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Quantum Electrodynamics

1

D. E. Soper

2

University of Oregon

Physics 666, Quantum Field Theory

April 2001

1 The action

We begin with an argument that quantum electrodynamics is a natural ex- tension of the theory of a free Dirac field, with action

S[¯ψ,ψ] =?

d

4x¯ψ(x){i/∂-m}ψ(x).(1)

Notice that this action is invariant under the transformation

ψ(x)→e-iQeαψ(x)

¯ψ(x)→¯ψ(x)eiQeα(2)

HereQeis a constant that tells us how much to rotate the fieldψunder the rotation specified by the parameterα. We sill later identifye= +|e| with the proton electric charge. ThenQewill be the charge of the particle annihilated by the fieldψ. That is,Q=-1 for an electron. This notation allows us to describe several fields,ψJ, each of which transforms as above with chargeQJe. Note that Peskin and Schroeder use the symboleto mean -|e|. I think that"s confusing. The action isnotinvariant under the transformation

ψ(x)→e-iQeα(x)ψ(x)

in whichαdepends on the space-time pointx. This is called a gauge thans- formation. In fact, we have

S[¯ψ,ψ]→?

d

μγμ-m}ψ(x).(4)1

Copyright, 2001, D. E. Soper

2soper@bovine.uoregon.edu

1 Let us try to extend tha theory so that it is invariant under gauge transfor- mations. We add a fieldAμ(x) with a transformation law A

μ(x)→Aμ(x) +∂α(x)∂x

μ.(5)

We take

S[¯ψ,ψ,A] =?

d

4x¯ψ(x){i/∂-Qe/A(x)-m}ψ(x) +···.(6)

The +···indicates that we are going to have to add something else. But we can note immediately that the terms we have so far are invariant under gauge thransformations. We need to add a "kinetic energy" term forAμ, something analogous to the integral of 12 ∂μφ∂μφfor a scalar field. Whatever we add should be gauge invariant by itself. We take

S[¯ψ,ψ,A] =?

d

Fμν(x)Fμν(x)?,(7)

whereFμν(x) is a convenient shorthand for F μν(x) =∂μAν(x)-∂νAμ(x).(8) We see thatFμν(x) is gauge invariant, so the added term is gauge invariant also. From this action, we get the equation of motion forψ: {i/∂-Qe/A(x)-m}ψ(x) = 0,(9) which is the Dirac equation for an electron in a potentialAμ(x). The equation of motion ofAμis

μFμν(x) =Jν(x),(10)

where J

ν(x) =Qe¯ψ(x)γνψ(x).(11)

This is the inhomogenious parts of the Maxwell equations for the electro- magnetic fields produced by a currentJν(x). The homogenious part of the Maxwell equations follows automatically becauseFμν(x) is expressed in terms of the potentialAμ(x).Exercise:Derive the equation of motion ofAμ.2

2 Coulomb gauge

We want to useAμ(x) as the canonical coordinates for the electromagnetic field. However, two of the four degrees of freedom per space point are illusory because of gauge invariance. For instance, if you had one solution of the eequations of motion based on initial conditions att= 0, you could always change if fort > t1>0 by making a gauge thansformation. For this reason, we will choose a gauge for the quantum theory, namely Coulomb gauge.quotesdbs_dbs7.pdfusesText_5