e i \u03b8 \u03bc J \u03bc 25 As usual it is convenient to work with the connected piece of

# E i θ μ j μ 25 as usual it is convenient to work

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] e i θ∂ μ J μ (25) As usual it is convenient to work with the connected piece of the generating function, Z [ J ] = e - iW [ J ] (26) where W [ J ] is the generator of all connected diagrams. We see that, W [ J μ - 1 ξ μ θ ] - θ ( x ) μ J μ = W [ J μ ] (27) 3
I’m considering infinitesimal θ so I can expand this and write, W [ J ] - d 4 x ∂W ∂J ( x ) 1 ξ μ θ - θ ( x ) μ J μ = W [ J ] (28) or, - d 4 x ∂W ∂J ( x ) 1 ξ μ θ - θ ( x ) μ J μ = 0 (29) This is a very strong constraint on the correlation functions. Notice that the derivative on W with respect to the source is pulling out a photon and we are dotting it into an incoming momentum. Let’s differentiate it once with respect to J ν ( y ) to get, 1 ξ x μ x 2 W ∂J μ ( x ) ∂J ν ( y ) + x μ δ μ ν δ ( x - y ) = 0 (30) evaluating at J μ = 0 this is simply the statement that, 1 ξ x μ x μν ( x - y ) = x ν δ ( x - y ) (31) In momentum space this is saying that, 1 ξ p 2 p μ μν ( p ) = p ν (32) Recall that the most general form of the propagator consistent with Lorentz invariance is, μν = 1 p 2 ( η μν - p μ p ν p 2 ) F ( p 2 ) + p μ p ν p 2 G ( p 2 ) (33) So we get from (32) the relation, G ( p 2 ) ξ = 1 (34) This implies that the non-transverse part must be ξ . The propagator is fixed to have the form μν = 1 p 2 ( η μν - p μ p ν p 2 ) F ( p 2 ) + ξ p μ p ν p 2 (35) This is true to any order in perturbation theory. There is no contribution to that coefficient. Let’s take a few more functional derivatives in (29), 1 xi x μ x n W ∂J μ ( x ) . . . ∂J ν N ( y n ) = 0 (36) This looks like, μ - BLOB - Crapcomingout (37) In momentum space this looks like the statement 1 ξ p μ p 2 1 p 2 ( η μν - p μ p ν p 2 ) F ( p 2 ) + ξ p μ p ν p 2 M ν ( p, . . . ) = 0 (38) and I can conclude that, p ν M ν ( p, . . . ) = 0 (39) 4
Exactly as I wanted to prove. However, in the example with only photons on the external states, this is sort of a little bit too trivial because it holds even off-shell (I haven’t assumed p μ to be on-shell). However, more generally, you must consider other external particles. Running through the same logic, the source term for the other fields is not gauge invariant. So you get that, W [ J μ - 1 ξ μ θ, Ke - ] - d 4 ( x ) μ J μ = W [ J, K ] (40) from which we can get the Ward identities and the fact that p μ M μ ( p, . . . ) = 0 only on shell. I would like to give you another way of thinking about it which for QED makes the argument very transparent. To begin I would like to tie up some loose end. We just said that our gauge fixing term depends on ξ but physical amplitudes cannot. We know that this is true from the general arguments we considered above. But, when we actually calculate scattering amplitudes we do it in

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