x), π†(x′)] = iδ3(x− x′)

The Hamiltonian operator will be

H =

∫R3

: (π†(x)π(x) + (∇φ†(x))(∇φ(x)) +m2φ†(x)φ(x)) : d3x

=

∫R3

ωp(a†(p)a(p) + b†(p)b(p))d3p

Note that the classical solutions to the Klein-Gordon equation have bothpositive and negative energy, but the quantization is chosen so that negativeenergy solutions correspond to antiparticle annihilation and creation operators,and all states of the quantum theory have non-negative energy.

436

42.2 Poincare symmetry and scalar fields

Momentum and energy operators, angular momentum operators. Discuss actionof Lorentz boosts.

The Poincare group action on the coordinates φ(p) on H1 will be given by

u(a,Λ)φ(p) = e−ipaφ(Λ−1p)


437

Chapter 43

U(1) Gauge Symmetry andElectromagnetic Fields

We have now constructed both relativistic and non-relativistic quantum fieldtheories for free scalar particles. In the non-relativistic case we had to usecomplex valued fields, and found that the theory came with an action of a U(1)group, the group of phase transformations on the fields. In the relativistic casereal-valued fields could be used, but if we took complex-valued ones (or usedpairs of real-valued fields), again there was an action of a U(1) group of phasetransformations. This is the simplest example of a so-called “internal symmetry”and it is reflected in the existence of an operator Q called the “charge”.

In this chapter we’ll see to to go beyond the theory of free quantized chargedparticles by introducing background electromagnetic fields that the charged par-ticles will interact with. We will see that this can be done using the U(1) groupaction, but now acting independently at each point in space-time, giving a large,infinite-dimensional group called the “gauge group”. This requires introducing anew sort of space-time dependent field, called a “vector potential” by physicists,a “connection” by mathematicians. Use of this field allows the construction ofa Hamiltonian dynamics invariant under the gauge group. This fixes the waycharged particles interact with electromagnetic fields, which are described bythe vector potential.

Most of our discussion will be for the case of the U(1) group, but we willalso indicate how to generalize to the case of non-Abelian group such as SU(2).

43.1 U(1) gauge symmetry

In sections 36.1.1 and 42.1 we saw that the existence of a U(1) group action byoverall phase transformations on the complex field values led to the existence ofan operator Q, which commuted with the Hamiltonian and acted with integraleigenvalues on the space of states. Instead of acting on fields by a multiplicationby a constant phase eiϕ, one can imagine multiplying by a phase that varies with

439

the coordinates x. Such phase transformations are called “gauge transforma-tions” and form an infinite dimensional group under pointwise multiplication:

Definition (Gauge group). The group G of functions on R4 with values in theunit circle U(1), with group law given by point-wise multiplication

eiϕ1(x) · eiϕ2(x) = ei(ϕ1(x)+ϕ2(x))

is called the U(1) gauge group, or group of U(1) gauge transformations.

Here x = (x0,x) ∈ R4, and ϕ is a real-valued function

φ(x) : R4 → R

The vector space of such functions is the Lie algebra of the Abelian Lie groupG, so has trivial Lie bracket.

The group G acts on complex functions ψ of space-time as

ψ(x)→ eiϕ(x)ψ(x), ψ(x)→ e−iϕ(x)ψ(x)

Note that, in quantum mechanics, this is a group action on the wave-functions,and it does not correspond to any group action on the finite-dimensional phasespace of coordinates and momenta. In quantum field theory though, where thesewave-functions make up the phase space to be quantized, this is a group actionon classical fields.

Terms in the Hamiltonian that just involve |ψ(x)|2 = ψ(x)ψ(x) will be in-variant under the group G, but terms with derivatives such as

|∇ψ|2

will not, since whenψ → eiϕ(x)ψ(x)

one has the inhomogeneous behavior

∂

∂xµψ(x)→ ∂

∂xµ(eiϕ(x)ψ(x)) = eiϕ(x)(i

∂

∂xµϕ(x) +

∂

∂xµ)ψ(x)

To deal with this problem, one introduces a new kind of field

Definition (Connection or vector potential). A U(1) connection (mathemati-cian’s terminology) or vector potential (physicist’s terminology) is a function Aon space-time R4 taking values in R4, with its components denoted

