lecture 10 - university of...

Lecture 10Lecture 10

The Dirac equationThe Dirac equation

WS2010/11WS2010/11: : ‚‚Introduction to Nuclear and Particle PhysicsIntroduction to Nuclear and Particle Physics‘‘

The Dirac equation is a relativistic quantum mechanical

wave equation formulated by British physicist Paul Dirac

in 1928.


• The Dirac equation provides a description of elementary spin-½

particles, such as electrons, consistent with both the principles of

quantum mechanics and the theory of special relativity.

• The equation demands the existence of antiparticles and actually

predated their experimental discovery.

This made the discovery of the positron, the antiparticle of the

electron, one of the greatest triumphs of modern theoretical physics.

1902 1902 -- 19841984

Dirac found the equation for free relativistic fermions under the requirement :

1) that the wave-equation – as in case of the Schrödinger equation – should be of

1st order in �/�t ��/�x0 and

2) to allow for a continuity equation with a positive density �*� .

3) Relativistic covariance (with respect to the Lorentz transformation) then

requires that the wave-equation also has to be of 1st order in the space

derivatives �/�xk (k = 1, 2, 3) .

Accordingly the equation has to be of the general form


(1)

(2)with a restmass M and an hermitean Hamiltonian

α, βα, βα, βα, β – unknown (yet!) coefficients

If the equation should describe free particles it‘s solutions should be plane waves

with the proper dispersion relation for particles with restmass M,

(3)with

The Dirac equation The Dirac equation

(4)

(5)

By comparing the quadratic form of (1), symmetrized in the indices k, l ,

with the Klein-Gordon-equation (4)

one has to require that the unknown coefficients �k (k = 1, 2, 3) and �

must have the properties

(6)

(7)

The conditions (6) can only be fulfilled for matrices. Due to Eq. (2) these

matrices have to be hermetian:

The eigenvalues then are real and according to (6) can only be +1 .

In this case the wavefunction � (or each component of �) has to be a solution also

of the Klein-Gordon-Equation:


These matrices must be traceless. Accordingly, the dimension of the matrices

has to be an even number. In 2 dimensions, however, there are only 3 linearly

independent matrices that anticommute, i.e. the Pauli matrices:

(9)

(10)

or unitary transformations of (10). Accordingly, the �k, � must be at least 4x4

matrices. In Pauli-Dirac standard notation (with 2x2 submatrices) these are

given by

(11)

Unitary equivalent representations are also possible.

(8)

Furthermore, since �k = −��k� – due to cyclic invariance of the trace ��


The wavefunction �(x) in (3) then must have 4 components

(12)

Its adjoint is

and fulfills the equation

(13)

(14)

By multiplication of (1) with �† from left, multiplication of (14) with � from

right and subtraction we obtain the continuity equation

(15)

with the positive definite density

(16)

as desired.

�-matrices and Dirac-algebra

For the analysis of transformation properties it is more convenient to introduce

an equivalent set of matrices which is obtained as follows:

(17)

(18)

The Dirac-equation then reads:.

with the Pauli-adjoint Spinor(19)

(20)

(21)

For the �-matrices Eqs. (6)

then become more compact:

and Eq. (7) leads to the pseudo-hermeticity


This can be easily proven in the standard representation (EX)

(22)

(23)

In the standard representation �1, �3 are real and �2 is imaginary.

and describes the vanishing of the four-divergence of the four-current j�(x).

With the help of the 4-� matrices one can construct a complete set of 4x4

matrices.

Since a general 4x4 has 16 independent elements there also have to be 16 basis

elements. For convenience one chooses the basis:


(23)

The continuity equation (15) then reads (EX)

1 matrix (4 x4 unit matrix)

4 matrixes

6 matrixes (antisymmetric)

4 matrixes

1 matrix

(24)

with


where �5 is the ‚chirality‘ matrix given by

(25)

This matrix can also be written as

(26)

(27)if (µ,ν,ρ,σ)(µ,ν,ρ,σ)(µ,ν,ρ,σ)(µ,ν,ρ,σ) is an even permutation of (0,1,2,3)

if (µ,ν,ρ,σ)(µ,ν,ρ,σ)(µ,ν,ρ,σ)(µ,ν,ρ,σ) is an odd permutation of (0,1,2,3)

in all other cases

if (µ,ν,ρ,σ)(µ,ν,ρ,σ)(µ,ν,ρ,σ)(µ,ν,ρ,σ) is an even permutation of (0,1,2,3)

if (µ,ν,ρ,σ)(µ,ν,ρ,σ)(µ,ν,ρ,σ)(µ,ν,ρ,σ) is an odd permutation of (0,1,2,3)

in all other cases

with the complete antisymmetric unit tensor of fourth order.

