Introduction to Gauge Theory

Bernd SchroersHeriot-Watt Universityb.j.schroers@hw.ac.uk

SMSTC Advanced Course on Gauge Theory, 2013/14

Outline of Lecture

Introduction

Schrodinger-Maxwell Theory

A Review of Differential Geometry

Gauge Theory on Open Sets

The inventors of gauge theory

Figure : James Clerk Maxwell and Hermann Weyl

A fruitful failure

Length scale (‘gauge’) depends on spacetime?

Parallel transport of a length scale ` ∈ R in terms of 1-formA = Atdt + A1dx1 + A2dx2 + A3dx3:

d` = −A`, (1)

Change the gauge `′ = λ`, λ : R4 → R+.Then

d`′ = λd`+ dλ` = (−λA + dλ)` = λ(−A + d lnλ)`.

In order to maintain the condition (1) in the new gauge werequire

A′ = A− d lnλ.

F = dA is unchanged!Electromagnetic field? Einstein: ruled out by experiment

What is gauge theory?

I All measurements depend conventions and ‘gauges’ -physics does not!

I Einstein’s General Relativity is a ‘gauge theory ofspacetime’: spacetime frame as gauge.

I Gauge theories now used in physics, mathematics,economics and finance.

I Here: gauge theory of ‘internal degrees of freedom’(compact Lie groups)

I The unreasonable effectiveness of gauge theories inmodern physics and mathematics. Why?

What is gauge theory?

I All measurements depend conventions and ‘gauges’ -physics does not!

I Einstein’s General Relativity is a ‘gauge theory ofspacetime’: spacetime frame as gauge.

I Gauge theories now used in physics, mathematics,economics and finance.

I Here: gauge theory of ‘internal degrees of freedom’(compact Lie groups)

I The unreasonable effectiveness of gauge theories inmodern physics and mathematics. Why?

What is gauge theory?

I All measurements depend conventions and ‘gauges’ -physics does not!

I Einstein’s General Relativity is a ‘gauge theory ofspacetime’: spacetime frame as gauge.

I Gauge theories now used in physics, mathematics,economics and finance.

I Here: gauge theory of ‘internal degrees of freedom’(compact Lie groups)

I The unreasonable effectiveness of gauge theories inmodern physics and mathematics. Why?

What is gauge theory?

I All measurements depend conventions and ‘gauges’ -physics does not!

I Einstein’s General Relativity is a ‘gauge theory ofspacetime’: spacetime frame as gauge.

I Gauge theories now used in physics, mathematics,economics and finance.

I Here: gauge theory of ‘internal degrees of freedom’(compact Lie groups)

I The unreasonable effectiveness of gauge theories inmodern physics and mathematics. Why?

What is gauge theory?

I All measurements depend conventions and ‘gauges’ -physics does not!

I Einstein’s General Relativity is a ‘gauge theory ofspacetime’: spacetime frame as gauge.

I Gauge theories now used in physics, mathematics,economics and finance.

I Here: gauge theory of ‘internal degrees of freedom’(compact Lie groups)

I The unreasonable effectiveness of gauge theories inmodern physics and mathematics. Why?

Outline of Lecture

Introduction

Schrodinger-Maxwell Theory

A Review of Differential Geometry

Gauge Theory on Open Sets

Schrodinger Equation

Wavefunction of free particle non-relativistic quantummechanics is a map ψ : R4 → C which obeys

i~∂tψ = − ~2

2m∆ψ. (2)

Normalised∫R3 |ψ(t ,x)|2d3x = 1, ∀t ∈ R, so that the

probability of the particle being in a region R ⊂ R3 at time t is

p(t ,R) =

∫R|ψ(t ,x)|2d3x .

The probability is invariant under a ‘phase change’

ψ 7→ ψ′ = eiχψ, χ : R4 → [0,2π).

Modifying the Schrodinger Equation

Introduce functions at ,a1,a2,a3 on R4 and

Dt = ∂t + iat , Dj = ∂j + iaj , j = 1,2,3.

Then

i~Dtψ = − ~2

2m

3∑j=1

D2j ψ, (3)

respects local phase rotations if we also map

at 7→ a′t = at − ∂tχ, aj 7→ a′j = aj − ∂jχ.