Aµ(x) = (A0(t,x),A(t,x))

and such that the gauge group G acts on the space of U(1) connections by

Aµ(t,x)→ Aµ(t,x) +∂

∂xµϕ(t,x)

440

The vector potential allows one to define a new kind of derivative, suchthat the derivative of the field ψ has the same homogeneous transformationproperties under G as ψ itself:

Definition (Covariant derivative). Given a connection A, the associated co-variant derivative in the µ direction is the operator

(DA)µ =∂

∂xµ− iAµ(x)

With this definition, the effect of a gauge transformation is

(DA)µψ → eiϕ(x)((DA)µ + i∂

∂xµ− i ∂

∂xµ)ψ = eiϕ(x)((DA)µψ

If one replaces derivatives by covariant derivatives, terms in a Hamiltonian suchas

|∇ψ|2

will become3∑j=1

((DA)jψ)((DA)jψ)

which will be invariant under the infinite-dimensional group G. The procedureof starting with a theory of complex fields, introducing a vector potential whilechanging derivatives to covariant derivatives in the Hamiltonian is called the“minimal coupling” prescription for how a theory of complex free fields de-scribing charged particles can be turned into a theory of fields coupled to abackground electromagnetic field. Note that the normalization of A is a matterof convention, with a common one introducing a constant e in the definition,taking

(DA)µ =∂

∂xµ− ieAµ(x)

In our convention the constant e will instead appear later, at the point wherethe normalization of the Hamiltonian governing the dynamics of A is chosen.

43.2 Electric and magnetic fields

While the connection or vector potential A is the fundamental geometrical quan-tity needed to construct theories with gauge symmetry, one often wants to workinstead with quantities derived from A which describe the information containedin A that does not change when one acts by a gauge transformation. To a math-ematician, this is the curvature of a connection, to a physicist, it is the electricand magnetic field strengths derived from a vector potential. One can define

Definition (Electric and magnetic fields). The electric and magnetic fields aretwo functions from R4 to R3, with components given by

Ej = −∂Aj∂t

+∂A0

∂xj, Bj = εjkl(

∂Al∂xk− ∂Ak∂xl

)

441

or, in vector notation

E = −∂A

∂t+ ∇A0, B = ∇×A

E is called the electric field, B the magnetic field.

These are invariant under gauge transformations since

E→ E− ∂

∂t∇ϕ+ ∇ ∂

∂tϕ = E

B→ B + ∇×∇φ = B

In the last equation we use the fact that

∇×∇f = 0 (43.1)

for any function f .

43.3 Field equations with background electro-magnetic fields

One can easily use the minimal coupling method described above to write downthe field equations for our free-particle theories, now coupled to electromagneticfields. They are:

• The Schrodinger equation for a non-relativistic particle coupled to a back-ground electromagnetic field is

i(∂

∂t− iA0)ψ = − 1

2m

3∑j=1

(∂

∂xj− iAj)2ψ

A special case of this is Coulomb potential problem discussed in chapter19, which corresponds to the choice of background field

A0 =e

r, A = 0

Another exactly solvable special case is that of a constant magnetic fieldB = (0, 0, B), which corresponds to the choice

A0 = 0, A = (−By, 0, 0)

• The Pauli-Schrodinger equation (31.1) describes a free spin-half non-relativisticquantum particle. Replacing derivatives by covariant derivatives one gets

i(∂

∂t− iA0)

(ψ1(q)ψ2(q)

)= − 1

2m(σ · (∇− iA))2

(ψ1(q)ψ2(q)

)442

Using the anticommutation

σjσk + σkσj = 2δjk

and commutation

σjσk − σkσj = iεjklσl

relations one finds

σjσk = δjk +1

2iεjklσl

This implies that

(σ · (∇− iA))2 =3∑j=1

(∂

∂xj− iAj)2 +

3∑j,k=1

(∂

∂xj− iAj)(

∂

∂xk− iAk)

i

2εjklσl

=

3∑j=1

(∂

∂xj− iAj)2 + σ ·B

and the Pauli-Schrodinger equation can be written

i(∂

∂t− iA0)

(ψ1(q)ψ2(q)

)= − 1

2m(

3∑j=1

(∂

∂xj− iAj)2 + σ ·B)

(ψ1(q)ψ2(q)

)

This two-component equation is just two copies of the standard Schrodingerequation and an added term coupling the spin and magnetic field which isexactly the one studied in chapter 7. Comparing to the discussion there,we see that the minimal coupling prescription here is equivalent to a choiceof gyromagnetic ratio g = 1.

• With minimal coupling to the electromagnetic field, the Klein-Gordonequation becomes

(−(∂

∂t− iA0)2 +

3∑j=1

(∂

∂xj− iAj)2 −m2)φ = 0

The first two equations are non-relativistic theories, and one can interpretthese equations as describing a single quantum particle (with spin 1

2 in thesecond case) moving in a background electromagnetic field. In the relativis-tic Klein-Gordon case, there is no consistent interpretation of solutions to theequation as describing a single particle. In all three cases, the equations are lin-ear and one can in principle define a quantum field theory by taking the spaceof the solutions of the equation as phase space, and applying the Bargmann-Fock quantization method (in practice this is difficult, since one no longer hastranslation invariance and a plane-wave basis of solutions).

443

43.4 The non-Abelian case

We saw in chapters 36 and 42 that one can construct quantum field theorieswith a U(m) group acting on the fields, just by taking m-component complexfields and a Hamiltonian that is the sum of the usual Hamiltonian for eachcomponent. One can generalize the constructions of this chapter from the U(1)to U(m) case, getting a gauge group G of maps from R4 to U(m). In this sectionwe’ll outline how this works, without going into full detail. The non-AbelianU(m) case is a relatively straightforward generalization, until one gets to thedefinition of the field strengths, where complications arise.

The diagonal U(1) ⊂ U(m) subgroup is treated exactly as above, and it isonly the SU(m) ⊂ U(m) subgroup which requires a separate treatment. Wewill do this just for the case m = 2, which is known as the Yang-Mills theorycase, since it was first investigated by the physicists Yang and Mills in 1954. Forthis case, we replace iϕ(x) by a matrix-valued function, taking values in the Liealgebra of SU(2) for each x, which can be written in terms of three functionsϕa as

iϕ(x) = i

3∑a=1

ϕa(x)σa2

using the Pauli matrices. The gauge group becomes the group GYM of mapsfrom space-time to SU(2), with Lie algebra the maps iϕ(x) from space-time tosu(2) (2 by 2 skew-adjoint matrices with trace zero). Unlike the U(1) case, thisLie algebra has a non-trivial Lie bracket, given just by the point-wise su(2) Liebracket (the commutator of matrices).

In the SU(2) case, the analog of the real-valued function Aµ will now bematrix-valued and one can write

Aµ(x) =

3∑a=1

Aaµ(x)σa2

Instead of one vector potential function for each µ we have three (the Aaµ(x)),and we will refer to this as the “gauge field”. The complex fields ψ are now twocomponent fields, and the covariant derivative is

(DA)µψ = (∂

∂xµ1− i

3∑a=1

Aaµ(x)σa2

)ψ

For the theories of complex fields with U(m) symmetry discussed in chapters 36and 42, for m = 2 one can replace derivatives by covariant derivatives, yieldingtheories of matter particles coupled to background gauge fields.

One thing that is significantly different in the non-Abelian case is the def-inition of the field strengths. We will not enter enough into the mathematicaltheory of the curvature of a connection to explain how to derive these, butthe equations for the electric and magnetic fields in the SU(2) case become

444

equations for matrix-valued functions:

Ej = −∂Aj∂t

+∂A0

∂xj− i[Aj , A0]

and

Bj = εjkl(∂Al∂xk− ∂Ak∂xl− i[Ak, Al])

43.5 The geometric significance of the gauge po-tential

Paths, graphicExplain the path-integral formalism, weighting of paths. Lagrangian form,

just minimal coupling.Wilson loop variables.

Digression. If one is familiar with differential forms, the definition of the cur-vature F of a connection A is most simply made by thinking of A as an elementof Ω(R4), the space of 1-forms on space-time R4. Then the curvature of A issimply the 2-form F = dA, where d is the de Rham differential. The gaugegroup acts on connections by

A→ A+ dϕ

Comment on SU(2) case.


