Phys 253a
1
Problem Set 12 Solutions
December 4, 2010
1.
(a) The kinetic term for
φ
is
(
D
μ
φ
)
*
D
μ
φ
=
(
∂
μ

ieA
μ
)
φ
*
(
∂
μ
+
ieA
μ
)
φ
If we switch
φ
and
φ
*
, then clearly we should take
A
μ
→ 
A
μ
to leave the
Lagrangian invariant.
(b) We have an expression for the correlation function in terms of a path integral
h
0

T
{
A
μ
1
(
q
1
)
. . . A
μ
n
(
q
n
)
}
0
i
=
1
Z
Z
D
A
D
φ
D
φ
*
n
Y
j
=1
A
μ
j
(
q
j
)
e
iS
=
1
Z
Z
D
A
D
φ
D
φ
*
n
Y
j
=1
Z
d
4
x
j
A
μ
j
(
x
j
)
e

ix
j
·
q
j
e
iS
where
Z
=
R
D
A
D
φ
D
φ
*
e
iS
(and we could take the fourier transforms out of
the path integral if we like). To prove Furry’s theorem, consider the change of
variables
φ
0
=
φ
*
, φ
*0
=
φ
, and
A
0
=

A
.
The action is invariant
S
0
=
S
by
construction. The measure is also invariant
D
[

A
]
D
φ
*
D
φ
=
D
A
D
φ
D
φ
*
.
1
Thus, we get
Z
D
A
D
φ
D
φ
*
n
Y
j
=1
A
μ
j
(
q
j
)
e
iS
=
Z
D
[

A
]
D
φ
*
D
φ
n
Y
j
=1

A
μ
j
(
q
j
)
e
iS
0
=
Z
D
A
D
φ
D
φ
*
n
Y
j
=1

A
μ
j
(
q
j
)
e
iS
=
(

1)
n
Z
D
A
D
φ
D
φ
*
n
Y
j
=1
A
μ
j
(
q
j
)
e
iS
1
This is true because path integrals are defined using a regulator, and a regulated path integral behaves
like a finite dimensional integral (for instance, a latticeregularized path integral
is
finitedimensional). The
measure is clearly invariant under the given transformation if the integral is finitedimensional (say if we
integrate over only a finite number of modes of
A
,
φ
, and
φ
*
), so the path integral is invariant too. There are
problems with this argument if the regulator itself somehow isn’t invariant under the given transformation.
We’ll learn about this phenomenon (called an anomaly) next semester.
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Phys 253a
2
So the correlation function vanishes if
n
is odd.
(c) The correlation function is well defined if the
q
j
are offshell, so Furry’s theorem
holds then too.
2
This is useful, since offshell correlators appear as internal parts
of Feynman diagrams.
(d) The proof works essentially the same as before; we use charge conjugation in
variance of the action and measure. The only difference now is that charge con
jugation is slightly more complicated. We use the transformation
ψ
→
Cγ
0
ψ
*
,
A
μ
→ 
A
μ
, where
C
is defined by
C

1
γ
μ
C
=

γ
μT
.
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 '10
 SCHWARTZ
 Quantum Field Theory, correlation function, momenta, charge conjugation invariance, counterterm

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