6.1 General Considerations
115
τ
Q
t (x) := (some topolog y) − lim
N →∞
τ
N
t (x)
(6.1.5)
exist for some t > 0 and for sufficiently many x ∈ A, so that τ
Q
t could be extended
to a one parameter group (resp. semigroup) of mappings of A (or of some of its
completions) representing in a reasonable manner some time translations. We shall
show that this is indeed the case, and not only for the spin systems. The resulting
family of transformations τ
Q does not consist, however, (for general Q) of automorphisms of the original (i.e. that one used at the determination of the infinite
system) quasilocal C
∗ -algebra A. The family of
∗ -isomorphisms of A, τ
Q , can be
extended to a one parameter group τ
Q of
∗ -automorphism of a C
∗ -subalgebra of A
∗∗
containing A as a C
∗ -subalgebra. The resulting picture of the τ
Q -time evolution has
the properties of the quantum mean-field evolutions according to the usual understanding, cf. also [40]. We shall write down explicit formulas for the evolution of
an arbitrary element of the extended algebra of observables (including also an algebra of classical—intensive—observables) in terms of solutions of finite dimensional
differential equations.
In the present section, we shall introduce some basic concepts used in the general
construction of the automorphism group τ
Q . We shall sketch here also a scheme
of the general construction. Details will be proved in the following sections of this
chapter.
6.1.2 Let (g
∗
, λ; ϕ G ) be a Poisson manifold with the Poisson action ϕ G of the
Lie group G the orbits of which coincide with the maximal integral submanifolds
of the Poisson structure λ, cf. [212], and 5.1.37, and 5.2.2. We shall assume, for
simplicity, that ϕ G := Ad
∗
(G) and λ F (d f, dg) := −F([d F f, d F g]) for all f, g ∈
C
∞
(g
∗
, R). Let Q ∈ C
∞
(g
∗
, R) be such a fixed function on g
∗ , that the corresponding
Hamiltonian vector field σ Q on g
∗ , (5.1.144), is complete. This means that there
is a one parameter group t → ϕ
Q
t (ϕ
Q
t+s = ϕ
Q
t ◦ ϕ
Q
s for all t, s ∈ R) of Poisson
morphisms of (g
∗
; λ) the derivative of which is σ Q . Remember that σ Q is complete for
any Q in the case of compact groups G, in which case the Ad
∗
(G)-orbits are compact.
The tangent spaces T F g
∗
(F ∈ g
∗
) will be identified with the linear manifold g
∗ in
the canonical way. Then we have also the canonical identification T
∗
F g
∗
= g of the
cotangent spaces in any point F ∈ g
∗ with the Lie algebra g of G. Let f ξ ∈ C
∞
(g
∗
, R)
(for any ξ ∈ g) be the linear function
f ξ : F → f ξ (F) := F(ξ) .
Any element ξ of the Lie algebra g determines also a covector field on g
∗ :
d f ξ : F → d F f ξ = ξ ∈ g = T
∗
F g
∗
.
(6.1.6)
115
τ
Q
t (x) := (some topolog y) − lim
N →∞
τ
N
t (x)
(6.1.5)
exist for some t > 0 and for sufficiently many x ∈ A, so that τ
Q
t could be extended
to a one parameter group (resp. semigroup) of mappings of A (or of some of its
completions) representing in a reasonable manner some time translations. We shall
show that this is indeed the case, and not only for the spin systems. The resulting
family of transformations τ
Q does not consist, however, (for general Q) of automorphisms of the original (i.e. that one used at the determination of the infinite
system) quasilocal C
∗ -algebra A. The family of
∗ -isomorphisms of A, τ
Q , can be
extended to a one parameter group τ
Q of
∗ -automorphism of a C
∗ -subalgebra of A
∗∗
containing A as a C
∗ -subalgebra. The resulting picture of the τ
Q -time evolution has
the properties of the quantum mean-field evolutions according to the usual understanding, cf. also [40]. We shall write down explicit formulas for the evolution of
an arbitrary element of the extended algebra of observables (including also an algebra of classical—intensive—observables) in terms of solutions of finite dimensional
differential equations.
In the present section, we shall introduce some basic concepts used in the general
construction of the automorphism group τ
Q . We shall sketch here also a scheme
of the general construction. Details will be proved in the following sections of this
chapter.
6.1.2 Let (g
∗
, λ; ϕ G ) be a Poisson manifold with the Poisson action ϕ G of the
Lie group G the orbits of which coincide with the maximal integral submanifolds
of the Poisson structure λ, cf. [212], and 5.1.37, and 5.2.2. We shall assume, for
simplicity, that ϕ G := Ad
∗
(G) and λ F (d f, dg) := −F([d F f, d F g]) for all f, g ∈
C
∞
(g
∗
, R). Let Q ∈ C
∞
(g
∗
, R) be such a fixed function on g
∗ , that the corresponding
Hamiltonian vector field σ Q on g
∗ , (5.1.144), is complete. This means that there
is a one parameter group t → ϕ
Q
t (ϕ
Q
t+s = ϕ
Q
t ◦ ϕ
Q
s for all t, s ∈ R) of Poisson
morphisms of (g
∗
; λ) the derivative of which is σ Q . Remember that σ Q is complete for
any Q in the case of compact groups G, in which case the Ad
∗
(G)-orbits are compact.
The tangent spaces T F g
∗
(F ∈ g
∗
) will be identified with the linear manifold g
∗ in
the canonical way. Then we have also the canonical identification T
∗
F g
∗
= g of the
cotangent spaces in any point F ∈ g
∗ with the Lie algebra g of G. Let f ξ ∈ C
∞
(g
∗
, R)
(for any ξ ∈ g) be the linear function
f ξ : F → f ξ (F) := F(ξ) .
Any element ξ of the Lie algebra g determines also a covector field on g
∗ :
d f ξ : F → d F f ξ = ξ ∈ g = T
∗
F g
∗
.
(6.1.6)