Lets show that the action is indeed invariant under

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Let’s show that the action is indeed invariant under this transformation. Let ψ = 0 η * . Then we can calculate ψγ μ ( μ - ieA μ ) ψ = η T γ 0 C γ 0 γ μ ( μ - ieA μ ) 0 η * = η T γ 0 γ 0 T γ μT γ 0 ( μ - ieA μ ) η * = - μ η γ 0 γ μ η + ieA μ η γ 0 γ μ η = - μ ηγ μ η + ieA μ ηγ μ η where we’ve used γ 0 T = - γ 0 . Thus inside of the action this term is Z d 4 xi ψγ μ ( μ - ieA μ ) ψ = Z d 4 xi ( - μ ηγ μ η + ieA μ ηγ μ η ) = Z d 4 xi ( ηγ μ μ η + ieA μ ηγ μ η ) = Z d 4 xi ηγ μ ( μ - ie ( - A μ )) η Since the kinetic term for the photon is obviously invariant under our trans- formation, we’ve found that the action is in fact invariant under ψ 0 ψ * , A μ → - A μ . One can also check that [ dψd ψ ] = [ dηd η ]. 2 This is actually a gauge dependent fact, since the correlator of a bunch of A μ ’s is not gauge invariant.
Phys 253a 3 Now we can carry through our argument from before, in a slightly slicker way: Z [ J ] = Z [ dAdψd ψ ] e iS ( A,ψ, ψ ) e R J μ A μ = Z [ dAdηd η ] e iS ( A,ψ, ψ ) e R J μ A μ = Z [ dAdηd η ] e iS ( - A,η, η ) e R J μ A μ = Z [ dAdηd η ] e iS ( A,η, η ) e - R J μ A μ = Z [ - J ] In the second line we used the invariance of the measure, in the third the invari- ance of the action shown above, and in the fourth we changed variables A → - A . Since a correlation function with an odd number of A ’s will involve an odd num- ber of derivatives with respect to J , it will vanish by the above identity, as desired. (e) Note that in our argument above we needed to use charge conjugation invari- ance of the lagrangian. Since this no longer holds in the electroweak theory, we expect violations of Furry’s theorem. Of course the violations should come from diagrams involving weak interaction vertices which violate the C symmetry. 2. (a) The amputated diagrams contributing to Compton scattering at one loop are I( s ) II( s ) III( s ) IV( s ) V VI
Phys 253a 4 plus u -channel versions of I through IV. (Note that switching the direction of the electron loop to get form diagram V to diagram VI is the same as switching the external photon lines.) (b) The sum of diagrams V and VI contains a photon three-point function, so
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