11.2 Quantization of the Klein–Gordon Field
361
Taking into account that the complex conjugation of a classical dynamic variable
must be replaced by the hermitian conjugate of the corresponding quantum operator,
we can also write the hermitian conjugate counterparts of (
11.2
):
ˆ
φ
†
α
(
x
,
t
),
ˆ
π
†
β
(
y
,
t
)
=
i
δ
α
β
δ
3
(
x
−
y
),
ˆ
φ
†
α
(
x
,
t
),
ˆ
φ
†
β
(
y
,
t
)
=
ˆ
π
†
α
(
x
,
t
),
ˆ
π
†
β
(
y
,
t
)
=
0
.
(11.3)
Note that the classical relation (
8.208
) now becomes
ˆ
π
α
(
x
,
t
)
=
∂
∂
t
ˆ
φ
†
α
(
x
,
t
)
;
π
†
α
(
x
,
t
)
=
∂
∂
t
ˆ
φ
α
(
x
,
t
).
(11.4)
The same replacement (
11.1
) implies that the classical Hamilton equations, given in
terms of the Poisson brackets in (
8.221
) and (
8.222
), at the quantum level become
i
∂
∂
t
ˆ
φ
α
(
x
,
t
)
=
ˆ
φ
α
(
x
,
t
),
ˆ
H
,
i
∂
∂
t
ˆ
π
α
(
x
,
t
)
=
ˆ
π
α
(
x
,
t
),
ˆ
H
,
(11.5)
where the Hamiltonian operator is obtained from the classical expression (
8.209
)
and (
8.210
) by promoting the field variables to quantum operators. We note that this
replacement implies time evolution in the quantum system to be described in the
Heisenberg picture
since the classical dynamic variables are time dependent. Thus
the quantum state of the system is time independent.
In this section we restrict our discussion to the dynamics of a
free complex scalar
field
, which, as discussed in
Sects.8.8.1
and
8.8.9
is equivalent to
two
real scalar
fields. Since by definition a scalar field sits in the trivial representation of the Lorentz
group, it corresponds to a spin-0 field, carrying no representation indices. Its classical
description is given in terms of the Lagrangian (
10.11
) from which the classical
Klein–Gordon equation (
10.12
) is derived. In that case, following (
11.1
), the Poisson
brackets (
8.226
) become the equal-time commutators (
11.2
) with no indices
α, β
:
ˆ
φ(
x
,
t
),
ˆ
π(
y
,
t
)
=
i
δ
3
(
x
−
y
),
ˆ
φ
†
(
x
,
t
),
ˆ
π
†
(
y
,
t
)
=
i
δ
3
(
x
−
y
),
(11.6)
all the other possible commutators being zero.