The matrix �5 has the properties (EX)

(28)


(29)

The nonzero elements of the matrices �µνµνµνµν are those from nondiagonal spacial

components (�, �) = (k, l) with k unequal l and (�, �) = (0, k). These elements can

be expressed as

(30)

(31)where

(32)

(33)

is the ε ε ε ε -tensor in 3 dimensions.

The three 4x4 matrices (29) have the properties of a set of spin-matrices,

i.e (EX)

which is proven explicitly in the standard representation


Where �m are again the Pauli spin-matrices. In this representation we have

furthermore

(34)

Hermetian properties are fixed by (21), (28) and read

The 16 matrices

(35)

(36)

build up (except for ΓΓΓΓ1) a traceless orthonormal set of matrices with the scalar

product

(37)

i.e.

(38)

and


(39)

In particular, these matrices are linearly independent.

Accordingly any 4x4 matrix M can be written as a linear combination in the

basis (36):

Products of arbitrary matrices thus may also be written in terms of an

expansion in the ΓΓΓΓ-matrices.

For a couple of such products we present the result in the following

‚multiplication table‘:


(40)


(41)

The system of matrices (36) with the multiplication table (40) is denoted as

Dirac-Clifford-algebra.

An application of (40) is the summation over spin-matrices in transition

amplitudes.

Further examples are often appearing products such as:

if n - odd

Contractions with arbitrary four-vectors a�, bv are written in shorthand form

e.g.

etc. In this context one also has to mention �-identities such as:


(42)

(43)

(44)

Nowadays the Dirac-Clifford-algebra is encoded in software routines

such as Mathemetica, REDUCE, FORM, DERIVE etc.

Transformation of Dirac spinors

(45)

In case of homogenuous Lorentz transformations

the Dirac-equation must be form-invariant according to the principle of special

relativity, i.e. read as

(46)

(47)

For ��(x�) we have to allow for a linear mixing of the components of �(x),

which requires to define a transformation in Hilbert space, i.e.

or

with

with a ΛΛΛΛ dependent 4 x4 matrix Sαβαβαβαβ((((ΛΛΛΛ)))).... ΤΤΤΤhe equations for S are obtained by

insertion of (47) in (46); multiplication from left by S−1 gives

(48)


Comparing (48) with the original Dirac-equation shows that the term [. . .] � �νννν,

i.e. the transformation in Hilbert space must be aquivalent to a Lorentz

transformation (49)

For Lorentz transformations with det (ΛΛΛΛ) = +1 and ΛΛΛΛ00 � 1 we employ the

Ansatz(50)

with an antisymmetric parameter-matrix r�v = −rvµµµµ.

For infinitesimal transformations ΛΛΛΛ=1 + r (|r|<< 1) we get

(51)

(52)

For the matrices µνµνµνµν we obtain according to (49) the condition


Using the multiplication table (40) one identifies the unique solution

(53)

A four-component field function �αααα(x), which transforms under homogeneous

Lorentz transformations (47) as

is called a Dirac-Spinor.

The 4x4 matrices S(ΛΛΛΛ) – acting in the Hilbert space – follow:

(54)

(55)

The group transformations are not unitary, but due to the real and antisymmetric

parameter-matrix r�v follow:

(56)


We now consider (54) in particular for rotations and Lorentz boosts for which we

obtain:

for a rotation with vector

for a boost with velocity

An expansion of the exp.-function (54) and using the matrix identities (40) gives:

(57)

(58)

1) For arbitrary complex vectors a,b a pure rotation leads to a

unitary transformation (with )


In standard representation this corresponds to a doubling of the Pauli matrices

for two-component spinors.

