12.5 Amplitudes in the Momentum Representation
481
particle with
m
2
=
k
2
.
While for a real photon only the two transverse polariza-
tions are physical, when a
virtual
photon is exchanged, all four polarization vectors
ε
(λ)
μ
(
k
), λ
=
0
,
1
,
2
,
3 contribute to the amplitude. It is then interesting to see what is
the role of the time-like and longitudinal photons
ε
(
0
)
μ
(
k
), ε
(
3
)
μ
(
k
)
in the interpretation
of the process.
Let us refer for concreteness to the second diagram of the Möller scattering whose
lowest order amplitude is given by the second term of (
12.163
). We observe that the
sum over the indices
μ
of the gamma-matrices is due to the
η
μν
factor of the photon
propagator
˜
D
F
μν
= −
i
η
μν
(
k
2
+
i
)
−
1
, which in turn comes from the completeness
relation (
12.166
). Therefore the amplitude corresponding to the first term of (
12.163
)
could have been alternatively written as
¯
u
(
q
2
,
s
2
)γ
μ
u
(
p
2
,
r
2
)
λ
=
3
λ
=
0
ε
(λ)
μ
(
k
)ε
(λ)
ν
(
k
)
−
i
(
p
1
−
q
1
)
2
¯
u
(
q
1
,
s
1
)γ
ν
u
(
p
1
,
r
1
).
(12.164)
For a virtual photon we must take as polarization vectors a set which for
k
2
→
0
reduces to the set used for a real photon in (11.236) and (11.237). As seen in Sect.
11.7, such set is obtained by simply replacing the longitudinal vector
ε
(
3
)
μ
(
k
)
of
(11.237) with
ε
(
3
)
μ
(
k
)
=
k
μ
−
η
μ
(
k
·
η)
(
k
·
η)
2
−
k
2
.
(12.165)
Let us now decompose the sum appearing in the completeness relation
3
λ
=
0
ε
(λ)
μ
(
k
)ε
(λ)ν
(
k
)
=
η
μν
,
(12.166)
into the sum over
λ
=
0
,
3, corresponding to the exchange of timelike and longitu-
dinal photons and the sum over the transverse polarizations
λ
=
1
,
2
.
In particular,
using (
12.165
) for
ε
(
3
)
μ
(
k
)
and the value
ε
(
0
)
μ
(
k
)
=
η
μ
of (11.239), we have
ε
(
0
)
μ
(
k
)ε
(
0
)
ν
(
k
)
−
ε
(
3
)
μ
(
k
)ε
(
3
)
ν
(
k
)
=
η
μ
η
ν
−
[
k
μ
−
η
μ
(
k
·
η)
][
k
ν
−
η
ν
(
k
·
η)
]
(
k
·
η)
2
−
k
2
.
Since we are interested in the contribution to the amplitude of the
λ
=
0
,
3 polariza-
tions, we substitute the right hand side of this expression into (
12.164
) with the sum
restricted to the values
λ
=
0
, λ
=
3
.
Since, as already seen in the previous section,
the terms proportional to
k
μ
do not contribute by virtue of gauge invariance
17
we
17
In this special case this can be also seen directly. Indeed
k
μ
¯
u
(
q
2
,
s
2
)γ
μ
u
(
p
2
,
r
2
)
= ¯
u
(
q
2
,
s
2
)(
p
2
−
q
2
)
μ
γ
μ
u
(
p
2
,
r
2
)
= −
m
¯
u
(
q
2
,
s
2
)
u
(
p
2
,
r
2
)
+
m
¯
u
(
q
2
,
s
2
)
u
(
p
2
,
r
2
)
=
0
,
(
12
.
167
)
and similarly for the other factor of (
12.168
).