362
11
Quantization of Boson and Fermion Fields
in products in (
11.6
), being computed at the
same space–time point x
μ
=
y
μ
,
do not
in general commute (see
Sect.11.4
below). This implies that a certain order must be
chosen. The convention we shall use will be shown in the sequel to lead to consistent
results in the development of the theory. It consists in the following substitutions:
(
classical fields
)
(
field operators
),
π
∗
(
x
)π(
x
)
→
ˆ
π(
x
)
ˆ
π
†
(
x
),
φ
∗
(
x
)φ(
x
)
→
ˆ
φ
†
(
x
)
ˆ
φ(
x
),
∇
φ
∗
(
x
)
·
∇
φ(
x
)
→
∇
ˆ
φ
†
(
x
)
·
∇
ˆ
φ(
x
).
(11.8)
The resulting Hamiltonian operator reads:
ˆ
H
=
d
3
x
ˆ
π(
x
)
ˆ
π
†
(
x
)
+
c
2
∇
ˆ
φ
†
(
x
)
·
∇
ˆ
φ(
x
)
+
m
2
c
4
2
ˆ
φ
†
(
x
)
ˆ
φ(
x
)
.
(11.9)
Let us now use this Hamiltonian in the quantum Hamilton’s equations (
11.5
)
i
∂
∂
t
ˆ
φ(
x
,
t
)
=
ˆ
φ(
x
,
t
),
ˆ
H
;
i
∂
∂
t
ˆ
π(
x
,
t
)
=
ˆ
π(
x
,
t
),
ˆ
H
.
(11.10)
and show that it reproduces the quantum version of the classical Klein–Gordon
equation.
Applying (
11.6
) to the the first of (
11.10
) we find
i
∂
∂
t
ˆ
φ(
y
,
t
)
=
ˆ
φ(
x
,
t
),
ˆ
H
(
t
)
=
d
3
y
ˆ
φ(
x
,
t
),
ˆ
π(
y
,
t
)
ˆ
π
†
(
y
,
t
)
=
i
ˆ
π
†
(
x
,
t
),
(11.11)
which is the expression (
11.4
) of the conjugate momentum operators. The same
computations applied to the last of (
11.10
) (or better to its hermitian conjugate),
yields
i
∂
∂
t
ˆ
π
†
(
x
,
t
)
=
c
2
d
3
y
ˆ
π
†
(
x
,
t
),
∂
ˆ
φ
†
∂
y
i
∂
ˆ
φ
∂
y
i
(
y
,
t
)
+
m
2
c
2
2
d
3
y
ˆ
π
†
(
x
,
t
),
ˆ
φ
†
(
y
,
t
)
ˆ
φ(
y
,
t
)
= −
c
2
d
3
y
ˆ
π
†
(
x
,
t
),
ˆ
φ
†
(
y
,
t
)
∇
2
ˆ
φ(
y
,
t
)
−
i
m
2
c
2
ˆ
φ(
x
,
t
)
=
i
c
2
∇
2
ˆ
φ
−
m
2
c
2
2
ˆ
φ
(
x
,
t
).
(11.12)
Substituting in the left hand side the value of
π
†
given by (
11.11
) we obtain