1 Abelian Gauge Theory

Remarks on Classical Gauge Theory

14.05.2012

As a start-up for gauge theory, let us collect some statements and properties. Let M be a mani- fold andGa group.

• For example, take the gauge groupU(1). Graphically, this is

M G

Figure 1: The group U(1) is a unitary group, denoted by a circle to emphasize the possible phases.

Define a product of the manifold with the group, P= M×G. On the group tube, we can locally define fibres F which are isomorphic toG, F'G, which are mapped to a point Π(F) in M. Π(F) is a projection of F on M. If we find a connection onG, then we can define a vertical vector (tangential to the fibre) and a horizontal vector.

G

M _Π (F)

• Let’s look atZ2×, a (rectangular) strip. Take the Möbius strip:

Figure 2: The Möbius Strip: A rectangle, twisted once and glued together at the ends; it only has one edge.

Locally, the shape and geometry of the möbius strip is just a rectangle. Globally, however, it is different and a lot more complicated.

1 Abelian Gauge Theory

FornDirac particles, the Lagrangian density is given by L=

n

X

j=1

iψ¯_jγ^µ∂_µψ_j−mψ¯_jψ_j

If the Lagragian density is left invariant under a transformation, this transformation is called a symmetry. This Lagrangian is invariant underglobal phase transformation,

ψj→ψ⁰_j(x)=exp (−qjΘ)ψj(x)

⇒ L → L⁰=L

Locally tranforming the phase can be interpreted as wandering along the fibre.

Noether’s theorem states that for every symmetry, there is a conserved quantity or current. For a global phase transformation, we find

j_µ(x)=X

k

qkψ¯kγ_µψk

There is also a conserved charge,

Q=Z

j₀(x)d³x

Now, if we do not only shift along the fibre, but also left and right, we get a local phase transformation, where the Lagrangian density is not preserved.

ψ_j→ψ⁰_j(x)=exp

q_jΘ(x) ψ_j(x)

⇒ L → L⁰,L

We need to find a new derivative such that the Lagrangian stays invariant: the covariant deriva- tiveDµ.

1.1 The covariant derivative

We are in need of a covariant derivative, such that (D_µψ_j)⁰=exp

q_iΘ(x)

D_µψ_j(x). WithD_µ, the Lagrangian will stay invariant,LL⁰=L.

Declare: D_µψ_j(x)B

h∂_µ+ieq_jA_µi

ψ_j(x), where e is a pasotive constant. Then we will have gauge invariance if we also transform the gauge field,A_µ:

A_µ(x)→A⁰_µ(x)=A_µ(x)+ f rac1e∂_µΘ(x) The commutator is

With this ansatz, D_µ transforms as it should. We find that h

∂_µA_ν−∂_νA_µi

= D_µν(x) is gauge invariant. Fµνis called Maxwell’s Field Strength in physics, or curvature in mathematics.

The Lagrangian density with the covariant derivative becomes L=ψ(i¯ D/−m)ψ−1

4F_µνF^µν with D/=X

µ

γ^µ(∂_µ+iea_µ)

Then, in fact,L⁰=L. The equations of motion are given by

∂_µF^µν=eψγ¯ ^νψ (i/∂−m)ψ=eAψ/ (iD/−m)ψ=0

The third equation is called the coraviant Dirac equation. These are classically Maxwell’s equations! We also call them Dirac-Maxwell equations of motion.

Remark If the gauge fieldA(”connection”) acquires a mass term in the Lagrangian,m²A_µA^µ, with the A_µ bosonic fields, gauge invariance in the Lagrangian would be destroyed. This for- malism allows only for massless bosons. Gauge invariance does not work for massive gauge particles! This seems like a problem because we know from experimental physics that there are massive gauge bosons in the theory of electroweak interaction. This is allowed by another process, namely spontaneous symmetry breaking, which leads to the Higgs formalism.

In a consistens theory, counterterms are local. Interactions are only dependent on γ_µ. In our case, the coupling term is ¯ψ /Aψ, and renormalizability allows only certain interactions between fermions and bosons.