445

Chapter 44

Quantization of theElectromagnetic Field: thePhoton

Understanding the classical field theory of coupled scalar fields and vector po-tentials is rather difficult, with the quantized theory even more so, due to thefact that the Hamiltonian is no longer quadratic in the field variables. If onesimplifies the problem by ignoring the scalar fields and just considering thevector potentials, one does get a theory with quadratic Hamiltonian that canbe readily understood and quantized. The classical equations of motion arethe Maxwell equation in a vacuum, with solutions electromagnetic waves. Thequantization will be a relativistic theory of free, massless particles of helicity±1, the photons.

To get a sensible, unitary theory of photons, one must take into accountthe infinite dimensional gauge group G that acts on the classical phase space ofsolutions to the Maxwell equations. We will see that there are various ways ofdoing this, each with its own subtleties.

44.1 Maxwell’s equations

We saw in chapter 43 that our quantum theories of free particles could becoupled to a background electromagnetic field by introducing vector potentialfields Aµ = (A0,A). Electric and magnetic fields are defined in terms of Aµ bythe equations

E = −∂A

∂t+ ∇A0, B = ∇×A

In this chapter we will see how to make the Aµ fields dynamical variables,although restricting to the special case of electromagnetic fields by themselves,without interaction with matter fields. The equations of motion will be

447

Definition (Maxwell’s equations in vacuo). The Maxwell equations for electro-magnetic fields in the vacuum are

∇ ·B = 0 (44.1)

∇×E = −∂B

∂t(44.2)

∇×B =∂E

∂t(44.3)

and Gauss’s law:∇ ·E = 0 (44.4)

Digression. In terms of differential forms these equations can be written verysimply as

dF = 0, d ∗ F = 0

where the first equation is equivalent to 44.1 and 44.2, the second (which usesthe Hodge star operator for the Minkowski metric) is equivalent to 44.4 and44.3. Note that the first equation is automatically satisfied since by definitionF = dA, and the d operator satisfies d2 = 0.

Writing out equation 44.1 in terms of the vector potential give

∇ ·∇×A

which is automatically satisfied for any vector field A. Similarly, in terms of thevector potential equation 44.2 is

∇× (−∂A

∂t+ ∇A0) = − ∂

∂t(∇×A)

which is automatically satisfied since

∇×∇f = 0

for any function f .Note that since Maxwell’s equations only depend on Aµ through the gauge

invariant fields B and E, if Aµ = (A0,A) is a solution, so is the gauge transform

Aϕµ = (A0 −∂ϕ

∂t,A + ∇ϕ)

44.2 Hamiltonian formalism for electromagneticfields

In order to quantize the electromagnetic field, we first need to express Maxwell’sequations in Hamiltonian form. These equations are second-order differentialequations in t, so we expect to parametrize solutions in terms of initial data

(A0(x, 0),A(x, 0)) and (∂

∂tA0(x, 0)|t=0,

∂

∂tA(x, 0)|t=0)

448

The problem with this is that gauge invariance implies that if Aµ is a solutionwith this initial data, so is Aϕµ for any function ϕ(x, t) such that

ϕ(x, 0) = 0,∂

∂tϕ(x, t)|t=0 = 0

So the solution is underdetermined by the initial data of the vector potentialand its time derivative at t = 0.

One way to deal with this problem is to try and find conditions on the vectorpotential which will remove this freedom to perform such gauge transformations,this is called making a “choice of gauge”, and we will begin with

Definition (Temporal gauge). A vector potential Aµ is said to be in temporalgauge if A0 = 0.

