208
7 Some Models of “Quantum Measurement”
side of (7.6.14) quickly converges to zero. All the vectors β N (ψ) (ψ ∈ L
2
(R
3
, d
3 x))
describe the states of the composite system:
{ the measured system & the rest of the spin chain & the Fermi field}
in which all the N + 1 spins “are pointing up”, which has to mimic the macroscopically different state from the initial state (c ↓ I A + c ↑ a
∗
0 ))
S
0 ⊗
F
0 ≡ (c ↓ I A +
c ↑ a
∗
0 )) 0 , as well as from 0 , of the compound system. For the wave function (7.6.13)
of the compound system we obtain asymptotically for large times t → ∞:
t = c ↓ 0 + c ↑ e
−it H
β 0 c ↓ 0 + c ↑ β N (ψ(t)),
(7.6.16)
which has the form of the formula (7.7.2) for the (approximate expression of the)
“measurement dynamics” in the conventional QM framework of considering of only
finite systems (as measuring apparatuses). The probabilities of the two different
“measurement results” corresponding to the states ϕ ↓ , resp. ϕ ↑ , occurring in the
orthogonal decomposition of the initial state ϕ 0 of the measured system are, as it
was expected, the numbers |c ↓ |
2 , resp. |c ↑ |
2 . By ‘tracing out’ the states of the environment ≡ the Fermi field, we obtain the density matrix for the spin chain, and by
tracing out the both {Fermi field & the spins 1,2,…N}, we obtain the density matrix
:= |c ↓ |
2 P ϕ ↓ + |c ↑ |
2 P ϕ ↑ , with P ϕ ↑ ≡ a
∗
0 a 0 and P ϕ ↓ ≡ a 0 a
∗
0 , in the state space of the
measured system (i.e. of the spin placed in the point 0 of the chain), corresponding
formally to the ‘collapse of its wave packet’ ϕ 0 := c ↓ ϕ ↓ + c ↑ ϕ ↑ , i.e. of its initial
state of the just described process. Neither of these density matrices can be, however,
interpreted as describing a ‘proper’, or ‘genuine’ probability distribution of quantal
states in the sense of classical statistics. To interpret them in that sense, and distinguish one decomposition of a density matrix as ‘more relevant’ (i.e. reflecting the
classical-type statistics), some another additional assumption is needed. We have had
in our interpretations of the infinite models in previous Sections the requirement of
disjointness of mutually noninterfering states, and this was ensured by existence of a
macroscopic quantity obtaining mutually different values in these states. For some
alternative approaches, we could go back again to the attempts in the ‘decoherence
programs’, [124, 279, 343, 347]. More detailed mathematical and interpretational
considerations on decompositions of states of a C
∗ -algebra can be found in our 1.2.3,
1.3.3, 1.3.4, 1.4.3, and citations therein, e.g. [53, Chap. 4].
7.6.8 Notes on irreversibility. This model of a radiating multispin system can be
also considered as a caricature reflecting one of the usual mechanisms of irreversible
behaviour of large physical systems: Large systems usually (resp. ‘almost always’)
are not isolated from their environment, and their interaction with (a ‘relative stable’,
and a ‘relative stationary’) environment leads to their motion to more stable stationary,
e.g. thermodynamic equilibrium, states. Some kind of radiation, as it was built in into
our model, is a usual form of interactions of large systems with their environment.
This approach reflects just one ‘aspect’ of irreversible behaviour of physical systems. Another often discussed ‘aspect’ of theoretical descriptions of irreversible
7 Some Models of “Quantum Measurement”
side of (7.6.14) quickly converges to zero. All the vectors β N (ψ) (ψ ∈ L
2
(R
3
, d
3 x))
describe the states of the composite system:
{ the measured system & the rest of the spin chain & the Fermi field}
in which all the N + 1 spins “are pointing up”, which has to mimic the macroscopically different state from the initial state (c ↓ I A + c ↑ a
∗
0 ))
S
0 ⊗
F
0 ≡ (c ↓ I A +
c ↑ a
∗
0 )) 0 , as well as from 0 , of the compound system. For the wave function (7.6.13)
of the compound system we obtain asymptotically for large times t → ∞:
t = c ↓ 0 + c ↑ e
−it H
β 0 c ↓ 0 + c ↑ β N (ψ(t)),
(7.6.16)
which has the form of the formula (7.7.2) for the (approximate expression of the)
“measurement dynamics” in the conventional QM framework of considering of only
finite systems (as measuring apparatuses). The probabilities of the two different
“measurement results” corresponding to the states ϕ ↓ , resp. ϕ ↑ , occurring in the
orthogonal decomposition of the initial state ϕ 0 of the measured system are, as it
was expected, the numbers |c ↓ |
2 , resp. |c ↑ |
2 . By ‘tracing out’ the states of the environment ≡ the Fermi field, we obtain the density matrix for the spin chain, and by
tracing out the both {Fermi field & the spins 1,2,…N}, we obtain the density matrix
:= |c ↓ |
2 P ϕ ↓ + |c ↑ |
2 P ϕ ↑ , with P ϕ ↑ ≡ a
∗
0 a 0 and P ϕ ↓ ≡ a 0 a
∗
0 , in the state space of the
measured system (i.e. of the spin placed in the point 0 of the chain), corresponding
formally to the ‘collapse of its wave packet’ ϕ 0 := c ↓ ϕ ↓ + c ↑ ϕ ↑ , i.e. of its initial
state of the just described process. Neither of these density matrices can be, however,
interpreted as describing a ‘proper’, or ‘genuine’ probability distribution of quantal
states in the sense of classical statistics. To interpret them in that sense, and distinguish one decomposition of a density matrix as ‘more relevant’ (i.e. reflecting the
classical-type statistics), some another additional assumption is needed. We have had
in our interpretations of the infinite models in previous Sections the requirement of
disjointness of mutually noninterfering states, and this was ensured by existence of a
macroscopic quantity obtaining mutually different values in these states. For some
alternative approaches, we could go back again to the attempts in the ‘decoherence
programs’, [124, 279, 343, 347]. More detailed mathematical and interpretational
considerations on decompositions of states of a C
∗ -algebra can be found in our 1.2.3,
1.3.3, 1.3.4, 1.4.3, and citations therein, e.g. [53, Chap. 4].
7.6.8 Notes on irreversibility. This model of a radiating multispin system can be
also considered as a caricature reflecting one of the usual mechanisms of irreversible
behaviour of large physical systems: Large systems usually (resp. ‘almost always’)
are not isolated from their environment, and their interaction with (a ‘relative stable’,
and a ‘relative stationary’) environment leads to their motion to more stable stationary,
e.g. thermodynamic equilibrium, states. Some kind of radiation, as it was built in into
our model, is a usual form of interactions of large systems with their environment.
This approach reflects just one ‘aspect’ of irreversible behaviour of physical systems. Another often discussed ‘aspect’ of theoretical descriptions of irreversible