158
6 Dynamics of Quantum Mechanical Macroscopic Systems
ω = ω of the states on A, hence the relation (6.4.18). [Warning: This does not imply
uniqueness of the τ
Q -KMS states on C, but we have proved uniqueness of the KMS
states on C with respect to one parameter groups σ(exp(tξ)) =: σ ξ (t). Different
extremal τ
Q -KMS states at the same temperature T give different values of F ω and
of β
Q
F ω
, hence lead to different one parameter groups σ ξ (ξ := -β
Q
F ω
).]
Let now ω
0
∈ S(A 0 ) be a given faithful normal KMS-state at the temperature
T > 0 corresponding to the group σ ξ with ξ := -β
Q
F ω
, where F ω ∈ g
∗ satisfies (6.4.19).
Then the product state ω from (6.4.18) is locally normal, since the finite product of
normal states is a normal state on the tensor product of W
∗ -algebras, [306, Sect.
IV.5]. The factoriality is trivial for product states, [53, 2.6.10]. According to the
Pusz-Woronowicz theorem, [54, 5.3.22], ω
0 satisfies the passivity condition (6.4.8)
with τ := σ ξ (ξ := -β
Q
F ω
). This implies the satisfaction of (6.4.8) with respect to the
same group by the state ω. The cluster property of the product state gives now the
KMS-property of ω with respect to the σ ξ . Since ω
0 satisfies σ ξ -KMS condition
with T = 0 positive, the same is true for ω. Since F ω is a fixed point of ϕ
Q , the
derivations of the σ ξ and of τ
Q coincide in the GNS-representations corresponding
to the states supported by E g (F ω ), cf. (6.4.14). The assumption (6.4.20) ensures,
that the macroscopic limit of the product state ω from (6.4.18) is concentrated on
F ω , hence the evolutions of proposition 6.4.7 τ
Q and σ ξ (ξ := -β
Q
F ω
) coincide in the
representation π ω corresponding to the state ω := ω from (6.4.18).
6.4.11 Corollary. Let A := A
and the system (A; σ(G); π(()) be defined according to Sect. 5.1, i.e. the G-measure E g is given by 5.1.33 and σ(G) is locally implementable in states ω ∈ S g . Let, with the assumptions of Theorem 6.4.10, ω be locally
normal extremal τ
Q -KMS state at T > 0. Let X ξ (ξ ∈ g) be the generators of the
(σ(G)-defining) representation U (G) on H 0 := H, A 0 = L(H), σ(exp(tξ))(y) :=
exp(−it X ξ ) y exp(it X ξ ) for all y ∈ A 0 . Then
ω
0
(ex p(it X ξ )) = exp(it F ω (ξ)), ∀ξ ∈ g,
(6.4.29)
where F ω is given by the (trivially fulfilled) ‘consistency condition’
ω(E g ( f ξ )) = F ω (ξ), ξ ∈ g.
(6.4.30)
Proof. Since exp(it X ξ ) ∈ A 0 , the generators of the restriction of σ(G) onto A p :=
π p (A 0 ) are π p (X ξ ), where
exp(itπ p (X ξ )) := π p (exp(it X ξ )).
(6.4.31)
The generators of the restriction of σ(G) onto A
J (finite J ⊂ ) are X
J
ξ :=
p∈J π p (X ξ ),
exp(it X
J
ξ ) :=
p∈J
exp(itπ p (X ξ )) ∈ A
J
.
(6.4.32)
6 Dynamics of Quantum Mechanical Macroscopic Systems
ω = ω of the states on A, hence the relation (6.4.18). [Warning: This does not imply
uniqueness of the τ
Q -KMS states on C, but we have proved uniqueness of the KMS
states on C with respect to one parameter groups σ(exp(tξ)) =: σ ξ (t). Different
extremal τ
Q -KMS states at the same temperature T give different values of F ω and
of β
Q
F ω
, hence lead to different one parameter groups σ ξ (ξ := -β
Q
F ω
).]
Let now ω
0
∈ S(A 0 ) be a given faithful normal KMS-state at the temperature
T > 0 corresponding to the group σ ξ with ξ := -β
Q
F ω
, where F ω ∈ g
∗ satisfies (6.4.19).
Then the product state ω from (6.4.18) is locally normal, since the finite product of
normal states is a normal state on the tensor product of W
∗ -algebras, [306, Sect.
IV.5]. The factoriality is trivial for product states, [53, 2.6.10]. According to the
Pusz-Woronowicz theorem, [54, 5.3.22], ω
0 satisfies the passivity condition (6.4.8)
with τ := σ ξ (ξ := -β
Q
F ω
). This implies the satisfaction of (6.4.8) with respect to the
same group by the state ω. The cluster property of the product state gives now the
KMS-property of ω with respect to the σ ξ . Since ω
0 satisfies σ ξ -KMS condition
with T = 0 positive, the same is true for ω. Since F ω is a fixed point of ϕ
Q , the
derivations of the σ ξ and of τ
Q coincide in the GNS-representations corresponding
to the states supported by E g (F ω ), cf. (6.4.14). The assumption (6.4.20) ensures,
that the macroscopic limit of the product state ω from (6.4.18) is concentrated on
F ω , hence the evolutions of proposition 6.4.7 τ
Q and σ ξ (ξ := -β
Q
F ω
) coincide in the
representation π ω corresponding to the state ω := ω from (6.4.18).
6.4.11 Corollary. Let A := A
and the system (A; σ(G); π(()) be defined according to Sect. 5.1, i.e. the G-measure E g is given by 5.1.33 and σ(G) is locally implementable in states ω ∈ S g . Let, with the assumptions of Theorem 6.4.10, ω be locally
normal extremal τ
Q -KMS state at T > 0. Let X ξ (ξ ∈ g) be the generators of the
(σ(G)-defining) representation U (G) on H 0 := H, A 0 = L(H), σ(exp(tξ))(y) :=
exp(−it X ξ ) y exp(it X ξ ) for all y ∈ A 0 . Then
ω
0
(ex p(it X ξ )) = exp(it F ω (ξ)), ∀ξ ∈ g,
(6.4.29)
where F ω is given by the (trivially fulfilled) ‘consistency condition’
ω(E g ( f ξ )) = F ω (ξ), ξ ∈ g.
(6.4.30)
Proof. Since exp(it X ξ ) ∈ A 0 , the generators of the restriction of σ(G) onto A p :=
π p (A 0 ) are π p (X ξ ), where
exp(itπ p (X ξ )) := π p (exp(it X ξ )).
(6.4.31)
The generators of the restriction of σ(G) onto A
J (finite J ⊂ ) are X
J
ξ :=
p∈J π p (X ξ ),
exp(it X
J
ξ ) :=
p∈J
exp(itπ p (X ξ )) ∈ A
J
.
(6.4.32)