6.3 Time Evolution in Generalized Mean-Field Theories
139
introduced in 6.1.2 and 6.1.3 will be used here. Let us note that the equation (6.1.16)
for g Q can be written in any continuous unitary representation U (G) in H in the
form
i
d
dt
U (g Q (t, F)) = X (β
Q
F t
)U (g Q (t, F)), F ∈ g
∗
, t ∈ R,
(6.3.2)
where F t := ϕ
Q
t (F); X (ξ) := X ξ (ξ ∈ g) are the selfadjoint generators of U (G),
and β
Q
F ∈ g was introduced in 6.1.3. The equation (6.3.2) is of the form of (linear) quantum-mechanical evolution equation with the time-dependent Hamiltonian
operator X (β
Q
F t
). The equation (6.3.2) describes, in the setting of Sect. 5.1, the time
evolution of any ‘individual’ quantum subsystem placed in any fixed site k ∈ in
the surrounding ‘mean field’ ϕ
Q
t (F) ∈ g
∗ generated by the whole collection of the
quantal subsystems (for all the sites k ∈ ) interacting by an ‘infinitely weak and of
infinitely long-range’ interaction with each other. The equation (6.3.2) will be useful
in the analysis of thermodynamic properties of the considered systems.
6.3.2 Definitions.
(i) C b := C b (supp E g , C) will denote the set of all uniformly bounded complexvalued continuous functions on supp E g ⊂ g
∗ , see 5.2.3 for the definition of supp E g .
(ii) The s
∗ -topology on A is determined by seminorms ˆ
p ω , ˆ
p
∗
ω (cf. (6.2.19)) for all
ω ∈ p G S(A).
(iii) Let C bs := C bs (supp E g , A) be the set of all A-va1ued, uniformly bounded s
∗ -
continuous functions on supp E g , i.e. f ∈ C bs means that the function
f : F(∈ supp E g ) → f (F) (∈ A)
(6.3.3)
is bounded in the sense
f := sup{{ f (F) : F ∈ supp E g } < ∞,
(6.3.4)
and all the functions
F → ω
( f (F) − f (F 0 ))
∗
( f (F) − f (F 0 ))
, ω ∈ p G S(A), F 0 ∈ supp E g ,
(6.3.5a)
F → ω
( f (F) − f (F 0 ))( f (F) − f (F 0 ))
∗
, ω ∈ p G S(A), F 0 ∈ supp E g ,
(6.3.5b)
converge to zero for F converging to F 0 in the norm-topology of g
∗ .