118
6 Dynamics of Quantum Mechanical Macroscopic Systems
Then we have for the Poisson bracket of two classical Hamiltonians Q 1 and Q 2 the
expression (called also the Berezin bracket):
{Q 1 , Q 2 }(F) := −F([d F Q 1 , d F Q 2 ]) = −c
j
km
∂ Q 1
∂ F k
∂ Q 2
∂ F m
F j .
(6.1.20)
6.1.4 Let us describe here, in a heuristic manner, the basic idea leading to the definition of the time evolutions τ
Q mentioned in 6.1.1 which will be described in Sect. 6.3
in details. It will be also shown in Sect. 6.3 that the evolutions obtained from the thermodynamic limits in the ‘polynomial cases’ (mentioned in 6.1.1 and investigated in
Sect. 6.2) are special cases of the general definition of τ
Q based on the following
general ideas.
The cocycle g Q reproduces an arbitrary classical Hamiltonian evolution on the
Poisson manifold (g
∗
, λ; Ad
∗
(G)) (since Q is an arbitrary Hamiltonian function) via
the given (fixed!) action Ad
∗
(G), cf. (6.1.9). We have given an action σ(G) ∈
∗ - Aut A
and also the corresponding dual action σ
∗
(G) on the set S(A) of states on A, cf.
(5.1.44). We have also a canonical decomposition of an arbitrary state ω ∈ S(A)
into the states ω m corresponding to classical phase space points m ∈ M, namely
(5.1.146), resp. the corresponding statement in 5.2.11. For ω ∈ S g := p G S(A), the
states ω m lying in the support of the corresponding measure ˆ
μ ω on S(A
∗∗
) can be
indexed by F m ∈ g
∗ , where the classical measure on g
∗ corresponding to the state
ω m ∈ E g is concentrated on the one point set {F m }, cf. 5.1.36 and 5.1.39. Hence we
can use the family of mappings
t → σ
∗
(g Q (t, F m )), t ∈ R, m ∈ M,
(6.1.21)
for a definition of time translations of the states ω m . Such a definition makes
sense since the projection measure E g (:= the G-macroscopic limit of the system
(A; σ(G)) in (g
∗
, λ; Ad
∗
(G)), 5.2.10) is G-equivariant, (5.2.5c), what implies that
the classical point-measure corresponding to σ
∗
(g Q (t, F m ))ω m ∈ E g is concentrated
on Ad
∗
(g Q (t, F m ))F m = ϕ
Q
t (F m ) ∈ g
∗ ; hence the cocycle identity (6.1.13) can be
used to prove the group property of mappings (6.1.21). A heuristic definition of the
time evolution τ
Q is then given with the help of the decomposition (5.1.146) by the
formula:
ω(τ
Q
t (x)) :=
M
σ
∗
(g Q (t, F m ))ω m (x) μ ω (dm), ∀t ∈ R, ω ∈ S g .
(6.1.22)
We shall see in Sect. 6.3 that this intuitive construction leads to a rigorously
defined group τ
Q of
∗ -automorphisms of a C
∗ -subalgebra of the W
∗ -algebra p G A
∗∗
containing the algebra A as well as an algebra N
c of classical observables in a natural
manner. The algebra N
c is then τ
Q -invariant: τ
Q
R (N
c
) = N
c
, contrary to the algebra
A (in a general case).
6.1.5 Remark. The general definition of mean-field time evolutions τ
Q based on
the formula (6.1.22) depends on a topology determined by the subset S g := p G S(A)
6 Dynamics of Quantum Mechanical Macroscopic Systems
Then we have for the Poisson bracket of two classical Hamiltonians Q 1 and Q 2 the
expression (called also the Berezin bracket):
{Q 1 , Q 2 }(F) := −F([d F Q 1 , d F Q 2 ]) = −c
j
km
∂ Q 1
∂ F k
∂ Q 2
∂ F m
F j .
(6.1.20)
6.1.4 Let us describe here, in a heuristic manner, the basic idea leading to the definition of the time evolutions τ
Q mentioned in 6.1.1 which will be described in Sect. 6.3
in details. It will be also shown in Sect. 6.3 that the evolutions obtained from the thermodynamic limits in the ‘polynomial cases’ (mentioned in 6.1.1 and investigated in
Sect. 6.2) are special cases of the general definition of τ
Q based on the following
general ideas.
The cocycle g Q reproduces an arbitrary classical Hamiltonian evolution on the
Poisson manifold (g
∗
, λ; Ad
∗
(G)) (since Q is an arbitrary Hamiltonian function) via
the given (fixed!) action Ad
∗
(G), cf. (6.1.9). We have given an action σ(G) ∈
∗ - Aut A
and also the corresponding dual action σ
∗
(G) on the set S(A) of states on A, cf.
(5.1.44). We have also a canonical decomposition of an arbitrary state ω ∈ S(A)
into the states ω m corresponding to classical phase space points m ∈ M, namely
(5.1.146), resp. the corresponding statement in 5.2.11. For ω ∈ S g := p G S(A), the
states ω m lying in the support of the corresponding measure ˆ
μ ω on S(A
∗∗
) can be
indexed by F m ∈ g
∗ , where the classical measure on g
∗ corresponding to the state
ω m ∈ E g is concentrated on the one point set {F m }, cf. 5.1.36 and 5.1.39. Hence we
can use the family of mappings
t → σ
∗
(g Q (t, F m )), t ∈ R, m ∈ M,
(6.1.21)
for a definition of time translations of the states ω m . Such a definition makes
sense since the projection measure E g (:= the G-macroscopic limit of the system
(A; σ(G)) in (g
∗
, λ; Ad
∗
(G)), 5.2.10) is G-equivariant, (5.2.5c), what implies that
the classical point-measure corresponding to σ
∗
(g Q (t, F m ))ω m ∈ E g is concentrated
on Ad
∗
(g Q (t, F m ))F m = ϕ
Q
t (F m ) ∈ g
∗ ; hence the cocycle identity (6.1.13) can be
used to prove the group property of mappings (6.1.21). A heuristic definition of the
time evolution τ
Q is then given with the help of the decomposition (5.1.146) by the
formula:
ω(τ
Q
t (x)) :=
M
σ
∗
(g Q (t, F m ))ω m (x) μ ω (dm), ∀t ∈ R, ω ∈ S g .
(6.1.22)
We shall see in Sect. 6.3 that this intuitive construction leads to a rigorously
defined group τ
Q of
∗ -automorphisms of a C
∗ -subalgebra of the W
∗ -algebra p G A
∗∗
containing the algebra A as well as an algebra N
c of classical observables in a natural
manner. The algebra N
c is then τ
Q -invariant: τ
Q
R (N
c
) = N
c
, contrary to the algebra
A (in a general case).
6.1.5 Remark. The general definition of mean-field time evolutions τ
Q based on
the formula (6.1.22) depends on a topology determined by the subset S g := p G S(A)