Reason: D′tψ′ = eiχDtψ

Gauging the Schrodinger Equation

Introduce gauge potential a on R4 and covariant derivatives

Dt = ∂t + iat , Dj = ∂j + iaj , j = 1,2,3.

Then the minimally coupled/gauged Schrodinger equation

i~Dtψ = − ~2

2m

3∑j=1

D2j ψ, (4)

is covariant if we apply gauge transformation

at 7→ a′t = at − ∂tχ, aj 7→ a′j = aj − ∂jχ.

Reason: Covariance D′tψ′ = eiχDtψ

Maxwell theory from gauging

In vector field notation, at 7→ a′t = at − ∂tχ, a 7→ a −∇χ,leaves invariant

e = ∇at − ∂ta, b = ∇× a.

Electric and magentic field? Yes:

∇ · b = 0, ∇× e + ∂tb = 0 :

Homogeneous Maxwell equations: ‘no magnetic monopoles’and Faraday’s law of induction. With electric charge density ρand a current density j :

∇ · e = ρ, ∇× b − ∂te = j .

Gauss and Ampere-Maxwell⇐ Schrodinger-Maxwell action.

Einstein’s objection revisited: Aharonov-Bohm

Outline of Lecture

Introduction

Schrodinger-Maxwell Theory

A Review of Differential Geometry

Gauge Theory on Open Sets

Notation

1. U ⊂ Rn an open set2. x = (x1, . . . , xn) Cartesian coordinates on U ⊂ Rn

3. u = (u1, . . . ,un) arbitrary coordinates - e.g. polar,cylindrical, ...

4. (t , x1, x2, x3) for Cartesian coordinates on R4 interpreted asspacetime

5. ∂i = ∂∂xi

etc6. V vector space

Vector field = directional derivative

DefinitionFor each p ∈ U, the tangent space TpU to U at p is the set ofvelocities of all curves that pass through p. An element of TpUis called a tangent vector and if c is a curve that passes throughp, we let c′(0) denote the corresponding element in TpU.

Think of vector fields on U ∈ Rn as directional derivatives andwrite the action of a vector field v on a function f ∈ C∞(U) asv [f ]:

v = 2x1∂1 − 3x1x2x3∂2, v [f ] = 2x1∂1f − 3x1x2x3∂2f

Differential 1-forms

Recall dual vector space = space of linear maps V → R

DefinitionA 1-form at p ∈ U ⊂ Rn is a linear map Tp(U)→ R. Adifferential 1-form on U ⊂ Rn is a smooth choice for each p ∈ Uof a 1-form at p. If α is a 1-form and v a vector field, we writeα(v) for the function which is the result of applying α to v .

dxj (∂k ) =

1, if j = k0, otherwise.

If α = 3dx1 + 2x3dx2 − dx3 then

α(x2∂1) = 3x2

Exterior Derivative

DefinitionGiven a function f on U, its exterior derivative is the 1-form dfdefined by

(df )(v) = v [f ].

It is easy to check in coordinates that

df =n∑

i=1

∂i f dxi .

Differential k -forms

DefinitionA k-form at p ∈ U ⊂ Rn is a map TpU × · · ·×TpU → R which is

I multilinear, i.e., linear in each of its k arguments,I antisymmetric in any two of its arguments.

A differential 1-form on U ⊂ Rn is a smooth choice for eachp ∈ U of a k -form at p. The first argument of a k -form denotesthe dependence on the point. Write Ωk (U) for the vector spaceof k -forms.The integer k is called the degree of the form. A zero-form is afunction on U.Important extension: Write Ωk (U,V ) for V -valued k -forms.

Wedge Product and Exterior Derivatives of k -forms

With

α = x3dx1 ∧ dx2, v1 = x2∂1 + ∂3, v2 = x2∂2 + x1∂1

we have

α(v1, v2) = x3(dx1(v1)dx2(v2)− dx1(v2)dx2(v1)) = x3(x22 − x1).

Also

d(sin(x3)dx1∧dx2) = cos(x3)dx3∧dx1∧dx2 = cos(x3)dx1∧dx2∧dx3.