Note that given any vector potential Aµ, we can find a gauge transformationϕ such that the gauge transformed vector potential will have A0 = 0 by justsolving the equation

∂

∂tϕ(t,x) = A0(t,x)

which has solution

ϕ(t,x) =

∫ t

0

A0(τ,x)dτ + ϕ0(x)

where ϕ0(x) is any function of the spatial variables x.In temporal gauge, initial data for a solution to Maxwell’s equations is given

by a pair of functions

A(x),∂

∂tA(x)

and we can take these as our coordinates on the phase space of solutions. Theelectric field is now

E = −∂A

∂t

so we can also write our coordinates on phase space as

(A(x),−E(x))

Assuming these coordinates behave just like position and momentum coordi-nates in the finite-dimensional case, we can specify the Poisson bracket andthus the symplectic form by

Aj(x), Ak(x′) = Ej(x), Ek(x′) = 0

Aj(x), Ek(x′) = −δjkδ3(x− x′)

If we then take as hamiltonian function

h =1

2

∫R3

(|E|2 + |B|2)d3x

449

Hamilton’s equations become

∂

∂tA(x, t) = A(x, t), h = −E(x)

and∂

∂tE(x, t) = E(x, t), h

which one can show is just the Maxwell equation 44.3

∂

∂tE = ∇×B

The final Maxwell’s equation, Gauss’s law (44.4) does not appear in theHamiltonian formalism as an equation of motion. In the next two sections wewill see two different ways of dealing with this problem.

Note that same problem in Lagrangian formalism, no conjugate variable toA0.

Non-abelian case: Hamiltonian has cubic and quartic terms. Equations ofmotion are non-linear.

44.3 Coulomb gauge quantization

For any vector potential satisfying the temporal gauge condition A0 = 0, one canfind a gauge transformation such that the transformed vector potential satisfies

Definition (Coulomb or radiation gauge). A vector potential Aµ is said to bein Coulomb gauge (also called radiation gauge) if ∇ ·A = 0.

To see that this is possible, note that under a gauge transformation one has

∇ ·A→∇ ·A + ∇2ϕ

so such a gauge transformation will put a vector potential in Coulomb gauge ifwe can find a solution to

∇2ϕ = −∇ ·A

It is a standard result in the theory of partial differential equations that one cansolve this equation for ϕ, with the result

ϕ(x) =1

4π

∫R3

1

|x− x′|(∇ ·A(x′))d3x′

Since in temporal gauge

∇ ·E = − ∂

∂t∇ ·A

the Coulomb gauge condition automatically implies that Gauss’s law (44.4) willhold. We thus can take as phase space the solutions to the Maxwell equations

450

satisfying the two conditions A0 = 0,∇ ·A = 0, with phase space coordinatesthe A(x) satisfying the constraint ∇ ·A = 0.

In Coulomb gauge the one Maxwell equation (44.3) that is not automaticallysatisfied is, in terms of the vector potential

− ∂2

∂t2A = ∇× (∇×A)

Using the vector calculus identity

∇× (∇×A) = ∇(∇ ·A)−∇2A

and the Coulomb gauge condition, this becomes the wave equation

(∂2

∂t2−∇2)A = 0 (44.5)

This is just three copies of the real Klein-Gordon equation for mass m = 0,although it needs to be supplemented by the Coulomb gauge condition.

One can thus proceed exactly as for the Klein-Gordon case, finding coordi-nates on the momentum space solutions, and quantizing with annihilation andcreation operators. The coordinates on the momentum space solutions are theFourier transforms Aj(p) and one can write a classical solution in terms of themby a simple generalization of equation 41.2 for the scalar field case

Aj(x, t) =1

(2π)3/2

∫R3

(αj(p)e−iωpteip·x + αj(p)eiωpte−ip·x)d3p√2ωp

where

αj(p) =Aj,+(p)√

2ωp

, αj(p) =Aj,−(−p)√

2ωp

Here ωp = |p|, and Aj,+, Aj,− are the positive and negative energy solutions of44.5.

The three components αj(p) make up a vector-valued function α(p). Thismust satisfy the Coulomb gauge condition, which in momentum space is

p ·α(p) = 0

The space of solutions of this will be two-dimensional for each value of p, andwe can choose some orthonormal basis

ε1(p), ε2(p)

of such solutions. These will satisfy

p · ε1(p) = p · ε2(p) = 0, ε1(p) · ε2(p = 0, |ε1(p)|2 = |ε2(p)|2 = 1

Write down the field in terms of these. Poisson brackets. Quantization interms on annihilation and creation operators. Formulas for electric and magneticfield operators. Hamiltonian operator in momentum space, normal-ordered.

Problems with the Coulomb gauge in the non-Abelian case.

