so that we have the following expression for the full Hamiltonian,
H=
∫ d (^3) k
(2π)^3
∑
α
|k|aα(k)∗αα(k) (19.36)
The Poisson brackets ofaα(k) withaα(k′) and withaα(k′)∗may be induced from the time evolution
equations for these oscillators, which are already available to us, namely,
aα(t,k) =aα(0,k)e−i|k|t (19.37)
As a result, we have
∂taα(t,k) =−i|k|aα(t,k) ={H,aα(t,k)} (19.38)
which is naturally realized in terms of
{aα(k),αβ(k′)} = 0
{aα(k),αβ(k′)∗} = iδαβδ(3)(k′−k) (19.39)
One recognizes this Hamiltonian, and associated Poisson brackets as those corresponding to an
infinite number of independent harmonic oscillatorsaα(k), labelled by the combined index (α,k).
One may pass, formally, to a more familiar notation for the harmonic oscillators in terms of real
degrees of freedomxα(k) andpα(k),
aα(k) =
1
√
2 ω(
+ipα(k) +ωxα(k))aα(k)∗ =1
√
2 ω(
−ipα(k) +ωxα(k))
(19.40)in terms of which the Hamiltonian takes the form
H=
∫ d (^3) k
(2π)^3
∑
α
(
1
2
pα(k)^2 +
1
2
ω^2 xα(k)^2)
(19.41)where the Poisson brackets take the form,
{xα(k),pα(k′)} = δαβδ(3)(k′−k)
{xα(k),xα(k′)} = 0
{pα(k),pα(k′)} = 0 (19.42)Of course, it must be pointed out thatxαandpαare not actual positions and momentum operators
of the photon. They are formal devices without direct physical meaning.