Some Important Rules

(1) ddα = 0 ∀ α ∈ Ωk (U). (Schwarz!)(2) α ∧ β = (−1)degα degββ ∧ α.(3) d(α ∧ β) = (dα) ∧ β + (−1)degαα ∧ dβ.

...and some terminology:

DefinitionA k -form α is called closed if dα = 0. It is called exact if thereexists a (k − 1)-form β so that α = dβExact forms are necessarily closed.

Integration of Forms over Open Sets

To integrate k -form over U ⊂ Rk , need an orientation:

DefinitionAn orientation of U ∈ Rk is determined by a nowhere vanishingk -form ω on U. A coordinate system u1, . . . ,uk is oriented if theassociated basis ∂1, . . . , ∂k of the tangent spaces TpUsatisfies

ω(p,∂

∂u1, . . .

∂

∂uk) > 0 ∀p ∈ U.

E.g. orientation on R2 is given by dx1 ∧ dx2Integrate a k -form α over an open set U with orientation:∫

Uα =

∫Uα

(u,

∂

∂u1, . . .

∂

∂uk

)du1 . . . duk .

Independent of coordinate choice!

Integration over Submanifolds

Assuming the submanifold R ⊂ RN is parametrised by an openset U ∈ Rk with coordinate u = (u1, . . . ,uk ) and by a mapγ : U → Rn, we define the integral by ‘pulling back’ the form tothe parameter space:∫

Rα =

∫α

(γ(u),

∂γ

∂u1, . . .

∂γ

∂uk

)du1 . . . duk .

Stokes’ Theorem

TheoremLet R be a compact, oriented k-dimensional submanifold of Rn

and α ∈ Ωk−1(U) for an open set U ⊂ Rn containing R. Denotethe boundary of R with the induced orientation by ∂R. Then∫

Rdα =

∫∂Rα.

k = 3: Gauss’ divergence theorem∫

R∇ · E =∫∂R E · ndA

k = 2: Stokes’ theorem∫

R(∇× E) · ndA =∫∂R E · d`

Metrics

DefinitionIf V is a real vector space, a metric η is a bilinear mapV × V → R which is

1. symmmetric, i.e.,∀v ,w ∈ V , η(v ,w) = η(w , v),2. non-degenerate, i.e., ∀v , η(v ,w) = 0⇒ w = 0

A metric on an open set U ⊂ Rn is a smooth choice of metricson TpU for each p ∈ U.Isomorphism v ∈ V 7→ η(v , ·) ∈ V ∗ ⇒ Induces a metric on V ∗

and tensor powers of V ∗!Minkowski space is R4 equipped with the metricη = dt2 − dx2

1 − dx22 − dx2

3

The Hodge Star Operator

DefinitionWith orientation ω of U, the Hodge star operator? : Ωk (U)→ Ωn−k (U) is defined via

α ∧ (?β) = η(α, β) ω ∀α, β ∈ Ωk (U).

In terms of an orthonormal and oriented basis e1, . . . ,en of1-forms:

?(ei1 ∧ ei2 ∧ · · · ∧ eik ) = ηi1i1 . . . ηik ik eik+1 ∧ eik+2 ∧ · · · ∧ ein ,

where i1, · · · ik , ik+1 · · · in is an even permutation of1,2, · · · n and ηij = η(ej ,ek )

Self-dual Forms

Note that ? ? α = (−1)k(n−k)+sα, ∀α ∈ Ωk (U)

DefinitionA k -form α is called self-dual if ?α = α. It is called anti-self-dual?α = −α

ExampleOn Minkowski space, with orientation ω = dt ∧ dx1 ∧ dx2 ∧ dx3

?(dt ∧ dx1) = −dx2 ∧ dx3, ?(dx2 ∧ dx3) = dt ∧ dx1

Outline of Lecture

Introduction

Schrodinger-Maxwell Theory

A Review of Differential Geometry

Gauge Theory on Open Sets

Maxwell Electrodynamics Revisited

Assemble gauge potentials at ,a1,a2,a3 into a 1-form on R4,

a = atdt + a1dx1 + a2dx2 + a3dx3,

to find electromagnetic field strength

f = da = −3∑

i=1

eidt∧dxi +b1dx2∧dx3+b2dx3∧dx1+b3dx1∧dx2,

Further defining a current 3-form as

j = ρdx1∧dx2∧dx3−j1dt∧dx2∧dx3−j2dt∧dx3∧dx1−j3dt∧dx1∧dx2,

Maxwell’s equations are df = 0, d ? f = j .