2 Non-abelian gauge theory

Let us look at a global phase transformation again, this time in a more general form, not re- stricted toU(1). CallT^athe hermitian generators of some gauge group (not necessarily abelian), then the glbal phase shift will be

ψ(x)→ψ⁰(x)=exp (−iθ^aT^a)ψ(x) The corresponding Lie algebra is generated by the commutator,

hT^a,T^bi

= f^abcT^c

From the covariant derivative, we demand that for a local phase transformation, it behaves like D_µψ(x)0

=exp −iθ^a(x)T^a

| {z }

U(x)∈gG

D_µψ(x)

gGbelongs to the Lie algebraR.

2.1 The covariant derivative in non-abelian gauge theory

Take the same definition for the coraviant derivative as before:

D_µψ(x)=

∂_µ+A_µ(x) ψ(x) Aµ(x)=igA^a_µT^a

The couplinggis a positive constant, like theein abelian gauge theory. The covariant derivative transforms sensibly if the gauge field transforms like

A⁰_µ(x)=U(x)A_µ(x)U⁻¹(x)−

∂_µU(x)

U⁻¹(x)

=U(x)

A_µ+∂_µ

U⁻¹(x) Infinitesimally, the transformation is

δA_µ(x)=A⁰_µ(x)−A_µ(x)=∂_µθ(x)−h

A_µ(x),θ(x)i with [·,·] being the commutator in the Lie algebra, and withθ(x)Biθ^a(x)T^a.

⇒ δA^a_µ(x)= 1

g∂_µθ^a(x)+ f^abcθ^b(c)A^c_µ(x) The commutator of two covariant derivatives is

hD_µ,D_νi

ψ(x)=

∂_µA_ν(x)−∂_νA_µ(x)+ h

A_µ(x),A_ν(x)i

| {z }

=AaµT a AbνT b−AbνT b AaµT a

=Aaµ(x)Abν(x)[T a,T b]

[T a,T b]=f abcT c

Define the covariant quantityGµνas the commutator:

G_µν(x)=∂_µA_ν−∂_νA_µ+h A_µ,A_νi which tranforms like

G⁰_µν(x)=U(x)G_µν(x)U⁻¹(x) δG_µν(x)=igG^a_µνT^a

G^a_µν=∂_µA^a_ν−∂_νA^a_µ−g f^abcA^b_µA^c_ν For the coraviant derivative, we write

Dρ=∂_ρ+Aρ

D_ρ(f g)=(D_ρf)f+ f(D_ρg)

The covariant derivative fulfills the Leibniz rule. As a short notation, we introduce f,g≡ f^aT^a,g^aT^a

Then we get

Finally, we have arrived at theLagrangian density of a non-abelian gauge field coupled to a fermionic field:

L= 1 2g²Tr

G_µνG^µν

+ψ(i¯ D/−m)ψ

=−1 4

∂µA^a_ν−∂νA^a_µ

+g f^abcA^b_µA^c_ν∂^µA^aν−1

4g²fabdfabeA^bA^cA^dµA^eν

WE still have to add kinetic energy terms, these are only the interaction terms. We see that there is not only a cubic interaction between the gauge boson field and the fermion field, but also a cubic and quartic interaction between gauge fields. The possible vertices for self-interaction, which is only possible in non-abelian gauge theory, can be read offdirectly from the Lagrangian and are:

cubic interaction:

quartic interactions:

Both quartic interactions are of the orderg².

2.2 Noether currents

Last but not least, here is a list of Noether currents and tranformations.

A_µ→A⁰_µ=A_µ+δA_µ δ∂_νA_µ=∂_νδA_µ

δA⁰_µ=UδA_µU⁻¹

D_µν →G⁰_µν=G_µν+DµδA_ν−D_νδA_µ j_µ= j^a_µT^a

j^a_µ=ψγ¯ _µT^aψ D_µj^µ=0

The last line clearly indicates that the Noether current is covariantly conserved. Naturally, for a conserved current, we ask for a conserved charge,R

j⁰d³x, but this is not so easy, since we went from∂_µtoD_µ, integration does not simply work. We need Ward identities to solve this problem and they are a lot more complicated in non-abelian gauge theory than in abelian gauge theory.