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6 Dynamics of Quantum Mechanical Macroscopic Systems
The group π(() is norm-asymptotically abelian, i.e.
lim
p→∞
[π( p)(x), y]] = 0, for all x, y ∈ C.
(6.4.4)
In more general cases, the abelianess properties of the action of on C can be
weaker. Systems with this structure will be denoted
(C; τ
Q
; π(()), or (A; σ(G); τ
Q
; π(()).
We shall use, as usual, β := T
−1
:= (kT )
−1 to denote the inverse temperature in
convenient units. The following definitions are found e.g. in [54, 5.3.1, 5.3.18, and
5.3.21], and [235, 8.12].
6.4.2 Definition. Let (C, τ ) be a C
∗ -dynamical system, i.e. the one parameter group
τ ⊂
∗ - Aut C is strongly continuous. The state ω ∈ S(C) is defined to be a τ -KMS
state at value β ∈ R, or a (τ , β)-KMS state, if
ω(x τ iβ (y)) = ω(yx), for all x, y, ∈ C
◦
τ ,
(6.4.5)
where C
◦
τ is a norm-dense, τ -invariant
∗ -subalgebra of the set C τ of the entire analytic
elements of C:
y ∈ C
◦
τ ⇔ the function z → τ z (y) is analytic for all z ∈ C.
(6.4.6)
Let δ τ be the generator of τ . Then ω ∈ S(C) is called a τ -ground state if
− i ω(y
∗
δ τ (y)) ≥ 0, for all y ∈ D(δ τ ).
(6.4.7)
In this case, ω is also called a τ -KMS state at value β = ∞.
6.4.3 Definition. Let (C; τ ) be a C
∗ -dynamical system with a unital C
∗ -algebra C,
and let δ τ be the infinitesimal generator of τ . Then ω ∈ S(C) is said to be a passive
state if
− i ω(u
∗
δ τ (u)) ≥ 0
(6.4.8)
for any u ∈ D(δ τ ) belonging also to the connected component of the identity of the
unitary group of Cin the norm topology.
6.4.4 Let us collect here some important properties of the sets K β of (τ , β)-KMS
states:
Proofs of the listed facts can be found in [54, Chap. 5], or in [275, 4.3]. We shall
consider β ∈ (0, ∞], the set K ∞ being the set of all ground states ω ∈ S(C). Let
(C, τ ) be a C
∗ -dynamical system. Then: