7.5 The X-Y Chain as a Measuring Device
201
and due to the properties (7.5.15) and (7.5.27) of ω 0 and due to commutation properties of the a j , a
∗
k we see that the terms with r = s are zeros. According to (7.5.14)
we have:
ω t (a
∗
j a j ) =
+∞
r =−∞
|C t ( j − r )|
2
ω 0 (a
∗
r a r ) =
+∞
r =1
|C t ( j + r )|
2
=
+∞
r =1
J
2
j+r (κt) ≡
+∞
r =1
J
2
| j+r | (κt).
(7.5.30)
According to the known formula [180, (21.8-26)]:
1 = J
2
0 (z) + 2
+∞
k=1
J
2
k (z),
(7.5.31)
and due to the asymptotic behaviour of Bessel functions
J m (t) O(t
−
1
2 ), m ∈ Z,
(7.5.32)
we have finally
ω(a
∗
j a j ) := lim
t→+∞
ω t (a
∗
j a j ) =
1
2
, for all j ∈ Z.
(7.5.33)
Returning to the formulas (7.5.24) and (7.5.25) of our main interest, we have
obtained:
ω ∞ (a
∗
j a j ) := lim
t→+∞
ω t (a
∗
j a j ) = |c + |
2
ω(a
∗
j a j ) + |c − |
2
ω 0 (a
∗
j a j ).
(7.5.34)
The last formula describes an (incoherent) mixture of two mutually macroscopically distinct, hence disjoint states ω 0 , ω on the C
∗ -algebra A of the infinite spin
chain. This can be checked in the explicit way by calculating values of a macroscopic
observable in the states ω 0 , resp. ω, e.g. of the observable constructed from (7.3.29)
γ := w- lim
N →∞
1
N
N
n=1
a
∗
n a n ∈ Z(A
∗∗
) ⊂ A
∗∗
.
(7.5.35)
According to (7.5.33) and (7.5.27), it is:
ω(γ) =
1
2
= ω 0 (γ) = 0.
(7.5.36)