17.2 Free Bose Gas. Bose–Einstein Condensation
115
by replacing the local algebra A L , e.g. by the essentially local algebra A l , generated by the Weyl exponentials of ψ( f ), ψ(g)
∗ , f , g ∈ S(R
s
), which is stable under
time evolution. It is easy to see that A l and its norm closure A satisfy asymptotic
abelianess.
Another interesting feature of the model is related to gauge invariance. The gauge
transformations
β
λ
(ψ(x)) = e
i λ
ψ(x), β
λ
(ψ
∗
(x)) = e
−i λ
ψ
∗
(x), λ ∈ [0, 2π]
define a one-parameter group of
∗ -automorphisms of A, which commutes with α t .
The ground state Ψ 0,θ is not gauge invariant, in the sense that its correlation functions
are not invariant under β
λ , since, e.g.
< β
λ
(ψ) > θ =< ψ > θ+λ .
In fact, under gauge transformations Ω θ → Ω θ+λ .
A gauge invariant state can be defined by averaging over θ
Ω(A) ≡ (2π)
−1
2π
0
dθ Ω θ (A), ∀A ∈ A.
(17.6)
One has
Ω(ψ
∗
1 . . . ψ
∗
k ψ k+1 . . . ψ k+ j ) = (2π)
−1
2π
0
dθ (n)
(k+ j)/2 e
−i(k− j)θ
,
which vanishes unless k = j; similarly for A = ψ 1 . . . ψ k ψ
∗
k+1 . . . ψ
∗
k+ j one has
Ω(A) = 0, if k = j, since Ω θ (A) = Ω 0 (β
θ
(A)) = e
iθ(k− j)
Ω 0 (A).
As displayed by (17.6), Ω is not a pure state on A and the GNS representation
defined by Ω is not irreducible. This can be explicitly seen by noting that
ψ( f ) ∞ = lim
V →∞
V
−1
V
d
s x (ψ( f )) x
commutes with A, by asymptotic abelianess, and exp i(ψ( f )) ∞ belongs to the centre
Z; on the other hand
Ω((ψ( f )) ∞ ) = 0, Ω((ψ( f )
∗
) ∞ (ψ( f )) ∞ ) = n,
so that exp i(ψ( f )) ∞ is not a multiple of the identity in the GNS representation
defined by Ω and this excludes irreducibility.
A simple computation gives
Ω θ ((ψ( f )) ∞ ) =
√
ne
iθ ˜
f (0), Ω θ (e
i(ψ( f )+ψ( f )
∗ ) ∞ ) = e
2i Re (<ψ> θ ˜
f (0))
,