176
6
Quantization of the Electromagnetic Field
We associate with each oscillator
(
k
, α)
, i.e. with each plane wave, a
state
of a particle
called
photon
and denoted by the symbol
γ
, carrying the quantum of momentum,
k
, and of energy,
ω
k
, and having polarization
α
. The state

N
k
,α
,
t
of the
(
k
, α)

oscillator is then then interpreted as describing
N
k
,α
photons in the state
(
k
, α)
.
Its energy and momentum are
N
k
,α
ω
k
and
N
k
,α
k
respectively, namely the sum
of the
N
k
,α
quanta of the two quantities associated with each photon. The state
{
N
k
,α
}
,
t
of the whole electromagnetic field then describes
N
k
,α
photons in each
state
(
k
, α)
and its energy and momentum, as given in (
6.63
), is the sum of the energy
and momenta of the photons in the various states. A photon with energy
E
=
ω
k
and momentum
p
=
k
has a rest mass given by:
m
2
γ
=
1
c
4
E
2
−
1
c
2

p

2
=
1
c
4
2
(ω
2
k
−
c
2

k

2
)
=
0
,
(6.64)
where we have used the definition of
ω
k
. As was anticipated in
Chap.5
, the photon is
therefore a massless particle. Its momentum fourvector
p
μ
is thus
times the wave
number fourvector
k
μ
associated with the corresponding plane wave and defined in
(
5.14
).
The action of
a
†
k
,α
or of
a
k
,α
on a state amounts to “creating” or “destroying”
a
(
k
, α)
photon since they increase or decrease the energy and momentum of the
corresponding oscillator state by one quantum respectively. This can be seen by
recalling, from elementary quantum mechanics, the following relations which hold
for the
(
k
, α)
oscillator:
a
†
k
,α

N
k
,α
,
t
=
N
k
,α
+
1

N
k
,α
+
1
,
t
,
a
k
,α

N
k
,α
,
t
=
N
k
,α

N
k
,α
−
1
,
t
.
(6.65)
Expressing the canonical operators in terms of
a
k
,α
,
a
†
k
,α
,
ˆ
P
k
,α
= −
i
ω
k
2
a
k
,α
−
a
+
k
,α
,
ˆ
Q
k
,α
=
2
ω
k
a
k
,α
+
a
+
k
,α
,
(6.66)
we find the following relations:
N
k
,α

ˆ
Q
k
,α

N
k
,α
−
1
=
N
k
,α
−
1

ˆ
Q
k
,α

N
k
,α
=
N
k
,α
2
ω
k
,
N
k
,α

ˆ
P
k
,α

N
k
,α
−
1
= −
N
k
,α
−
1

ˆ
P
k
,α

N
k
,α
=
i
ω
k
N
k
,α
2
.
(6.67)
The representation of the states of the electromagnetic field in terms of occupation
number eigenstates associated with the constituent harmonic oscillators, is called