148
6 Dynamics of Quantum Mechanical Macroscopic Systems
δ Q (E g ( f )) :=
d
dt
t=0
τ
Q
t (E g ( f )) =
(6.3.43)
=
n
j=1
∂ j f (F){Q, F j }(F) − ∂ j Q(F) δ ξ j ( f (F))
E g (dF),
where the derivation is taken in the σ(C, S g )-topology, the symbol ∂ j f (F) means
the derivative of a function on g
∗ with respect to the variable F j in the point F ∈ g
∗ ,
and the meaning of the integral is explained in Proposition 6.3.6. {Q, F j } is here the
Poisson bracket on g
∗ , 6.1.2.
(iv) If the group σ(G) is strongly continuous (i.e. g → σ g (x) is continuous in norm
for each x ∈ A), and if K is compact, then the group τ
Q
t will be strongly continuous.
Proof. The group τ
Q is considered here as an automorphism group of the τ
Q -
invariant subalgebra E g (C
G
bs ) =: C of p G A
∗∗ .
(i) The invariance of N
c is given by the invariance of C b with respect to the transformations (6.3.36), which is valid due to the invariance of scalars in A with respect to
σ(G) : σ g (λ id A ) = λ id A , λ ∈ C, ∀g ∈ G. Similarly, the relation σ(G)(A
J
) = A
J
gives the τ
Q -invariance of C
J .
(ii) The continuity of the functions t → ω(τ
Q
t (y)) (ω ∈ S g , y ∈ C) can be obtained
from the definition of the evolution f → f t in C
G
bs as well as from the definition
(6.3.16) of E g ( f ) as follows:
Due to the s
∗ -bicontinuity of the mappings (6.3.35) and due to the (bi-)continuity
of the functions g Q and ϕ
Q , the functions
(m) : t → t (m) := ω m ( f t (F m )), m ∈ supp μ ω ,
(6.3.44)
are continuous for any fixed ω ∈ S g and f ∈ C
G
bs . We have proved in (6.3.19) the measurability of all the functions t : m → t (m). Since | t (m)| ≤ ≤ f (t ∈ R, m ∈
supp μ ω ) and μ ω is finite, an application of the Lebesgue dominated convergence
theorem gives
lim
t→0
ω(τ
Q
t (E g ( f )) = lim
t→0
t (m) μ ω (dm) =
0 (m) μ ω (dm) = ω(E g ( f )).
(6.3.45)
This gives the desired continuity.
Any τ
Q
t can be considered as a
∗ -automorphism of the W
∗ -algebra p G A
∗∗ , and
each such automorphism is σ( p G A
∗∗
, S g ) − σ( p G A
∗∗
, S g )-continuous, cf. [274,
4.1.23]. This implies that the state ω ◦ τ
Q
t is a normal state on p G A
∗∗ together with
ω, hence ω ∈ S g implies that ω ◦ τ
Q
t ∈ S g .
6 Dynamics of Quantum Mechanical Macroscopic Systems
δ Q (E g ( f )) :=
d
dt
t=0
τ
Q
t (E g ( f )) =
(6.3.43)
=
n
j=1
∂ j f (F){Q, F j }(F) − ∂ j Q(F) δ ξ j ( f (F))
E g (dF),
where the derivation is taken in the σ(C, S g )-topology, the symbol ∂ j f (F) means
the derivative of a function on g
∗ with respect to the variable F j in the point F ∈ g
∗ ,
and the meaning of the integral is explained in Proposition 6.3.6. {Q, F j } is here the
Poisson bracket on g
∗ , 6.1.2.
(iv) If the group σ(G) is strongly continuous (i.e. g → σ g (x) is continuous in norm
for each x ∈ A), and if K is compact, then the group τ
Q
t will be strongly continuous.
Proof. The group τ
Q is considered here as an automorphism group of the τ
Q -
invariant subalgebra E g (C
G
bs ) =: C of p G A
∗∗ .
(i) The invariance of N
c is given by the invariance of C b with respect to the transformations (6.3.36), which is valid due to the invariance of scalars in A with respect to
σ(G) : σ g (λ id A ) = λ id A , λ ∈ C, ∀g ∈ G. Similarly, the relation σ(G)(A
J
) = A
J
gives the τ
Q -invariance of C
J .
(ii) The continuity of the functions t → ω(τ
Q
t (y)) (ω ∈ S g , y ∈ C) can be obtained
from the definition of the evolution f → f t in C
G
bs as well as from the definition
(6.3.16) of E g ( f ) as follows:
Due to the s
∗ -bicontinuity of the mappings (6.3.35) and due to the (bi-)continuity
of the functions g Q and ϕ
Q , the functions
(m) : t → t (m) := ω m ( f t (F m )), m ∈ supp μ ω ,
(6.3.44)
are continuous for any fixed ω ∈ S g and f ∈ C
G
bs . We have proved in (6.3.19) the measurability of all the functions t : m → t (m). Since | t (m)| ≤ ≤ f (t ∈ R, m ∈
supp μ ω ) and μ ω is finite, an application of the Lebesgue dominated convergence
theorem gives
lim
t→0
ω(τ
Q
t (E g ( f )) = lim
t→0
t (m) μ ω (dm) =
0 (m) μ ω (dm) = ω(E g ( f )).
(6.3.45)
This gives the desired continuity.
Any τ
Q
t can be considered as a
∗ -automorphism of the W
∗ -algebra p G A
∗∗ , and
each such automorphism is σ( p G A
∗∗
, S g ) − σ( p G A
∗∗
, S g )-continuous, cf. [274,
4.1.23]. This implies that the state ω ◦ τ
Q
t is a normal state on p G A
∗∗ together with
ω, hence ω ∈ S g implies that ω ◦ τ
Q
t ∈ S g .