130
6 Dynamics of Quantum Mechanical Macroscopic Systems
τ
Q
t 1 (τ
Q
t 2 (x)) = τ
Q
t 1 (s- lim
K →∞
τ
K
t 2
(x)) =
∞
m=0
(it 2 )
m
m!
τ
Q
t 1 (s- lim
K →∞
[Q
K
, x]
(m)
), (6.2.53)
where the norm continuity of τ
Q
t 1 and the norm-convergence of the series were used.
(We write here s- lim instead of s
∗ - lim, where the s(s G A
∗∗
, s G A
∗
)-topology is generated by the seminorms ˆ
p ω from (6.2.19). This notation is used for brevity only; the
existence and equality of both the limits s- lim and s
∗ - lim is clear from the proof
of Lemma 6.2.5.) Considering the structure of the multiple commutators in (6.2.53)
according to the discussion in the proof of 6.2.11, by the morphism property of τ
Q
t 1
on C
N as well as the definition (6.2.23) with (6.2.32) we obtain:
τ
Q
t 1 (s- lim
K
[Q
K
, x]
(m)
) = s- lim
K
τ
Q
t 1 ([Q
K
, x]
(m)
)
(6.2.54)
= s- lim
K
[τ
Q
t 1 (Q
K
), τ
Q
t 1 (x)]
(m)
.
(6.2.55)
Since any
∗ -morphism τ
Q is a contraction, the bounds from 6.2.4 are valid also
for the multiple commutators in (6.2.55). From the norm-convergence of the sums
we obtain consequently:
τ
Q
t 1 (τ
Q
t 2 (x)) = s- lim
K
∞
m=0
(it 2 )
m
m!
[τ
Q
t 1 (Q
K
), τ
Q
t 1 (x)]
(m)
= s- lim
K
τ
Q
t 1 (τ
K
t 2
(x)).
(6.2.56)
One has also
τ
K
t 2
(x) ∈ B
N for all x ∈ A
N
, and for all K ∈ .
(6.2.57)
Then, according to the 6.2.6 and the formula (6.2.18), one obtains:
τ
Q
t 1 (τ
K
t 2
(x)) = s- lim
L
τ
L
t 1
(τ
K
t 2
(x))
= s- lim
L
∞
k,m=0
(it 1 )
k
k!
(it 2 )
m
m!
[Q
L
, [Q
K
, x]
(m)
]
(k)
. (6.2.58)
The norms of the multiple commutators in (6.2.58) for L ≥ K ≥ N are bounded
from above according to the estimate (cf. also 6.2.3)
[Q
L
, [Q
K
, x]
(m)
]
(k)
<
b(x)
q
(m + k − 1)! (Mpq
2 b
q−1 a N )
m+k
,
(6.2.59)
what can be obtained by the considerations analogous to those used in the proof of
6.2.4. Hence the sum in (6.2.58) converges in norm, uniformly in (K ; L) ∈ ×
6 Dynamics of Quantum Mechanical Macroscopic Systems
τ
Q
t 1 (τ
Q
t 2 (x)) = τ
Q
t 1 (s- lim
K →∞
τ
K
t 2
(x)) =
∞
m=0
(it 2 )
m
m!
τ
Q
t 1 (s- lim
K →∞
[Q
K
, x]
(m)
), (6.2.53)
where the norm continuity of τ
Q
t 1 and the norm-convergence of the series were used.
(We write here s- lim instead of s
∗ - lim, where the s(s G A
∗∗
, s G A
∗
)-topology is generated by the seminorms ˆ
p ω from (6.2.19). This notation is used for brevity only; the
existence and equality of both the limits s- lim and s
∗ - lim is clear from the proof
of Lemma 6.2.5.) Considering the structure of the multiple commutators in (6.2.53)
according to the discussion in the proof of 6.2.11, by the morphism property of τ
Q
t 1
on C
N as well as the definition (6.2.23) with (6.2.32) we obtain:
τ
Q
t 1 (s- lim
K
[Q
K
, x]
(m)
) = s- lim
K
τ
Q
t 1 ([Q
K
, x]
(m)
)
(6.2.54)
= s- lim
K
[τ
Q
t 1 (Q
K
), τ
Q
t 1 (x)]
(m)
.
(6.2.55)
Since any
∗ -morphism τ
Q is a contraction, the bounds from 6.2.4 are valid also
for the multiple commutators in (6.2.55). From the norm-convergence of the sums
we obtain consequently:
τ
Q
t 1 (τ
Q
t 2 (x)) = s- lim
K
∞
m=0
(it 2 )
m
m!
[τ
Q
t 1 (Q
K
), τ
Q
t 1 (x)]
(m)
= s- lim
K
τ
Q
t 1 (τ
K
t 2
(x)).
(6.2.56)
One has also
τ
K
t 2
(x) ∈ B
N for all x ∈ A
N
, and for all K ∈ .
(6.2.57)
Then, according to the 6.2.6 and the formula (6.2.18), one obtains:
τ
Q
t 1 (τ
K
t 2
(x)) = s- lim
L
τ
L
t 1
(τ
K
t 2
(x))
= s- lim
L
∞
k,m=0
(it 1 )
k
k!
(it 2 )
m
m!
[Q
L
, [Q
K
, x]
(m)
]
(k)
. (6.2.58)
The norms of the multiple commutators in (6.2.58) for L ≥ K ≥ N are bounded
from above according to the estimate (cf. also 6.2.3)
[Q
L
, [Q
K
, x]
(m)
]
(k)
<
b(x)
q
(m + k − 1)! (Mpq
2 b
q−1 a N )
m+k
,
(6.2.59)
what can be obtained by the considerations analogous to those used in the proof of
6.2.4. Hence the sum in (6.2.58) converges in norm, uniformly in (K ; L) ∈ ×