114
17 Examples
operators ψ F , ψ
∗
F defined by
ψ F (x) ≡ ψ(x)− < ψ >
and one can easily compute the correlation functions of ψ, ψ
∗ in terms of those of
ψ F , ψ
∗
F ,
96 e.g.
< ψ(x)
∗
ψ(y) >= | < ψ > |
2
, < ψ(x) ψ(y) >=< ψ >
2
, etc.
From the above equations it follows that < ψ > is related to the average density
| < ψ > |
2
=< ψ(x)
∗
ψ(x) >= n, < ψ >=
√
n e
iθ
, θ ∈ [0, 2π).
The ground state can be thought as labeled by the “order parameter” < ψ > and, in
order to spell this out, we shall denote the ground state by Ψ 0,n,θ or, briefly, by Ψ θ
and the corresponding state on A by Ω θ , (θ ground state).
For any θ the GNS representation defined by Ω θ is irreducible because the algebras
generated by ψ, ψ
∗ and by ψ F , ψ
∗
F coincide and the latter is irreducible. It is not
difficult to see that different values of < ψ > label inequivalent representations of A
(see also the discussion below). Properties I-III of Chap. 15 are obviously satisfied.
The localization properties of the model deserve a few comments, since we have
a very simple example of the conceptual problem of identifying an algebra with
localization properties stable under time evolution. As a matter of fact, the quasilocal algebra obtained as the norm closure of the local algebra A L , generated by the
Weyl exponentials U ( f ), V (g), f , g ∈ D(R
s
) (see Sect. 14.1), is not stable under
time evolution. The point is that the Schrödinger time evolution does not map D(R
s
)
into D(R
s
) and, therefore, strictly localized operators at t = 0 are no longer so at
any subsequent time. This implies that α t (U ( f )) = U ( f t ), ˜
f t (k) ≡ ˜
f (k) e
ik
2 t/2m , is
not in the norm closure of A L .
97 Thus, the non-relativistic approximation and the
corresponding time evolution require to weaken slightly the condition of localization
96 They provide much more detailed information than the mere probability distribution of the occupation numbers, as it is done in the standard elementary treatments of the free Bose gas: see e.g.
R.P. Feynman, Introduction to Statistical Mechanics, Benjamin 1972, Sect. 1.9.
97 D.A. Dubin and G.L. Sewell, J. Math. Phys. 11, 2990 (1970); G.L. Sewell, Comm. Math. Phys.
33, 43 (1973). The point is that
||U ( f t ) − U (g n )|| = ||e
iIm ( ft ,gn ) U ( f t − g n ) − 1|| = 2,
unless || f t − g n || L 2 = 0, because A ≡ ψ(h) + ψ(h) ∗ is an unbounded operator with continuous
spectrum (linear in h) and therefore
||e
i A − 1||
2 = sup
λ∈σ(A)
|e
iλ − 1|
2 = 4.
On the other hand, if f , g n ∈ D(R s ), || f t − g n || L 2 cannot vanish, since f t /
∈ D.
17 Examples
operators ψ F , ψ
∗
F defined by
ψ F (x) ≡ ψ(x)− < ψ >
and one can easily compute the correlation functions of ψ, ψ
∗ in terms of those of
ψ F , ψ
∗
F ,
96 e.g.
< ψ(x)
∗
ψ(y) >= | < ψ > |
2
, < ψ(x) ψ(y) >=< ψ >
2
, etc.
From the above equations it follows that < ψ > is related to the average density
| < ψ > |
2
=< ψ(x)
∗
ψ(x) >= n, < ψ >=
√
n e
iθ
, θ ∈ [0, 2π).
The ground state can be thought as labeled by the “order parameter” < ψ > and, in
order to spell this out, we shall denote the ground state by Ψ 0,n,θ or, briefly, by Ψ θ
and the corresponding state on A by Ω θ , (θ ground state).
For any θ the GNS representation defined by Ω θ is irreducible because the algebras
generated by ψ, ψ
∗ and by ψ F , ψ
∗
F coincide and the latter is irreducible. It is not
difficult to see that different values of < ψ > label inequivalent representations of A
(see also the discussion below). Properties I-III of Chap. 15 are obviously satisfied.
The localization properties of the model deserve a few comments, since we have
a very simple example of the conceptual problem of identifying an algebra with
localization properties stable under time evolution. As a matter of fact, the quasilocal algebra obtained as the norm closure of the local algebra A L , generated by the
Weyl exponentials U ( f ), V (g), f , g ∈ D(R
s
) (see Sect. 14.1), is not stable under
time evolution. The point is that the Schrödinger time evolution does not map D(R
s
)
into D(R
s
) and, therefore, strictly localized operators at t = 0 are no longer so at
any subsequent time. This implies that α t (U ( f )) = U ( f t ), ˜
f t (k) ≡ ˜
f (k) e
ik
2 t/2m , is
not in the norm closure of A L .
97 Thus, the non-relativistic approximation and the
corresponding time evolution require to weaken slightly the condition of localization
96 They provide much more detailed information than the mere probability distribution of the occupation numbers, as it is done in the standard elementary treatments of the free Bose gas: see e.g.
R.P. Feynman, Introduction to Statistical Mechanics, Benjamin 1972, Sect. 1.9.
97 D.A. Dubin and G.L. Sewell, J. Math. Phys. 11, 2990 (1970); G.L. Sewell, Comm. Math. Phys.
33, 43 (1973). The point is that
||U ( f t ) − U (g n )|| = ||e
iIm ( ft ,gn ) U ( f t − g n ) − 1|| = 2,
unless || f t − g n || L 2 = 0, because A ≡ ψ(h) + ψ(h) ∗ is an unbounded operator with continuous
spectrum (linear in h) and therefore
||e
i A − 1||
2 = sup
λ∈σ(A)
|e
iλ − 1|
2 = 4.
On the other hand, if f , g n ∈ D(R s ), || f t − g n || L 2 cannot vanish, since f t /
∈ D.