408
11
Quantization of Boson and Fermion Fields
U
(
,
x
0
)
†
ˆ
ψ(
x
)
U
(
,
x
0
)
=
p
,
r
mc
2
E
p
V
U
†
c
(
p
,
r
)
Uu
(
p
,
r
)
e
−
i
p
·
x
+
U
†
d
†
(
p
,
r
)
U
v(
p
,
r
)
e
i
p
·
x
=
p
,
r
,
s
mc
2
E
p
V
R
(
,
−
1
p
)
r
s
×
c
(
−
1
p
,
s
)
u
(
p
,
r
)
e
−
i
p
·
(
x
+
x
0
)
+
d
†
(
−
1
p
,
s
)v(
p
,
r
)
e
i
p
·
(
x
+
x
0
)
=
p
,
r
,
s
mc
2
E
p
V
R
(
,
p
)
r
s
×
c
(
p
,
s
)
u
(
p
,
r
)
e
−
i
(
p
)
·
(
x
+
x
0
)
+
d
†
(
p
,
s
)v(
p
,
r
)
e
i
(
p
)
·
(
x
+
x
0
)
=
p
,
s
mc
2
E
p
V
c
(
p
,
s
)
S
(
)
u
(
p
,
s
)
e
−
i
p
·
x
+
d
†
(
p
,
s
)
S
(
)v(
p
,
s
)
e
i
p
·
x
=
S
(
)
p
,
s
mc
2
E
p
V
c
(
p
,
s
)
u
(
p
,
s
)
e
−
i
p
·
x
+
d
†
(
p
,
s
)v(
p
,
s
)
e
i
p
·
x
=
S
(
)
ˆ
ψ(
x
),
(11.196)
where we have changed summation variable from
p
to
p
=
−
1
p
and, as usual,
wrote
x
=
−
1
(
x
+
x
0
).
We have moreover used the transformation properties
(
10.149
).
11.6.3 Discrete Transformations
Let us now consider the three discrete transformations corresponding to
parity P
,
charge conjugation C
and
time-reversal T
for the Dirac quantum field. In the previous
Chapterwehaveseenthatforthe
classical
Diracfieldthespacereflectioncorresponds
to the active transformation (see (10.242)):
ψ(
x
,
t
)
→
ψ (
x
,
t
)
=
η
P
γ
0
ψ(
−
x
,
t
),
(11.197)
with respect to which it is easily verified that the Dirac equation is invariant. For the
quantized field we must seek a unitary operator
U
(
P
)
such that
16
U
(
P
)
†
ψ(
x
,
t
)
U
(
P
)
=
η
P
γ
0
ψ(
−
x
,
t
).
(11.198)
We can define through its action on the operators
c
(
p
,
r
),
d
†
(
p
,
r
),
which should
reproduce (
11.198
). Using the properties
16
We suppress here and in the following the “hat” symbol for the field operator.