It has the characteristic properties for fermions, i.e. for

we get: (59)

2) In case of pure Lorentz boosts we obtain the non-unitary transformation:

(60)

or with the four-unit vector

(61)with

or


For parity transformations we have to allow for a phase � = ±1 (� is the eigenvalue

of the parity transformation) as in case of the Klein-Gordon field . The condition

(49) ( ) leads to:

(62)

which is solved by (63)

Eq. (47) then reads:(64)

�� The Dirac equation is form-invariant under these parity transformations.

For the particular matrix �5 this implies:

(65)

whereas – due to the matrices �µνµνµνµν - one finds:

(66)


From the transformation � � S� follows for the Pauli-adjoint spinor:

(67)

Accordingly a term with arbitrary Dirac spinors is always a Lorentz

scalar:

(68)

while is invariant under ordinary Lorentz transformations

but changes sign under parity transformations, i.e. it is a pseudo-scalar:

(69)


scalar

4-vector

tensor of 2nd order

pseudovector

pseudoscalar

In particular the current (23) is a 4-vector.

By analyzing the residual matrices of the basis (24) and applying (49) one

finds five different bilinear covariants with the particular transformation

properties:

(70)

Solution of the free Dirac equation

The solution of the Dirac equation are plane waves (3).

In order to find the yet unknown spinors �(0) (or a suitable basis) we consider

solutions for positive and negative energies, separately:

with

(71)

By insertion in Eq. (1) we obtain:

(72)

(73)or in �-notation:

In order to specify the spinors u and υυυυ we consider first the case p = 0 (i.e. we

evaluate the spinor in the rest frame of the particle). In this frame (73) reads:

(74)


(75)

Now we can choose for u(0) and υυυυ(0) two orthogonal eigenspinors to �0 with

eigenvalues +1 and −1 , which are denoted by .

Since Tr (�0) = 0 each eigenvalue appears twice. Furthermore, the matrices

(29) commute with �0 and we can choose ur(0) and υυυυr(0) as eigenspinors of

with eigenvalues s= +1, where aa is an arbitrary unit vector because according to

(57) we have :

In standard representation one chooses a = (0, 0, 1) which leads to

(76)

The 4 basis spinors are normalized according to:

(77)

Accordingly we get the orthogonality relation:

(78)

since u and υυυυ belong to different eigenvalues of �0 .


The completeness relation then reads:

(79)

and any bilinear expansion of �0 wrt. eigenspinors reads as:

(80)

Consequently, the matrices

with

(81)

(82)

have the properties of projection operators on solutions (with p = 0) for positive

and negative energies +M .


In standard representation these relations become immediately apparent:

(83)

(84)

and the 4x4 matrices read

In order to get solutions for we have to apply a Lorentz transformation,

i.e. the transformation (61) with the proper Lorentz boost:

(85)

The appropriate boost velocity is given by:

(86)


According to (60) one gets with the parameters of (61) :

and

with

(87)

(88)

The solution is according to (60), (61):

(89)

Now the spinors for Dirac particles with momentum p read:

(90)


These are solutions of (73) since

(91)

Since the lefthand sides of (77) are Lorentz invariants the normalizations read:

(92)

In order to evaluate the general projectors on solutions with positive and

negative energy

(93)

we use

(94)


and obtain (EX)(95)

(96)

The completeness relation (79) then reads:

Note: the eigenvalue equation (76) does not hold for finite p, since ΣΣΣΣ3 does not

commute with Sp .

In standard representation the 4 basis spinors are given by:

(97)

or explicitly:


(98)

(99)

The matrix elements with these u- and υυυυ-spinors are easily calculated to give (EX)

(100)

such that the matrix elements of the Dirac-equation (73) (after multiplication with

M) read:

(101)

which is equivalent to the free dispersion relation for a

particle of mass M.


(102)

In the representation (100) – in the nonrelativistic limit |p| << M – the lower

components of us become smaller by a factor |p|/(2M) than the upper

components, while for υυυυs the relations are opposite.

The relation between us and υυυυs may be expressed according to (25) as (EX):

i.e. the chirality matrix �5 exchanges u- and υυυυ-spinors.

Finally, the explicit form of the Lorentz boost for the transformation (85) reads

(using NE = M(Ep +M)) (EX):

(103)

lecture 10 - university of...

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