Maxwell Electrodynamics is ...

1. First gauge theory2. First special relativistic field theory3. Prototype for Standard Model4. First arena of ‘duality’ in physics

Ingredients of Schrodinger-Maxwell Theory

1. The ‘phase rotation’ eiχ, with χ ∈ [0,2π),2. The wavefunction ψ ∈ Ω0(R4,C),3. The gauge field a ∈ Ω1(R4),4. The modified derivative d + ia of ψ,5. The field strength f = da ∈ Ω2(R4),6. Local gauge changes eiχ, for χ : R4 → [0,2π) on ψ and a.

Comparing Schrodinger-Maxwell and Weyl’s theory

I Phase rotation eiχ are elements of U(1). Weyl’s scalinggroup R+ is also a Lie group,

I The Lie algebra (tangent space at the identity) of U(1) isspanned by i. Interpret A = ia as a Lie-algebra valued1-form, so DA = d + A

I Acting on a k -form,

(d + A)2α = (d + A)(dα+ A∧α) = dA∧α−A∧dα+ A∧dα

Writing F = dA or F = if , we thus have D2Aα = F ∧ α.

Gauge ingredients

In order to define a gauge theory on U ⊂ Rn we require, ingeneral,

1. A Lie group G, whose Lie algebra we denote by g.2. A representation ρ of G on a vector space V . We denote

the associated representation of g on V by ρ∗.

Gauge recipe

1. Gauge changes at a point p ∈ U are implemented byelements of G: gauge group

2. ‘Wavefunctions’ become local sections: Ωk (U,V ).3. A local gauge field A ∈ Ω1(U, g),4. The covariant exterior derivative DA := d + A acts on local

sections

DA : Ωk (U,V )→ Ωk+1(U,V ), φ 7→ dφ+ ρ∗(A) ∧ φ.

5. The curvature/field strength is defined via the covariantderivative

(DA)2φ = ρ∗(FA) ∧ φ.

6. Local changes of gauge γ : U → G actA 7→ A′ = γAγ−1 + γdγ−1 and φ 7→ φ′ = ρ(γ)φ.

Working with vector-valued forms

Expand A ∈ Ω1(U, g) as A =∑d

α=1 Aαtα, for ordinary 1-formsAα. Then,

[A,A] =d∑

α,β=1

Aα ∧ Aβ[tα, tβ].

Applying the representation ρ∗ we note the useful rule

ρ∗([A,A]) =d∑

α,β=1

Aα ∧ Aβ[ρ∗(tα), ρ∗(tβ)]

=d∑

α,β=1

Aα ∧ Aβ(ρ∗(tα)ρ∗(tβ)− ρ∗(tβ)ρ∗(tα)

= 2ρ∗(A) ∧ ρ∗(A). (5)

Curvature and Gauge Transformations

(DA)2φ

= (d + ρ∗(A))(dφ+ ρ∗(A) ∧ φ)

= ρ∗(A) ∧ dφ+ ρ∗(dA) ∧ φ− ρ∗(A) ∧ dφ+ ρ∗(A) ∧ ρ∗(A) ∧ φ

= ρ∗(dA) ∧ φ+12ρ∗([A,A]) ∧ φ, (6)

ThusFA = dA +

12

[A ∧ A]

or FA = dA + A ∧ A for matrix groups.Under gauge change FA′ = γFAγ

−1, and DA′φ′ = ρ(γ)DAφ.

Gauge Theories in Physics

Write down gauge-invariant action

S = −∫

Uκ(FA ∧ ?FA) + tr (DAφ ∧ ?DAφ) + W (φ)

where κ is the (negative definite) Killing form on g andW : V → R is a potential.Require G-invariance W (ρ(g)(φ)) = W (φ)‘Mexican hat potential’ for a field φ in the definingrepresentation of a matrix group G:

W (φ) = (λ+ tr (φ2))2, (7)

where λ is a parameter which sets a mass scale in the theoryvia the Higgs effect.