]
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 nontransverse 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 offshell (I haven’t assumed
p
μ
to be
onshell). 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

iθ
]

d
4
xθ
(
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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 Fall '05
 Photon, Quantum Field Theory, Quantum chromodynamics, Gauge theory