190
7 Some Models of “Quantum Measurement”
It follows from (7.4.44) that the right hand side of (7.4.49) converges with t →
+∞ to zero, hence also the right hand side in (7.4.48) converges to zero (for all ψ j ).
Hence
lim
t→∞
ω
A&B
t
(P 0 |ψ 1 ψ 2 |) = 0, for all ψ 1,2 ∈ L
2
(R
3
) = H B .
(7.4.50)
Let us note that a different situation appeared in the case b := I H B in which case the
equation (7.4.45b) is valid.
It remained to find the limit of the expressions ω
A&B
t
(|ψ 1 ψ 2 |) ≡ ω
A&B
t
(I H vac ⊗
|ψ 1 ψ 2 |). Because we are working in the time-invariant subspace K ⊂ H vac ⊗ H B ,
and the projection onto it is P K =
∞
m=0 P m , we shall write this sum instead of I H vac
in ω
A&B
t
. The summation over m in its argument should be done, however, before
performing the limit lim t→∞ ω
A&B
t
(P m |ψ 1 ψ 2 |).
With the help of (7.4.33), we can obtain
ω
A&B
t
(P m |ψ 1 ψ 2 |) = γ
2
t
0 dt
t
0 dt
¯
F(t
)F(t
)F
0
ϕ (ψ 1 )(t
− t)
¯
F
0
ϕ (ψ 2 )(t
− t) ¯
f m (t − t
) f m (t − t
).
(7.4.51)
Let us introduce the functions G(t
, t
) and g(ψ 1 , ψ 2 )(t
, t
) of {t
, t
} ∈ R
2 :
G(t
, t
) := ¯
F + (t
)F + (t
);
g(ψ 1 , ψ 2 )(t
, t
) := f 1 (t
− t
)F
0
ϕ (ψ 1 ) − (−t
) ¯
F
0
ϕ (ψ 2 ) − (−t
)
where e.g. F
0
ϕ (ψ) − (−t) := θ(−t)F
0
ϕ (ψ)(−t).
A use of (7.4.40) leads us to:
ω
A&B
t
((P K − P 0 )|ψ 1 ψ 2 |) = γ
2
t
0
dt
t
0
dt
¯
F(t
)F(t
)F
0
ϕ (ψ 1 )(t
− t)
¯
F
0
ϕ (ψ 2 )(t
− t) f 1 (t
− t
)
= γ
2 G ∗ g(ψ 1 , ψ 2 )(t, t),
(7.4.52)
where ∗ denotes the 2-dimensional convolution. From the given properties of the
entering functions (cf. also our Lemma 7.4.6, and the L
p -estimates in [262, 298]),
and with the use of (7.4.50), we obtain the desired result:
lim
t→∞
ω
A&B
t
(|ψ 1 ψ 2 |) = 0, ψ j ∈ H B .
(7.4.53)
The existence of a limit state ω := w
∗ - lim t→+∞ ω
A&B
t
according to (7.4.26) is
proved; its form as a product state (7.4.26) in S(A ⊗ B) can be seen by checking
its values on elements of A ⊗ B, cf. also [38] and [90, I.4.5. Proposition 2]. By
comparing the definition in 7.4.3 of the no-particle state ω
B
0 on B with our results,
and considering the results (7.4.45), (7.4.47), and (7.4.50) (together with (7.4.53))
we finally obtain: