206
7 Some Models of “Quantum Measurement”
F[θ(t)φ, e
it H
ψ](ξ) =
i
√
2π
φ, R H 0 (ξ)ψ −
i
√
2π
φ, R H 0 (ξ)V R H (ξ)ψ.
(7.6.6)
Important for the following analysis are the matrix elements
F mn := =β m , R H (ξ), β n ,
(7.6.7)
since e.g.:
F[θ(t)β m , e
it H
β n ](ξ) =
i
√
2π
F mn (ξ).
(7.6.8)
Now, the proper choice of the parameters of the model is, according to [39]:
ε(p) := a|p|
2
, a > 0,
(7.6.9a)
F(σ)(p) = 0 (|p| < b), F(σ)(p) > 0 for all |p| > b > 0, σ ∈ S(R
3
),
(7.6.9b)
ε 0 > ab
2
+ 2,
(7.6.9c)
where S(R
3
) is the set of all rapidly decreasing Schwartz complex valued functions
on R
3 , the symbol F(σ) again means the Fourier transform (i.e. the transition to
“p-representation”), and the constants a, b occurring in (7.6.9c) are the same as the
ones occurring in (7.6.9a) and (7.6.9b).
After making this choice it is possible, after a series of considerations and calculations [39], to show that (cf. [39, (4.33)])
F[[β m , e
it H
β n ]( p) = −
2
π
lim
ν→0 +
Im F mn ( p − iν) ∈ S(R).
(7.6.10)
But the Schwartz set S(R) of rapidly decreasing smooth functions is invariant with
respect to the Fourier transform, hence the function t → →β m , e
it H
β n also belongs
to S(R), what proves the ‘almost exponential decay’ in time of this matrix element.
This result is crucial for the proof of Theorem 7.6.6.
To formulate the main result as a theorem, let us introduce also the notation:
μ((−∞, λ]) :=
ε(p)<λ
|F(σ)(p)|
2 d
3 p,
(7.6.11a)
ρ μ (λ) :=
dμ((−∞, λ])
dλ
.
(7.6.11b)
7.6.6 Theorem. In the above described model of finite spin chain QD interacting
with nonrelativistic scalar Fermi field, with the parameters specified in (7.6.9), for
either all such ε 0 with possibly one exception, or for all
7 Some Models of “Quantum Measurement”
F[θ(t)φ, e
it H
ψ](ξ) =
i
√
2π
φ, R H 0 (ξ)ψ −
i
√
2π
φ, R H 0 (ξ)V R H (ξ)ψ.
(7.6.6)
Important for the following analysis are the matrix elements
F mn := =β m , R H (ξ), β n ,
(7.6.7)
since e.g.:
F[θ(t)β m , e
it H
β n ](ξ) =
i
√
2π
F mn (ξ).
(7.6.8)
Now, the proper choice of the parameters of the model is, according to [39]:
ε(p) := a|p|
2
, a > 0,
(7.6.9a)
F(σ)(p) = 0 (|p| < b), F(σ)(p) > 0 for all |p| > b > 0, σ ∈ S(R
3
),
(7.6.9b)
ε 0 > ab
2
+ 2,
(7.6.9c)
where S(R
3
) is the set of all rapidly decreasing Schwartz complex valued functions
on R
3 , the symbol F(σ) again means the Fourier transform (i.e. the transition to
“p-representation”), and the constants a, b occurring in (7.6.9c) are the same as the
ones occurring in (7.6.9a) and (7.6.9b).
After making this choice it is possible, after a series of considerations and calculations [39], to show that (cf. [39, (4.33)])
F[[β m , e
it H
β n ]( p) = −
2
π
lim
ν→0 +
Im F mn ( p − iν) ∈ S(R).
(7.6.10)
But the Schwartz set S(R) of rapidly decreasing smooth functions is invariant with
respect to the Fourier transform, hence the function t → →β m , e
it H
β n also belongs
to S(R), what proves the ‘almost exponential decay’ in time of this matrix element.
This result is crucial for the proof of Theorem 7.6.6.
To formulate the main result as a theorem, let us introduce also the notation:
μ((−∞, λ]) :=
ε(p)<λ
|F(σ)(p)|
2 d
3 p,
(7.6.11a)
ρ μ (λ) :=
dμ((−∞, λ])
dλ
.
(7.6.11b)
7.6.6 Theorem. In the above described model of finite spin chain QD interacting
with nonrelativistic scalar Fermi field, with the parameters specified in (7.6.9), for
either all such ε 0 with possibly one exception, or for all