528
12
Fields in Interaction
μ
(
p
,
p
)
=
(
1
+
L
)γ
μ
+
f
μ
(
p
,
p
).
(12.302)
The ambiguity in the definition of L is fixed as follows. Let us first show that, on the
general grounds of Lorentz covariance, if
p
=
p
,
the current
V
μ
(
p
,
p
)
= ¯
u
(
p
)
μ
(
p
,
p
)
u
(
p
)
is proportional, through a constant, to
¯
u
(
p
)γ
μ
u
(
p
).
By Lorentz
covariance we can indeed convince ourselves that
μ
(
p
,
p
),
which is a spinorial
matrix depending on
p
, can only be combination of the matrices
p
μ
1
and
γ
μ
.
Using
then the property
38
¯
u
(
p
)γ
μ
u
(
p
)
=
p
μ
m
¯
u
(
p
)
u
(
p
),
(12.304)
we conclude that
V
μ
(
p
,
p
)
= ¯
u
(
p
)
μ
(
p
,
p
)
u
(
p
)
=
f
0
¯
u
(
p
)γ
μ
u
(
p
),
(12.305)
f
0
being a constant. Actually, using Lorentz covariance and the gauge invariance con-
dition (
12.299
), one can show that the current
V
μ
(
p
,
p
)
can only have the following
general form
V
μ
(
p
,
p
)
= ¯
u
(
p
)
F
1
(
k
2
)γ
μ
+
F
2
(
k
2
)γ
μν
k
ν
u
(
p
),
(12.306)
where
k
=
p
−
p
.
39
It follows that
V
μ
(
p
,
p
)
=
F
1
(
0
)
¯
u
(
p
)γ
μ
u
(
p
)
, so that
F
1
(
0
)
=
f
0
. We now fix the ambiguity in
L
by requiring
L
=
f
0
, which implies
u
(
p
,
s
)
f
μ
(
p
,
p
)
u
(
p
,
s
)
=
0
.
(12.307)
Let us observe that the vertex
μ
(
p
,
p
)
contains in general the coupling constant
e
0
and a factor
Z
2
Z
1
2
3
originating from the wave function renormalization of the electron
and photon fields
L
I
0
=
e
0
¯
ψ
0
γ
μ
ψ
0
A
0
μ
=
e
0
Z
2
Z
1
2
3
¯
ψγ
μ
ψ
A
μ
.
(12.308)
We conclude that the logarithmic divergence in the vertex part correction can be
absorbed in a
charge (or coupling constant) renormalization
as follows:
38
To show this use the general identity
¯
u
(
p
)γ
μ
u
(
p
)
=
1
2
m
u
(
p
)γ
μ
pu
(
p
)
+
u
(
p
)
p
γ
μ
u
(
p
)
=
u
(
p
)
p
μ
+
p
μ
2
m
−
γ
μν
(
p
ν
−
p
ν
)
2
m
u
(
p
),
(
12
.
303
)
where we have written
γ
μ
γ
ν
=
η
μν
+
γ
μν
,
and
γ
μν
being defined as
[
γ
μ
, γ
ν
]
/
2
.