6.3 Time Evolution in Generalized Mean-Field Theories
151
from the present section in the case of the UHF-algebra A := A
(cf. [53, 2.6.12],
[235, 6.4.1]; UHF:=“uniformly hyperfinite”) with the polynomial Q. This shows
also that the derivation δ Q for the case of a nonseparable A
and unbounded X ξ is
described by the same formulas as δ π is.
6.4 Equilibrium States
6.4.1 Let us consider in this section those states of physical systems which describe
the situations corresponding to the thermodynamic equilibrium at a given temperature
T ≥ 0. For quantal systems these states are specified usually by the KMS-condition,
cf. e.g. [54, 106, 235, 271]. We shall investgate here the KMS states
3 of systems
considered in this chapter, i.e. the systems specified by the triple (A; σ(G); τ
Q
),
cf. also [41]. To avoid possible technical complications, we shall concentrate our
attention here on the cases of strongly continuous time evolutions τ
Q including,
e.g. the cases described in 6.3.12(iv). Let us use the notation of Theorem 6.3.12,
hence C := E g (C
G
bs ) be the C
∗ -algebra of (generalized) observables describing the
considered system with the dynamics τ
Q . Instead of the above mentioned triple, we
shall use also the couple (C; τ
Q
) for denoting the system. In most of the analysis
of this section an additional structure of the system will be used. Let be a locally
compact noncompact group and π(() be its representation on C, i.e. π( p) ∈
∗ - Aut C
for all p ∈ . Let π(() commutes with τ
Q
:
τ
Q
t ◦ π( p) = π( p) ◦ τ
Q
t
for all t ∈ R, p ∈ .
(6.4.1)
We shall assume usually that π(() has some asymptotic abelianess properties.
As an example of such a π(() consider the situations described in Sect. 5.1. (i.e.
A := A
is a tensor product of the mutually commuting ‘local algebras’ A p :=
L(H p )), where the set Z + \ {0} is replaced by := Z
r (with easy modifications
of the whole formalism). Let us write π p : L(H) → L(H ) for the isomorphism
defined in (5.1.12), p ∈ . Now we define π( p) ∈
∗ - Aut A
by
π( p)(π j (A)) := π j+ p (A), for all A ∈ L(H), p, j ∈ .
(6.4.2)
Since the elements π j (A) ( j ∈ , A ∈ L(H)) generate A
, (6.4.2) determines an
automorphism π( p) of A
uniquely. This automorphism can be extended naturally
to an (equally denoted) automorphism group π(() of C := E g (C
G
bs ) by the relation
π( p)(E g ( f )) :=
π( p)( f (F)) E g (dF).
(6.4.3)
3 KMS is for Kubo, Martin and Schwinger.
151
from the present section in the case of the UHF-algebra A := A
(cf. [53, 2.6.12],
[235, 6.4.1]; UHF:=“uniformly hyperfinite”) with the polynomial Q. This shows
also that the derivation δ Q for the case of a nonseparable A
and unbounded X ξ is
described by the same formulas as δ π is.
6.4 Equilibrium States
6.4.1 Let us consider in this section those states of physical systems which describe
the situations corresponding to the thermodynamic equilibrium at a given temperature
T ≥ 0. For quantal systems these states are specified usually by the KMS-condition,
cf. e.g. [54, 106, 235, 271]. We shall investgate here the KMS states
3 of systems
considered in this chapter, i.e. the systems specified by the triple (A; σ(G); τ
Q
),
cf. also [41]. To avoid possible technical complications, we shall concentrate our
attention here on the cases of strongly continuous time evolutions τ
Q including,
e.g. the cases described in 6.3.12(iv). Let us use the notation of Theorem 6.3.12,
hence C := E g (C
G
bs ) be the C
∗ -algebra of (generalized) observables describing the
considered system with the dynamics τ
Q . Instead of the above mentioned triple, we
shall use also the couple (C; τ
Q
) for denoting the system. In most of the analysis
of this section an additional structure of the system will be used. Let be a locally
compact noncompact group and π(() be its representation on C, i.e. π( p) ∈
∗ - Aut C
for all p ∈ . Let π(() commutes with τ
Q
:
τ
Q
t ◦ π( p) = π( p) ◦ τ
Q
t
for all t ∈ R, p ∈ .
(6.4.1)
We shall assume usually that π(() has some asymptotic abelianess properties.
As an example of such a π(() consider the situations described in Sect. 5.1. (i.e.
A := A
is a tensor product of the mutually commuting ‘local algebras’ A p :=
L(H p )), where the set Z + \ {0} is replaced by := Z
r (with easy modifications
of the whole formalism). Let us write π p : L(H) → L(H ) for the isomorphism
defined in (5.1.12), p ∈ . Now we define π( p) ∈
∗ - Aut A
by
π( p)(π j (A)) := π j+ p (A), for all A ∈ L(H), p, j ∈ .
(6.4.2)
Since the elements π j (A) ( j ∈ , A ∈ L(H)) generate A
, (6.4.2) determines an
automorphism π( p) of A
uniquely. This automorphism can be extended naturally
to an (equally denoted) automorphism group π(() of C := E g (C
G
bs ) by the relation
π( p)(E g ( f )) :=
π( p)( f (F)) E g (dF).
(6.4.3)
3 KMS is for Kubo, Martin and Schwinger.