12 Fock Representation
81
and
[N j , a k ] = −δ jk a k .
(12.4)
Proposition 12.1 In an irreducible representation of the Heisenberg algebra with
domain D, the following properties are equivalent
1) the total number operator N =
j N j exists in the sense that ∀α ∈ R
strong − lim
K →∞
e
iα
K
j a
∗
j a j ≡ e
iαN
≡ T (α), ∀α ∈ R
(12.5)
exists on D and defines a one-parameter group of unitary operators T (α)
strongly continuous in α, leaving D stable, so that its generator N exists;
2) there exists a vector Ψ 0 , called the Fock (vacuum) vector, such that
a j Ψ 0 = 0, ∀ j.
(12.6)
In this case the representation is called a Fock representation.
Proof. Property 1) and the commutation relations imply that
T (α)a j T (α)
−1
= e
−iα a j
and therefore [T (2π), A H ] =0. By the irreducibility of A H it follows that T (2π) =
1 exp iθ, so that T
(α) ≡ T (α) exp(−iαθ/2π) satisfies T
(2π) = 1. By using this
condition in the spectral representation of T
(α)
T
(α) =
σ(N )
d E(λ)e
iαλ
, N
≡ N − θ/2π,
where σ(N
) denotes the spectrum of N
, one concludes that the projection valued
spectral measure must be supported on a subset of Z, i.e. the spectrum of N
and
therefore of N is discrete. Now, if λ > 0 is a point of the spectrum of N and Ψ λ a
corresponding eigenvector, then
0 < λ||Ψ λ ||
2
= (Ψ λ , N Ψ λ ) =
j
||a j Ψ λ ||
2
,
so that there must be at least one j such that a j Ψ λ = 0 and one has
T (α)a j Ψ λ = e
i(λ−1)α a j Ψ λ .