114
6 Dynamics of Quantum Mechanical Macroscopic Systems
multiplication of variables in such a way that the element Q(ξ 1 , ξ 2 , . . . ξ n ) of the Lie
algebra envelope has the following property:
(SA) Let Q ≡
q
k=0 Q k , where Q k is a homogeneous polynomial of degree k. In
any continuous unitary representation U of the group G on a separable Hilbert space
H, the operators Q k (X 1 , X 2 , . . . , X n ) defined on analytic elements of U are essentially selfadjoint, for all k = 1, 2, . . . , q.
This property (SA) is fulfilled e.g. if all the Q k (ξ 1 , ξ 2 , . . . , ξ n ) are symmetric
and elliptic, cf. [13, Chap. 11]. Then we define the local Hamiltonians Q
N , with
(5.1.14), denoting by N also an N -point subset of :
Q
N
:= N Q(X 1N , X 2N , . . . , X nN ), X j N :=
1
N
X
N
j , N = 1, 2, . . . ; (6.1.1)
cf. (5.1.118), i.e.
X
N
j :=
|N |
k=1
π k (X j ), j = 1, 2, . . . n, k ∈ N ⊂ ,
which can be considered as (essentially) selfadjoint operators on H (= H N ⊗
H \N ≡ -tuple tensor product). For any x ∈ A (:= A
) ⊂ L(H ) we set
τ
N
t (x) := exp(it Q
N
) x exp(−it Q
N
), t ∈ R,
(6.1.2)
and these mappings τ
N
t clearly form a one parameter group of
∗ -automorphisms
of A for each finite N . Systems of this type were introduced in [155] for the case
of spin systems (i.e. dim H was finite). It was shown in [40, 155] that the sequence
{τ
N
: N = 1, 2, . . . } determines an evolution τ
Q of the observables of the form X ξ ,
cf. 5.1.7 and 5.1.9, which is expressed in our notation by the formula
τ
Q
t (X ξ ) := w
∗
0 - lim
N →∞
τ
N
t (X ξ N ) =
f ξ (ϕ
Q
t F) E g (dF),
(6.1.3)
where w
∗
0 -topology on a von Neumann algebra containing A and X ξ (ξ ∈ g) is
determined by the set of the ‘classical states’. The integral in (6.1.3) corresponds to
the integral in [155, (2.29)], which, specified to our case, reads:
lim
N →∞
ω(τ
N
t (X ξ N )) =
f ξ (ϕ
Q
t F) ω(E g (dF)), ω ∈ S g .
(6.1.4)
We have used notation f ξ (F) := F(ξ) (ξ ∈ g, F ∈ g
∗
), and ϕ
Q is the classical
flow on g
∗ corresponding to the Hamiltonian function Q ∈ C
∞
(g
∗
, R), Q(F) :=
Q(F 1 , F 2 , . . . F n ), with F j := f ξ j (F) = F(ξ j ), (F ∈ g
∗
); the introduction of the
flow ϕ
Q will be discussed later in this section. The natural question is, however.
whether the limits
6 Dynamics of Quantum Mechanical Macroscopic Systems
multiplication of variables in such a way that the element Q(ξ 1 , ξ 2 , . . . ξ n ) of the Lie
algebra envelope has the following property:
(SA) Let Q ≡
q
k=0 Q k , where Q k is a homogeneous polynomial of degree k. In
any continuous unitary representation U of the group G on a separable Hilbert space
H, the operators Q k (X 1 , X 2 , . . . , X n ) defined on analytic elements of U are essentially selfadjoint, for all k = 1, 2, . . . , q.
This property (SA) is fulfilled e.g. if all the Q k (ξ 1 , ξ 2 , . . . , ξ n ) are symmetric
and elliptic, cf. [13, Chap. 11]. Then we define the local Hamiltonians Q
N , with
(5.1.14), denoting by N also an N -point subset of :
Q
N
:= N Q(X 1N , X 2N , . . . , X nN ), X j N :=
1
N
X
N
j , N = 1, 2, . . . ; (6.1.1)
cf. (5.1.118), i.e.
X
N
j :=
|N |
k=1
π k (X j ), j = 1, 2, . . . n, k ∈ N ⊂ ,
which can be considered as (essentially) selfadjoint operators on H (= H N ⊗
H \N ≡ -tuple tensor product). For any x ∈ A (:= A
) ⊂ L(H ) we set
τ
N
t (x) := exp(it Q
N
) x exp(−it Q
N
), t ∈ R,
(6.1.2)
and these mappings τ
N
t clearly form a one parameter group of
∗ -automorphisms
of A for each finite N . Systems of this type were introduced in [155] for the case
of spin systems (i.e. dim H was finite). It was shown in [40, 155] that the sequence
{τ
N
: N = 1, 2, . . . } determines an evolution τ
Q of the observables of the form X ξ ,
cf. 5.1.7 and 5.1.9, which is expressed in our notation by the formula
τ
Q
t (X ξ ) := w
∗
0 - lim
N →∞
τ
N
t (X ξ N ) =
f ξ (ϕ
Q
t F) E g (dF),
(6.1.3)
where w
∗
0 -topology on a von Neumann algebra containing A and X ξ (ξ ∈ g) is
determined by the set of the ‘classical states’. The integral in (6.1.3) corresponds to
the integral in [155, (2.29)], which, specified to our case, reads:
lim
N →∞
ω(τ
N
t (X ξ N )) =
f ξ (ϕ
Q
t F) ω(E g (dF)), ω ∈ S g .
(6.1.4)
We have used notation f ξ (F) := F(ξ) (ξ ∈ g, F ∈ g
∗
), and ϕ
Q is the classical
flow on g
∗ corresponding to the Hamiltonian function Q ∈ C
∞
(g
∗
, R), Q(F) :=
Q(F 1 , F 2 , . . . F n ), with F j := f ξ j (F) = F(ξ j ), (F ∈ g
∗
); the introduction of the
flow ϕ
Q will be discussed later in this section. The natural question is, however.
whether the limits