6.4 Equilibrium States
159
ω is expressed by (6.4.18), hence according to (5.1.120):
exp(it F ω (ξ)) = ω(exp(it X ξ )) = lim
J
ω(exp(it
1
|J |
X
J
ξ )) =
= lim
J
p∈J
ω
0
(exp(
it
|J |
X ξ )) = lim
|J |→∞
[ω
0
(exp(
it
|J |
X ξ ))]
|J |
. (6.4.33)
The result (6.4.29) is now obtained from (6.4.33) by the ‘law of large numbers’
([112, II.Chap. XVII.1. Theorem 1]) applied to the arithmetic means of |J | copies
of independent real-valued variables with equal distributions μ
0
ξ . The probability
measure μ
0
ξ on R is given here by the projection-valued spectral measure P ξ of X ξ :
X ξ =
R
λ P ξ (dλ).
(6.4.34)
Then we set
μ
0
ξ (dλ) := ω
0
(P ξ (dλ)),
(6.4.35)
and we can write:
[ω
0
(exp(
it
|J |
X ξ ))]
|J |
=
R |J |
exp
⎛
⎝ it
|J |
p∈J
λ p
⎞
⎠
m∈J
μ
0
ξ (dλ m ),
(6.4.36)
where ⊗ m∈J μ
0
ξ (dλ m ) is the tensor product of |J | copies of the measures (6.4.35)
describing the simultaneous probability distribution of the |J | independent random
variables. Combining (6.4.33) and (6.4.36) gives the wanted result (6.4.29).
6.4.12 Proposition. Let us consider the system (A; σ(G); τ
Q
; π(()) as in Theorem
6.4.10. Assume that ω
p
( p ∈ ) are ground states for the restriction of the group
σ(exp(−tβ
Q
F ω
)) to the subalgebras A p . Let the product-state
ω :=
p∈
ω
p
(6.4.37)
satisfy the ‘consistency condition’
ω(E g ( f ξ )) = F ω (ξ), f or all ξ ∈ g.
(6.4.38)
Then ω is a factor ground state of the evolution τ
Q . If all the ω
p are pure, then ω is
an extremal τ
Q -ground state.
Proof. The factoria1ity of ω is a consequence of cluster properties, cf. e.g. [53,
54]. The condition (6.4.7) is fulfilled for τ t := σ(exp(−β
Q
F ω
t)). An application of
159
ω is expressed by (6.4.18), hence according to (5.1.120):
exp(it F ω (ξ)) = ω(exp(it X ξ )) = lim
J
ω(exp(it
1
|J |
X
J
ξ )) =
= lim
J
p∈J
ω
0
(exp(
it
|J |
X ξ )) = lim
|J |→∞
[ω
0
(exp(
it
|J |
X ξ ))]
|J |
. (6.4.33)
The result (6.4.29) is now obtained from (6.4.33) by the ‘law of large numbers’
([112, II.Chap. XVII.1. Theorem 1]) applied to the arithmetic means of |J | copies
of independent real-valued variables with equal distributions μ
0
ξ . The probability
measure μ
0
ξ on R is given here by the projection-valued spectral measure P ξ of X ξ :
X ξ =
R
λ P ξ (dλ).
(6.4.34)
Then we set
μ
0
ξ (dλ) := ω
0
(P ξ (dλ)),
(6.4.35)
and we can write:
[ω
0
(exp(
it
|J |
X ξ ))]
|J |
=
R |J |
exp
⎛
⎝ it
|J |
p∈J
λ p
⎞
⎠
m∈J
μ
0
ξ (dλ m ),
(6.4.36)
where ⊗ m∈J μ
0
ξ (dλ m ) is the tensor product of |J | copies of the measures (6.4.35)
describing the simultaneous probability distribution of the |J | independent random
variables. Combining (6.4.33) and (6.4.36) gives the wanted result (6.4.29).
6.4.12 Proposition. Let us consider the system (A; σ(G); τ
Q
; π(()) as in Theorem
6.4.10. Assume that ω
p
( p ∈ ) are ground states for the restriction of the group
σ(exp(−tβ
Q
F ω
)) to the subalgebras A p . Let the product-state
ω :=
p∈
ω
p
(6.4.37)
satisfy the ‘consistency condition’
ω(E g ( f ξ )) = F ω (ξ), f or all ξ ∈ g.
(6.4.38)
Then ω is a factor ground state of the evolution τ
Q . If all the ω
p are pure, then ω is
an extremal τ
Q -ground state.
Proof. The factoria1ity of ω is a consequence of cluster properties, cf. e.g. [53,
54]. The condition (6.4.7) is fulfilled for τ t := σ(exp(−β
Q
F ω
t)). An application of