344
10
Relativistic Wave Equations
To this end let us consider the positive energy plane wave described by the spinor
u
(
p
,
r
).
Its Dirac conjugate
¯
u
will satisfy the following equation:
¯
u
(
p
) (
p
−
mc
)
=
0
.
By transposition we have
γ
T
μ
p
μ
−
mc
¯
u
T
(
p
)
=
0
If we now multiply the above equation to the left by the
C
matrix and use the property
(
10.184
) we obtain
(
p
+
mc
)
C
u
T
(
p
)
=
0
,
(10.187)
which shows that charge-conjugate spinor
u
c
(
p
)
=
C
u
T
(
p
)
satisfies the second
of (
10.148
) and should therefore coincide with a spinor
v(
p
)
defining the negative
energy solution
ψ
(
−
)
−
p
with opposite momentum
−
p
.
Besides changing the value of
the momentum, charge-conjugation also reverses the spin orientation. Going, for
the sake of simplicity, to the rest frame, where a positive energy solution with spin
projection
/
2 along a given direction, is described by
u
(
0
,
1
)
=
(
1
,
0
,
0
,
0
)
T
,
see (
10.152
), we find for the charge conjugate spinor
u
c
≡
C
γ
0
u
∗
(note that
γ
0
T
=
γ
0
)
u
c
(
0
,
r
)
=
C
γ
0
u
∗
(
0
,
r
=
1
)
=
(
0
,
0
,
0
,
1
)
T
=
v(
0
,
r
=
2
),
that is a negative energy spinor with spin projection
−
/
2
.
In general the reader can
verify that
u
c
(
0
,
r
)
=
rs
v(
0
,
s
),
(10.188)
where summation over
s
=
1
,
2 is understood, and
(
rs
)
is the matrix
i
σ
2
:
11
=
22
=
0
,
12
= −
21
=
1.
Let us now evaluate
u
c
(
p
,
r
)
using the explicit form of
u
(
p
,
r
)
given in (
10.154
):
u
c
(
p
,
r
)
=
C
γ
0
u
(
p
,
r
)
∗
=
C
γ
0
p
∗
+
mc
2
m
(
mc
2
+
E
p
)
u
(
0
,
r
)
∗
=
C
p
T
+
mc
2
m
(
mc
2
+
E
p
)
γ
0
u
(
0
,
r
)
∗
=
−
p
+
mc
2
m
(
mc
2
+
E
p
)
u
c
(
0
,
r
)
=
rs
−
p
+
mc
2
m
(
mc
2
+
E
p
)
v(
0
,
s
)
=
rs
v(
p
,
s
).
(10.189)