348
10
Relativistic Wave Equations
In the rest frame the above projector reads:
P
(
0
)
r
≡
1
2
1
+
ε
r
γ
5
n
i
α
i
=
1
2
+
ε
r
n
·
σ
0
0
1
2
+
ε
r
n
·
σ
.
(10.206)
The matrices
P
r
project on both positive and negative energy solutions with the same
spin component along
n
.
Let us now define two operators
+
,
r
,
−
,
r
projecting on
positive and negative solutions with a given spin component
r
, respectively:
+
,
r
u
(
p
,
s
)
=
δ
rs
u
(
p
,
s
)
;
+
,
r
v(
p
,
s
)
=
0
,
−
,
r
u
(
p
,
s
)
=
0
;
−
,
r
v(
p
,
s
)
=
δ
rs
v(
p
,
s
).
(10.207)
They have the following general form:
(
+
,
r
)
α
β
=
u
α
(
p
,
r
)
¯
u
β
(
p
,
r
)
;
(
−
,
r
)
α
β
= −
v
α
(
p
,
r
)
¯
v
β
(
p
,
r
),
(10.208)
as it follows from the orthogonality properties (
10.168
) and (
10.169
). To find the
explicit expression of these matrices in terms of
p
and
n
, we notice that they are
obtained by multiplying to the right and to the left the projectors
P
r
on the spin state
r
by the projectors
±
on the positive and negative energy states:
±
,
r
=
±
P
r
±
=
±
1
2
(
1
±
ε
r
γ
5
n
)
= ±
1
4
mc
(
p
±
mc
)(
1
±
ε
r
γ
5
n
),
where we have used the property:
1
+
ε
r
1
mc
γ
5
np
(
p
±
mc
)
=
(
p
±
mc
)(
1
±
ε
r
γ
5
n
),
(10.209)
which can be easily verified using the fact that
p
and
n
anticommute:
np
= −
pn
.
10.7 Dirac Equation in an External Electromagnetic Field
We shall now study the coupling of the Dirac field to the electromagnetic field
A
μ
.
To this end, as we did for the complex scalar field in
Sect.10.2.1
, we apply the
minimal coupling
prescription, namely we substitute in the free Dirac equation
p
μ
→
p
μ
+
e
c
A
μ
,
(10.210)
that is, in terms of the quantum operator
i
∂
μ
→
i
∂
μ
+
e
c
A
μ
.
(10.211)
In the convention which we adopt throughout the book, the electron has charge
e
= −
e

<
0
.
The coupled Dirac equation takes the following form: