6.4 Equilibrium States
155
The definition of the time evolution in Proposition 6.3.8 and the ϕ
Q -invariance
of F ω shows the identity of the time evolution of π ω (A) with the action of the
one-parameter group g
−1
Q (t, F ω ), cf. (6.1.13), with the generator -β
Q
F ω
, cf. (6.1.10).
According to (6.1.17) and (6.1.18), we obtain the remaining assertions of the proposition.
6.4.8 Note. The generator of the mean-field time evolution τ
Q of local perturbations
of an extremal equilibrium state ω given in (6.4.12) is usually called the BogoliubovHaag Hamiltonian, cf. [23, 140, 312].
6.4.9 We shall assume in the following that A is a quasilocal C
∗ -algebra generated
by a net {A
J
: J ⊂ , J finite} of local subalgebras A
J commuting with each other
for disjoint J ’s:
x ∈ A
J
, y ∈ A
J
, J ∩ J
= ∅ ⇒ [x, y] = 0.
(6.4.15)
Here is a countable infinite commutative group acting on A by the representation π : π p ∈
∗ - Aut A, in such a way that π p : A
J
→ A
J + p is an isomorphism
for any J ⊂ . This is the situation from (6.4.2), where L(H) is identified with
L(H u ), π 0 = π(0) = id L(A) (0 is here the identity of the group ), hence π( p) = π p
( p ∈ ).
It will be assumed in the following that each A
J
(J ⊂ ) is σ(G)-invariant, and
that the action of σ(G) commutes with π((). Then also (6.4.1) will be fulfilled (π(()
is naturally extended to the equally denoted automorphism groups of C and of A
∗∗ ).
In this situation, let ω ∈ S(A) be a factor state which is invariant with respect to
the action of π(():
ω(π p (x)) = ω(x), for all x ∈ A, p ∈ .
(6.4.16)
The locally normal factor states have short range correlations, [193], [53, Theorem 2.6.10], hence they are weakly π(()-clustering, and
lim
p→∞
ω(π p (x)y) = ω(x)ω(y), for all x, y, ∈ A.
(6.4.17)
If A
J are faithfully represented in Hilbert spaces H J , as it was the case of Sect. 5.1,
then π p will be used also for translations of unbounded operators acting on H J
to unitarily equivalent operators acting on H J + p (e.g. by translating their spectral
projectors belonging to A
J ); this can be done if the isomorphisms of A
J
⊂ L(H J )
with A
J + p
⊂ L(H J + p ) (J ⊂ , p ∈ ) are spatial. We shall write also A p := A
J
with J := { p} := the one-point set, p ∈ . Let all the A
J
(J ⊂ ) have common
unit and let the C
∗ -algebras A p with p ∈ J generate A
J
(J ⊂ ).
155
The definition of the time evolution in Proposition 6.3.8 and the ϕ
Q -invariance
of F ω shows the identity of the time evolution of π ω (A) with the action of the
one-parameter group g
−1
Q (t, F ω ), cf. (6.1.13), with the generator -β
Q
F ω
, cf. (6.1.10).
According to (6.1.17) and (6.1.18), we obtain the remaining assertions of the proposition.
6.4.8 Note. The generator of the mean-field time evolution τ
Q of local perturbations
of an extremal equilibrium state ω given in (6.4.12) is usually called the BogoliubovHaag Hamiltonian, cf. [23, 140, 312].
6.4.9 We shall assume in the following that A is a quasilocal C
∗ -algebra generated
by a net {A
J
: J ⊂ , J finite} of local subalgebras A
J commuting with each other
for disjoint J ’s:
x ∈ A
J
, y ∈ A
J
, J ∩ J
= ∅ ⇒ [x, y] = 0.
(6.4.15)
Here is a countable infinite commutative group acting on A by the representation π : π p ∈
∗ - Aut A, in such a way that π p : A
J
→ A
J + p is an isomorphism
for any J ⊂ . This is the situation from (6.4.2), where L(H) is identified with
L(H u ), π 0 = π(0) = id L(A) (0 is here the identity of the group ), hence π( p) = π p
( p ∈ ).
It will be assumed in the following that each A
J
(J ⊂ ) is σ(G)-invariant, and
that the action of σ(G) commutes with π((). Then also (6.4.1) will be fulfilled (π(()
is naturally extended to the equally denoted automorphism groups of C and of A
∗∗ ).
In this situation, let ω ∈ S(A) be a factor state which is invariant with respect to
the action of π(():
ω(π p (x)) = ω(x), for all x ∈ A, p ∈ .
(6.4.16)
The locally normal factor states have short range correlations, [193], [53, Theorem 2.6.10], hence they are weakly π(()-clustering, and
lim
p→∞
ω(π p (x)y) = ω(x)ω(y), for all x, y, ∈ A.
(6.4.17)
If A
J are faithfully represented in Hilbert spaces H J , as it was the case of Sect. 5.1,
then π p will be used also for translations of unbounded operators acting on H J
to unitarily equivalent operators acting on H J + p (e.g. by translating their spectral
projectors belonging to A
J ); this can be done if the isomorphisms of A
J
⊂ L(H J )
with A
J + p
⊂ L(H J + p ) (J ⊂ , p ∈ ) are spatial. We shall write also A p := A
J
with J := { p} := the one-point set, p ∈ . Let all the A
J
(J ⊂ ) have common
unit and let the C
∗ -algebras A p with p ∈ J generate A
J
(J ⊂ ).