156
6 Dynamics of Quantum Mechanical Macroscopic Systems
With the introduced notation and assumptions, we shall prove now the following:
6.4.10 Theorem. Let us consider a system (A; σ(G); τ
Q
; π(()) with simple
C
∗ -algebra A and ‘local’ subalgebras A
J
⊂ A being factors for all finite J . Let
ω ∈ S(A) and let ω
0 be the restriction of ω to the subalgebra A 0 (:= A
J with the
one-point set J containing the identity 0 ∈ ). Then the following two statements
are equivalent:
(i) ω is a locally normal extremal τ
Q -KMS state at a positive temperature β
−1
> 0.
(ii) ω= ω, where ω is the π(()-invariant product state determined by the relation
ω(π p 1 (x 1 )π p 2 (x 2 ) . . . π p m (x m )) =
m
j=1
ω
0
(x j ),
(6.4.18)
with x j ∈ A 0 , p j ∈ (p j = p k for j = k), j = 1, 2, . . . m, ∀m ∈ N and ω
0 is the
faithful normal KMS-state at β on A 0 corresponding to the one-parameter subgroup
{σ(exp(−tβ
Q
F ω
)) : t ∈ R} of
∗ - Aut A 0 with
ϕ
Q
t (F ω ) = F ω , for all t ∈ R
(6.4.19)
for some element F ω ∈ g
∗ . Moreover, the ‘consistency condition’
ω(E g ( f ξ )) = F ω (ξ), (ξ ∈ g, f ξ (F) := F(ξ) for F ∈ g
∗
)
(6.4.20)
is fulfilled.
4
Proof. (i) implies π ω (τ
Q
t (x)) = π ω (σ(exp(−tβ
Q
F ω
))(x)) according to (6.4.11). Hence
ω satisfies the KMS-condition with respect to the group σ(exp(−tβ
Q
F ω
)) at T
−1 and
the same is true for ω
0 , since σ(G)(A 0 ) = A 0 . Let X (β
Q
F ω
) be the restriction of
X ω (β
Q
F ω
) onto π ω (A 0 )) ω . ω is faithful on A (A is simple) and the cyclic vector ω
is separating for π ω (A)
, cf. [54, 5.3.9]. Hence ω(x
∗ x) = 0 for x = 0, and ω
0 is
faithful on A 0 . The local normality of ω implies normality of ω
0 . According to the
Takesaki’s theorem [54, 5.3.10], the one-parameter automorphism group of π ω (A 0 ):
t → exp(it X (β
Q
F ω
))π ω (x) exp(−it X (β
Q
F ω
)), x ∈ A 0 ,
(6.4.21)
coincides with the corresponding modular automorphism group of π ω (A 0 ) determined by the state ω
0 (up to a resca1ing of time t). According to [54, 5.3.29], the
KMS state at β := T
−1
∈ R on the factor A 0 corresponding to its automorphism
group σ(exp(−tβ
Q
F ω
)) is uniquely determined faithful normal state on A 0 .
4 The stationarity (6.4.19) is a consequence of the “consistency condition” (6.4.29), i.e. of (6.4.20);
hence (6.4.19), and (6.4.20) can be replaced by (6.4.29).
6 Dynamics of Quantum Mechanical Macroscopic Systems
With the introduced notation and assumptions, we shall prove now the following:
6.4.10 Theorem. Let us consider a system (A; σ(G); τ
Q
; π(()) with simple
C
∗ -algebra A and ‘local’ subalgebras A
J
⊂ A being factors for all finite J . Let
ω ∈ S(A) and let ω
0 be the restriction of ω to the subalgebra A 0 (:= A
J with the
one-point set J containing the identity 0 ∈ ). Then the following two statements
are equivalent:
(i) ω is a locally normal extremal τ
Q -KMS state at a positive temperature β
−1
> 0.
(ii) ω= ω, where ω is the π(()-invariant product state determined by the relation
ω(π p 1 (x 1 )π p 2 (x 2 ) . . . π p m (x m )) =
m
j=1
ω
0
(x j ),
(6.4.18)
with x j ∈ A 0 , p j ∈ (p j = p k for j = k), j = 1, 2, . . . m, ∀m ∈ N and ω
0 is the
faithful normal KMS-state at β on A 0 corresponding to the one-parameter subgroup
{σ(exp(−tβ
Q
F ω
)) : t ∈ R} of
∗ - Aut A 0 with
ϕ
Q
t (F ω ) = F ω , for all t ∈ R
(6.4.19)
for some element F ω ∈ g
∗ . Moreover, the ‘consistency condition’
ω(E g ( f ξ )) = F ω (ξ), (ξ ∈ g, f ξ (F) := F(ξ) for F ∈ g
∗
)
(6.4.20)
is fulfilled.
4
Proof. (i) implies π ω (τ
Q
t (x)) = π ω (σ(exp(−tβ
Q
F ω
))(x)) according to (6.4.11). Hence
ω satisfies the KMS-condition with respect to the group σ(exp(−tβ
Q
F ω
)) at T
−1 and
the same is true for ω
0 , since σ(G)(A 0 ) = A 0 . Let X (β
Q
F ω
) be the restriction of
X ω (β
Q
F ω
) onto π ω (A 0 )) ω . ω is faithful on A (A is simple) and the cyclic vector ω
is separating for π ω (A)
, cf. [54, 5.3.9]. Hence ω(x
∗ x) = 0 for x = 0, and ω
0 is
faithful on A 0 . The local normality of ω implies normality of ω
0 . According to the
Takesaki’s theorem [54, 5.3.10], the one-parameter automorphism group of π ω (A 0 ):
t → exp(it X (β
Q
F ω
))π ω (x) exp(−it X (β
Q
F ω
)), x ∈ A 0 ,
(6.4.21)
coincides with the corresponding modular automorphism group of π ω (A 0 ) determined by the state ω
0 (up to a resca1ing of time t). According to [54, 5.3.29], the
KMS state at β := T
−1
∈ R on the factor A 0 corresponding to its automorphism
group σ(exp(−tβ
Q
F ω
)) is uniquely determined faithful normal state on A 0 .
4 The stationarity (6.4.19) is a consequence of the “consistency condition” (6.4.29), i.e. of (6.4.20);
hence (6.4.19), and (6.4.20) can be replaced by (6.4.29).