146
6 Dynamics of Quantum Mechanical Macroscopic Systems
ϕ
∗
: f → ϕ
∗ f , ϕ
∗ f (F) := f (ϕF), F ∈ K ⊂ g
∗
,
(6.3.37)
as well as of σ(g) ∈
∗ - Aut A. The group property follows immediately from the
group property of the flow ϕ
Q and from the cocyc1e property (6.1.13) of g Q . The
group property implies invertibility, hence isometry of the considered mappings.
6.3.9 We have just proved existence of a certain ‘time evolution’ in the C
∗ -algebra
C bs containing A and N
c . This evolution is determined by an arbitrary classical
Hamiltonian function Q and by the representation σ(G) of the group G of ‘macroscopic symmetries’ with the help of the formula (6.3.36). To have possibility to see
eonnections with the ‘mean-field evolutions’ discussed in Sect. 6.2, we shall transfer
this evolution into A
∗∗ by a use of the isomorphism E g from (6.3.15). We shall see
that the time evolutions defined by a limiting procedure in Sect. 6.2 can be defined
directly by the formula (6.3.36) (transferred into p G A
∗∗ ). The same possibility of a
definition of ‘mean-field evolutions’ arises in all the systems considered in Sect. 5.1.
To make this possibility clear, let us prove the property (6.3.1) for those systems.
6.3.10 Lemma. Let us consider the systems determined with a help of infinite tensor
product considered in Sect. 5.1. Then the group σ(G) ⊂
∗ - Aut A (A := A
) has the
property (6.3.1): The functions g → σ g (x) on G are s
∗ -continuous for all x ∈ A,
the s
∗ -continuity being determined by the seminorms ˆ
p ω and ˆ
p
∗
ω from (6.2.19) with
ω ∈ p G S(A), and p G was defined in 5.1.29.
Proof. The implication “(6.3.1) ⇒ s
∗ -continuity” was proved in Lemma 6.3.4. Since
the set of local elements x ∈ ∪ N ⊂ A
N is norm-dense in A, it suffices to prove the
continuity in (6.3.1) for x local. We have assumed in 5.1.29 the existence of the
generators X
N
ξ (ξ ∈ g, N ⊂ ) of all one parameter subgroups of the unitary group
V N (G) acting in H N , cf. 4.3.8 and 5.1.2, as well as the existence of (equally denoted)
generators for the unitary groups p G π u (V N (exp(ξt))) for all ξ ∈ g. For ω ∈ p G S(A)
and x ∈ A
N we have
ω(σ(exp(ξt))(x)) = (( ω , exp(−it X
N
ξ )π u (x) exp(it X
N
ξ )) ω ),
(6.3.38)
what continuously depends on t. We have to prove the strong-continuity of the group
U (g) := p G π u (V N (g)) from the strong continuity of all one parameter subgroups
U (exp ξt) =: exp(−it X ξ ), (ξ ∈ g); we write here X ξ instead of X
N
ξ . Let ξ j ∈ g,
j = 1, 2, . . . n be a fixed basis in g and set X j := X ξ j . Let us parametrize g ∈ G in a
neighbourhood of the unity e ∈ G by t := (t 1 , t 2 , . . . t n ) ∈ R
n in the following way,
cf. [152, Lemma II.2.4]:
g ≡ g(t) := exp(t 1 ξ 1 ) exp(t 2 ξ 2 ) . . . exp(t n ξ n ).
(6.3.39)
Now we can prove weak continuity of U (g(t)) in t = 0 ∈ R
n from the known
strong continuity of U j (t) := U (exp ξ j t) = ex p(−it X j ), for all j = 1, 2, . . . n.
Since U is a representation of G, we can write
6 Dynamics of Quantum Mechanical Macroscopic Systems
ϕ
∗
: f → ϕ
∗ f , ϕ
∗ f (F) := f (ϕF), F ∈ K ⊂ g
∗
,
(6.3.37)
as well as of σ(g) ∈
∗ - Aut A. The group property follows immediately from the
group property of the flow ϕ
Q and from the cocyc1e property (6.1.13) of g Q . The
group property implies invertibility, hence isometry of the considered mappings.
6.3.9 We have just proved existence of a certain ‘time evolution’ in the C
∗ -algebra
C bs containing A and N
c . This evolution is determined by an arbitrary classical
Hamiltonian function Q and by the representation σ(G) of the group G of ‘macroscopic symmetries’ with the help of the formula (6.3.36). To have possibility to see
eonnections with the ‘mean-field evolutions’ discussed in Sect. 6.2, we shall transfer
this evolution into A
∗∗ by a use of the isomorphism E g from (6.3.15). We shall see
that the time evolutions defined by a limiting procedure in Sect. 6.2 can be defined
directly by the formula (6.3.36) (transferred into p G A
∗∗ ). The same possibility of a
definition of ‘mean-field evolutions’ arises in all the systems considered in Sect. 5.1.
To make this possibility clear, let us prove the property (6.3.1) for those systems.
6.3.10 Lemma. Let us consider the systems determined with a help of infinite tensor
product considered in Sect. 5.1. Then the group σ(G) ⊂
∗ - Aut A (A := A
) has the
property (6.3.1): The functions g → σ g (x) on G are s
∗ -continuous for all x ∈ A,
the s
∗ -continuity being determined by the seminorms ˆ
p ω and ˆ
p
∗
ω from (6.2.19) with
ω ∈ p G S(A), and p G was defined in 5.1.29.
Proof. The implication “(6.3.1) ⇒ s
∗ -continuity” was proved in Lemma 6.3.4. Since
the set of local elements x ∈ ∪ N ⊂ A
N is norm-dense in A, it suffices to prove the
continuity in (6.3.1) for x local. We have assumed in 5.1.29 the existence of the
generators X
N
ξ (ξ ∈ g, N ⊂ ) of all one parameter subgroups of the unitary group
V N (G) acting in H N , cf. 4.3.8 and 5.1.2, as well as the existence of (equally denoted)
generators for the unitary groups p G π u (V N (exp(ξt))) for all ξ ∈ g. For ω ∈ p G S(A)
and x ∈ A
N we have
ω(σ(exp(ξt))(x)) = (( ω , exp(−it X
N
ξ )π u (x) exp(it X
N
ξ )) ω ),
(6.3.38)
what continuously depends on t. We have to prove the strong-continuity of the group
U (g) := p G π u (V N (g)) from the strong continuity of all one parameter subgroups
U (exp ξt) =: exp(−it X ξ ), (ξ ∈ g); we write here X ξ instead of X
N
ξ . Let ξ j ∈ g,
j = 1, 2, . . . n be a fixed basis in g and set X j := X ξ j . Let us parametrize g ∈ G in a
neighbourhood of the unity e ∈ G by t := (t 1 , t 2 , . . . t n ) ∈ R
n in the following way,
cf. [152, Lemma II.2.4]:
g ≡ g(t) := exp(t 1 ξ 1 ) exp(t 2 ξ 2 ) . . . exp(t n ξ n ).
(6.3.39)
Now we can prove weak continuity of U (g(t)) in t = 0 ∈ R
n from the known
strong continuity of U j (t) := U (exp ξ j t) = ex p(−it X j ), for all j = 1, 2, . . . n.
Since U is a representation of G, we can write