QuantumPhysics.dvi

The only novelty here, as compared with systems of finite numbers of harmonic oscillators, is that
here, the oscillators are indexed by a continuous variablek, and the Kroneckerδin the commutation
relations is replaced by a Diracδ-function.

The Hamiltonian and momentum operators take the following values,

H =

∫ d (^3) k
(2π)^3
ω
∑
α
a†α(k)aα(k)

P =

∫

d^3 k

(2π)^3

k

∑

α

a†α(k)aα(k) (19.51)

Note that the Hamiltonian, given above, involves a choice. The classical Hamiltonian was expressed
in terms ofaα(k)∗aα(k), a quantity whose translation into an operator via the correspondence
principle, is ambiguous. In general, either ordering of theoperators might have occurred, or any
linear combination of the two orderings, such as

Hλ=

∫ d (^3) k
(2π)^3
ω
∑
α
(
(1−λ)a†α(k)aα(k) +λaα(k)a†α(k)
)
(19.52)
so that the original HamiltonianHcorresponds toH=H 0. Using the canonical commutations,
the difference between two Hamiltonians may be deduced from
Hλ−H=λ
∫ d (^3) k
(2π)^3
ω
∑
α
[aα(k),a†α(k)] = 2λ
∫
d^3 kω δ(3)(0) (19.53)
This energy shift is infinite due to the space-volume infinityδ(3)(0), as well as due to the momentum
volume space infinity resulting from the integration overkextending out to∞values ofk. More
importantly, however, the shift is a constant independent ofaα(k) anda†α(k), which is unobservable
(only differences in energy are observable). For the momentum, the analogous shift involves an
integral overk, which is odd and vanishes.

19.6 Photons – the Hilbert space of states

It remains to construct the Hilbert space of states. We have identified the observables xα(k)
andpα(k), and their non-self-adjoint equivalentsaα(k) andaα(k)†, as the fundamental mutually
independent quantities to which the electro-magnetic fields are equivalent. Thus, the Hilbert space
may be faithfully constructed in terms of these harmonic oscillators.

19.6.1 The ground state orvacuum

The ground state, orthe vacuum, is defined to be the state of lowest energy of the total system.
Thus it must be the lowest energy state for each one of the harmonic oscillators, which requires
that this state must be annihilated by the operatorsaα(k) for allα= 1,2 and for allk∈R^3 ,

aα(k)|∅〉= 0 (19.54)