451

44.4 Gauss’s law as an operator condition

Compute moment map for time-independent gauge transformation. Quantize,get action of gauge group on states. Gauss’s law says invariant states (or, moregenerally, ones corresponding to fixed background charge).

44.5 Covariant quantization methods

Say something about these. Lorenz gauge. Comment on BRST.

44.6 The Poincare group and the electromag-netic field

The action of time translations is already given by the Hamiltonian function h.Show that spatial translations are given by

PEM =

∫R3

E×B d3x

In Coulomb gauge show that in momentum space this is the sum of the expectedmomentum operators for the two helicity states of the photon.

Action of the Lorentz group: first explain expected action of the Lorentzgroup on vector fields. Note problem for boosts caused by use of the temporalgauge.

Spatial rotations. Calculate the angular momentum operator. As in thespin-half Pauli-equation case, get an orbital angular momentum and spin an-gular momentum term. Show that the helicity is ±1. Left and right circularlypolarized photons.


The topic of this chapter is treated in some version in every quantum fieldtextbook. For a good example, see chapter 7 of [24].

452

Chapter 45

The Dirac Equation andSpin-1/2 Fields

The space of solutions to the Klein-Gordon equation gives an irreducible repre-sentation of the Poincare group corresponding to a relativistic particle of massm and spin zero. Elementary matter particles (quarks and leptons) are spin 1/2particles, and we would like to have a relativistic wave equation that describesthem, suitable for building a quantum field theory. This is provided by a re-markable construction that uses the Clifford algebra and its action on spinorsto find a square root of the Klein-Gordon equation. The result, the Dirac equa-tion, requires that our fields take not just real or complex values, but values in aspinor ( 1

2 , 0)⊕ (0, 12 ) representation of the Lorentz group. In the massless case,

the Dirac equation decouples into two separate Weyl equations, for left-handed(( 1

2 , 0) representation of the Lorentz group) and right-handed ((0, 12 ) represen-

tation) Weyl spinor fields. As with the scalar fields discussed earlier, for nowwe are just considering free quantum fields, which describe non-interacting par-ticles. In following chapters we will see how to couple these fields to U(1) gaugefields, allowing a description of electromagnetic particle interactions.

45.1 The Dirac and Weyl Equations

Recall from our discussion of supersymmetric quantum mechanics that we foundthat in n-dimensions the operator

n∑j=1

P 2j

has a square root, given byn∑j=1

γjPj

453

where the γj are generators of the Clifford algebra Cliff(n). The same thing istrue for Minkowski space, where one takes the Clifford algebra Cliff(3, 1) whichis generated by elements γ0, γ1, γ2, γ3 satisfying

γ20 = −1, γ2

1 = γ22 = γ2

3 = +1, γjγk + γkγj = 0 forj 6= k

which we have seen can be realized in various ways an an algebra of 4 by 4matrices.

We have seen that −P 20 +P 2

1 +P 22 +P 2

3 is a Casimir operator for the Poincaregroup, and that we get irreducible representations of the Poincare group by usingthe condition that this acts as a scalar −m2

−P 20 + P 2

1 + P 22 + P 2

3 = m2

Using the Clifford algebra generators, we find that this Casimir operator has asquare root

±(−γ0P0 + γ1P1 + γ2P2 + γ3P3)

so one could instead look for solutions to

±(−γ0P0 + γ1P1 + γ2P2 + γ3P3) = im

If the operators Pj continue to act on functions φ(x) on Minkowski space as−i ∂

∂xj, then this square root of the Casimir acts on the tensor product of the

spinor space S ' C4 and a space of functions on Minkowski space. We willdenote such objects by Ψ; they are C4-valued functions on Minkowski space.Taking the negative square root, we have

Definition (Dirac operator and the Dirac equation). The Dirac operator is theoperator

/D = −γ0∂

∂x0+ γ1

∂

∂x1+ γ2

∂

∂x2+ γ3

∂

∂x3

and the Dirac equation is the equation

/DΨ = mΨ

This rather simple looking equation contains an immense amount of non-trivial mathematics and physics, providing a spectacularly successful descriptionof the behavior of physical elementary particles. Writing it more explicitlyrequires a choice of a specific representation of the γj as complex matrices, andwe will use the chiral or Weyl representation, here written as 2 by 2 blocks

γ0 = −i(

0 11 0

), γ1 = −i

(0 σ1

−σ1 0

), γ2 = −i

(0 σ2

−σ2 0

), γ3 = −i

(0 σ3

−σ3 0

)which act on

Ψ =

(ψLψR

)454

(recall that using γ-matrices this way, ψL transforms under the Lorentz groupas the SL representation, ψR as the dual of the SR representation).

