138
6 Dynamics of Quantum Mechanical Macroscopic Systems
s- lim
K
δ π ([i Q
K
, x]
(m)
) = δ π (s- lim
K
[i Q
K
, x]
(m)
) = δ
m+1
π
(x).
(6.2.92)
Insertion from (6.2.91) and (6.2.92) into (6.2.45), cf. the note following (6.2.53),
gives for x ∈ B
N
, |t| ≤ κ N the norm-convergent series:
τ
π
t (x) =
∞
m=0
t
m
m!
δ
m
π (x).
(6.2.93)
This proves that the operator δ π from (6.2.83) determines τ
π .
6.3 Time Evolution in Generalized Mean-Field Theories
6.3.1 We shall construct in this section a general class of time evolutions τ
Q of the
infinite quantum systems (A; σ(G)) defined in Sect. 5.2. The time evolution τ
Q is
determined in a canonical way by an arbitrary classical Hamiltonian function Q on the
(generalized) homogeneous classical phase space g
∗ as well as by the automorphism
group σ(G) of A. It will be shown later that the here presented construction leads to
the same evolution what was denoted by τ
Q in Sect. 6.2 in the case of A := A
, σ(G)
being defined according to 5.1.5, and with Q being a polynomial in a basis of g
∗ dual to
any fixed basis {ξ j , j = 1, 2, . . . n}; the generators of the continuous representation
U (G) in the ‘one-spin space’ H, 5.1.3, are supposed to be bounded in this special
case.
We shall start with the general case, the specifications to the cases considered in
Sect. 5.1, and the further specification to the cases of Sect. 6.2 will be made later on.
Let us fix here some general assumptions valid throughout of this section.
Using the notation of Sect. 5.2, let E g be a fixed nontrivial G-measure associated
with the system (A; σ(G)) such that, with p G := E g (g
∗
), the following implication
is valid:
ω ∈ p G S(A) =: S g ⇒ g (∈ G) → ω(σ(g)(x)) is continuous for all x ∈ A.
(6.3.1)
It will be shown in 6.3.10 that this assumption is fulfilled by σ(G) from 5.1.5 with
p G from 5.1.29. We shall assume that A is a unital C
∗ -algebra which is simple (this
last assumption is made only for brevity of our expression). The nontriviality of E g
means a certain ‘breaking of symmetries’ occurring in the system, cf. our 5.2.3 for
basic definitions, and for illustration of the phenomenon of “spontaneous symmetry
breaking” see [106, 265], [53, Sect. 4.3.4], [41, IV.A], 6.5.5.
The time evolution τ
Q will be defined with a help of the group-valued function
g Q (t, F) on R × g
∗ defined in 6.1.3 with (6.1.17) (another possible choice of β
Q
F will
not change the general construction of τ
Q , so that the nonuniqueness of β
Q
F leads
to various possibilities for the definition of the time evolutions τ
Q ). The notation
6 Dynamics of Quantum Mechanical Macroscopic Systems
s- lim
K
δ π ([i Q
K
, x]
(m)
) = δ π (s- lim
K
[i Q
K
, x]
(m)
) = δ
m+1
π
(x).
(6.2.92)
Insertion from (6.2.91) and (6.2.92) into (6.2.45), cf. the note following (6.2.53),
gives for x ∈ B
N
, |t| ≤ κ N the norm-convergent series:
τ
π
t (x) =
∞
m=0
t
m
m!
δ
m
π (x).
(6.2.93)
This proves that the operator δ π from (6.2.83) determines τ
π .
6.3 Time Evolution in Generalized Mean-Field Theories
6.3.1 We shall construct in this section a general class of time evolutions τ
Q of the
infinite quantum systems (A; σ(G)) defined in Sect. 5.2. The time evolution τ
Q is
determined in a canonical way by an arbitrary classical Hamiltonian function Q on the
(generalized) homogeneous classical phase space g
∗ as well as by the automorphism
group σ(G) of A. It will be shown later that the here presented construction leads to
the same evolution what was denoted by τ
Q in Sect. 6.2 in the case of A := A
, σ(G)
being defined according to 5.1.5, and with Q being a polynomial in a basis of g
∗ dual to
any fixed basis {ξ j , j = 1, 2, . . . n}; the generators of the continuous representation
U (G) in the ‘one-spin space’ H, 5.1.3, are supposed to be bounded in this special
case.
We shall start with the general case, the specifications to the cases considered in
Sect. 5.1, and the further specification to the cases of Sect. 6.2 will be made later on.
Let us fix here some general assumptions valid throughout of this section.
Using the notation of Sect. 5.2, let E g be a fixed nontrivial G-measure associated
with the system (A; σ(G)) such that, with p G := E g (g
∗
), the following implication
is valid:
ω ∈ p G S(A) =: S g ⇒ g (∈ G) → ω(σ(g)(x)) is continuous for all x ∈ A.
(6.3.1)
It will be shown in 6.3.10 that this assumption is fulfilled by σ(G) from 5.1.5 with
p G from 5.1.29. We shall assume that A is a unital C
∗ -algebra which is simple (this
last assumption is made only for brevity of our expression). The nontriviality of E g
means a certain ‘breaking of symmetries’ occurring in the system, cf. our 5.2.3 for
basic definitions, and for illustration of the phenomenon of “spontaneous symmetry
breaking” see [106, 265], [53, Sect. 4.3.4], [41, IV.A], 6.5.5.
The time evolution τ
Q will be defined with a help of the group-valued function
g Q (t, F) on R × g
∗ defined in 6.1.3 with (6.1.17) (another possible choice of β
Q
F will
not change the general construction of τ
Q , so that the nonuniqueness of β
Q
F leads
to various possibilities for the definition of the time evolutions τ
Q ). The notation