The Dirac equation is then(0 ( ∂

∂x0− σ ·∇)

( ∂∂x0

+ σ ·∇) 0

)(ψLψR

)= m

(ψLψR

)or, in components

(∂

∂x0− σ ·∇)ψR = mψL

(∂

∂x0+ σ ·∇)ψL = mψR

In the case that m = 0, these equations decouple and we get

Definition (Weyl equations). The Weyl wave equations for two-componentspinors are

(∂

∂x0− σ ·∇)ψR = 0, (

∂

∂x0+ σ ·∇)ψL = 0

To find solutions to these equations we Fourier transform, using

ψ(x0,x) =1

(2π)2

∫d4p ei(−p0x0+p·x)ψ(p0,p)

and see that the Weyl equations are

(p0 + σ · p)ψR = 0

(p0 − σ · p)ψL = 0

Since(p0 + σ · p)(p0 − σ · p) = p2

0 − (σ · p)2 = p20 − |p|2

both ψR and ψL satisfy(p2

0 − |p|2)ψ = 0

so are functions with support on the positive (p0 = |p|) and negative (p0 = −|p|)energy null-cone. These are Fourier transforms of solutions to the masslessKlein-Gordon equation

(− ∂2

∂x20

+∂2

∂x21

+∂2

∂x22

+∂2

∂x23

)ψ = 0

Recall that our general analysis of irreducible representations of the Poincaregroup showed that we expected to find such representations by looking at func-tions on the positive and negative energy null-cones, with values in representa-tions of SO(2), the group of rotations preserving the vector p. Acting on thespin-1/2 representation, the generator of this group is given by the followingoperator:

455

Definition (Helicity). The operator

h =σ · p|p|

is called the helicity operator. It has eigenvalues ± 12 , and its eigenstates are

said to have helicity ± 12 . States with helicity + 1

2 are called “left-handed”, thosewith helicity − 1

2 are called “right-handed”.

We see that a continuous basis of solutions to the Weyl equation for ψL isgiven by the positive energy (p0 = |p|) solutions

uL(p)ei(−p0x0+p·x)

where uL ∈ C2 satisfies

huL = +1

2uL

so the helicity is + 12 , and negative energy (p0 = −|p|) solutions

uL(p)ei(−p0x0+p·x)

with helicity − 12 . After quantization, this wave equation gives a field theory

describing masslessleft-handed particles and right-handed antiparticles. TheWeyl equation for ψR will give a description of massless right-handed particlesand left-handed antiparticles.

Show Lorentz covariance.Canonical formalism Hamiltonian, LagrangianSeparate sub-sections for the Dirac and Majorana cases

45.2 Quantum Fields

Start by recalling fermionic harmonic oscillatorWrite down expressions for the quantum fields in the various cases

45.3 Symmetries

Write down formulas for Poincare and internal symmetry actions.

45.4 The propagator

45.5 Quantum electrodynamics

Write down Hamiltonian, note problem of non-linearity of equations of motion.


Maggiore

456

Chapter 46

An Introduction to theStandard Model

46.1 Non-Abelian gauge fields

Explain non-abelian case.Note no longer quadratic.Asymptotic freedom.Why these groups and couplings?

46.2 Fundamental fermions

mention the possibility of a right-handed neutrinoSO(10) spinor.Problem, why these representations? Why three generations

46.3 Spontaneous symmetry breaking

Explain general idea with U(1) case. Mention superconductors.Higgs potential.Mass terms. Kobayashi-Maskawa.


457

Chapter 47

Further Topics

There’s a long list of topics that should be covered in a quantum mechanicscourse that aren’t discussed here due to lack of time in the class and lack ofenergy of the author. Two important ones are:

• Scattering theory. Here one studies solutions to Schrodinger’s equationthat in the far past and future correspond to free-particle solutions, witha localized interaction with a potential occuring at finite time. This isexactly the situation analyzed experimentally through the study of scat-tering processes. Use of the representation theory of the Euclidean group,the semi-direct product of rotations and translations in R3 provides in-sight into this problem and the various functions that occur, includingspherical Bessel functions.

• Perturbation methods. Rarely can one find exact solutions to quantummechanical problems, so one needs to have at hand an array of approxima-tion techniques. The most important is perturbation theory, the study ofhow to construct series expansions about exact solutions. This techniquecan be applied to a wide variety of situations, as long as the system inquestion is not too dramatically of a different nature than one for whichan exact solution exists.

More advanced topics where representation theory is important:

• Conformal geometry. For theories with massless particles, conformal groupas extension of Poincare group. Twistors.

• Infinite dimensional non-abelian groups: loop groups and affine Lie alge-bras, the Virasoro algebra. Conformal quantum field theories.

459

Appendix A

Conventions

I’ve attempted to stay close to the conventions used in the physics literature,leading to the choices listed here. Units have been chosen so that ~ = 1.

To get from the self-adjoint operators used by physicists as generators ofsymmetries, multiply by −i to get a skew-adjoint operator in a unitary repre-sentation of the Lie algebra, for example

• The Lie bracket on the space of functions on phase space M is given bythe Poisson bracket, determined by

q, p = 1

Quantization takes 1, q, p to self-adjoint operators 1, Q, P . To make thisa unitary representation of the Heisenberg Lie algebra h3, multiply theself-adjoint operators by −i, so they satisfy

[−iQ,−iP ] = −i1, or [Q,P ] = i1

In other words, our quantization map is the unitary representation of h3

that satisfies

Γ′(q) = −iQ, Γ′(p) = −iP, Γ′(1) = −i1

• The classical expressions for angular momentum quadratic in qj , pj , forexample

l1 = q2p3 − q3p2

under quantization go to the self-adjoint operator

L1 = Q2P3 −Q3P2

and −iL1 will be the skew-adjoint operator giving a unitary representationof the Lie algebra so(3). The three such operators will satisfy the Liebracket relations of so(3), for instance

[−iL1,−iL2] = −iL3

461

For the spin 12 representation, the self-adjoint operators are Sj =

σj2 ,

the Xj = −iσj2 give the Lie algebra representation. Unlike the integerspin representations, this representation does not come from the bosonicquantization map Γ.

Given a unitary Lie algebra representation π′(X), the unitary group actionon states is given by

|ψ〉 → π(eθX)|ψ〉 = eθπ′(X)|ψ〉

Instead of considering the action on states, one can consider the action onoperators by conjugation

O → O(θ) = e−θπ′(X)Oeθπ

′(X)

or the infinitesimal version of this

d

dθO(θ) = [O, π′(X)]

If a group G acts on a space M , the representation one gets on functions onM is given by

π(g)(f(x)) = f(g−1 · x)

Examples include

• Space translation (q → q + a). On states one has

|ψ〉 → e−iaP |ψ〉

which in the Schrodinger representation is

e−ia(−i ddq )ψ(q) = e−addqψ(q) = ψ(q − a)

So, the Lie algebra action is given by the operator −iP = − ddq . On

operators one hasO(a) = eiaPOe−iaP

or infinitesimallyd

daO(a) = [O,−iP ]

• Time translation (t → t − a). The convention for the Hamiltonian H isopposite that for the momentum P , with the Schrodinger equation sayingthat

−iH =d

dt

On states, time evolution is translation in the positive time direction, sostates evolve as

|ψ(t)〉 = e−itH |ψ(0)〉

462

Operators in the Heisenberg picture satisfy

O(t) = eitHOe−itH

or infinitesimallyd

dtO(t) = [O,−iH]

which is the quantization of the Poisson bracket relation in Hamiltonianmechanics

d

dtf = f, h

Conventions for special relativity.Conventions for representations on field operators.Conventions for anticommuting variables. for unitary and odd super Lie

algebra actions.

463

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Documents

algebra representations

unitary group representations

spin groups

unitary groups

representations of so3

characters of representations

symmetry groups

translation groups