202
7 Some Models of “Quantum Measurement”
Hence, again here, a microscopic system interacting with the macroscopic X-Y chain
changed the chain’s initial state ω 0 into a new, macroscopically distinct state ω ∞ =
|c + |
2
ω + |c − |
2
ω 0 . Here the probabilities |c ± |
2 of occurrence of the mutually disjoint
states ω 0 , ω in the proper (resp. ‘genuine’, cf. 1.1.4) mixture ω ∞ are exactly the
probabilities of appearing of the states |±± of the microsystem in its initial state ϕ 0 ,
cf. (7.5.20). This corresponds again to the “ideal measurement”, as it was discussed
in 7.1.3, 7.3.6 and 7.4.2.
7.6 Radiating Finite Spin Chain
7.6.1 We shall present very briefly in this section, without proofs, the dynamics of
a model of a large but finite system interacting with a Fermi field.
10 The system’s
initial state is stationary but unstable, as it was also the case of the models presented
in the preceding sections. After an initial perturbation, the model evolves quickly
into a new stationary state by simultaneous radiation of a Fermi particle, which
escapes into infinity. The process is very quick in contrast to the time evolutions in
the case of the models described in the previous Sects. 7.3, 7.4 and 7.5. The three
preceding models might, however, serve as clear mathematical pictures of “quantum
measurement” in the sense that the time evolution of a large system led with the time
growing to infinity to the state “macroscopically different” from its initial state. The
“macroscopic difference” between states of the system is mathematically expressed
there as disjointness of the states on the C
∗ -algebra of observable quantities of the
large system. The disjointness implies that if those states are represented as vectors
in a Hilbert space, their mutual linear combinations do not lead to any interference
(the C
∗ -algebra of observables representing all possible observations on the model
system is fixed!) and such a linear combination is physically equivalent to a “proper
mixture”, or “genuine mixture” (cf. 1.1.4), i.e. to a classical statistical description
of an ensemble in which the individual copies of the large system are distributed
between the uniquely determined ‘classical’ states under consideration. This unique
decomposability to pure states on the algebra of classical (macroscopic) observables
is a consequence of the fact that the states of a classical system form a simplex.
This differs from “mixed” quantum states described by density matrices of standard
QM of finite-size systems having multiple convex decompositions to extremal (pure)
states.
Since the model of a “large” system described in this section is finite (corresponding by physical intuition to that consisting of finite number of some “elementary” or
“small” subsystems, each of them described by elementary QM in separable Hilbert
space H v with the algebra of its observables coinciding with the whole L(H v )), there
is no possibility of emergence of any disjoint states, hence there is no unambiguously
10 The formulation and main features of the dynamics of this model were presented first time in
[33]. The technical details are described in [39].
7 Some Models of “Quantum Measurement”
Hence, again here, a microscopic system interacting with the macroscopic X-Y chain
changed the chain’s initial state ω 0 into a new, macroscopically distinct state ω ∞ =
|c + |
2
ω + |c − |
2
ω 0 . Here the probabilities |c ± |
2 of occurrence of the mutually disjoint
states ω 0 , ω in the proper (resp. ‘genuine’, cf. 1.1.4) mixture ω ∞ are exactly the
probabilities of appearing of the states |±± of the microsystem in its initial state ϕ 0 ,
cf. (7.5.20). This corresponds again to the “ideal measurement”, as it was discussed
in 7.1.3, 7.3.6 and 7.4.2.
7.6 Radiating Finite Spin Chain
7.6.1 We shall present very briefly in this section, without proofs, the dynamics of
a model of a large but finite system interacting with a Fermi field.
10 The system’s
initial state is stationary but unstable, as it was also the case of the models presented
in the preceding sections. After an initial perturbation, the model evolves quickly
into a new stationary state by simultaneous radiation of a Fermi particle, which
escapes into infinity. The process is very quick in contrast to the time evolutions in
the case of the models described in the previous Sects. 7.3, 7.4 and 7.5. The three
preceding models might, however, serve as clear mathematical pictures of “quantum
measurement” in the sense that the time evolution of a large system led with the time
growing to infinity to the state “macroscopically different” from its initial state. The
“macroscopic difference” between states of the system is mathematically expressed
there as disjointness of the states on the C
∗ -algebra of observable quantities of the
large system. The disjointness implies that if those states are represented as vectors
in a Hilbert space, their mutual linear combinations do not lead to any interference
(the C
∗ -algebra of observables representing all possible observations on the model
system is fixed!) and such a linear combination is physically equivalent to a “proper
mixture”, or “genuine mixture” (cf. 1.1.4), i.e. to a classical statistical description
of an ensemble in which the individual copies of the large system are distributed
between the uniquely determined ‘classical’ states under consideration. This unique
decomposability to pure states on the algebra of classical (macroscopic) observables
is a consequence of the fact that the states of a classical system form a simplex.
This differs from “mixed” quantum states described by density matrices of standard
QM of finite-size systems having multiple convex decompositions to extremal (pure)
states.
Since the model of a “large” system described in this section is finite (corresponding by physical intuition to that consisting of finite number of some “elementary” or
“small” subsystems, each of them described by elementary QM in separable Hilbert
space H v with the algebra of its observables coinciding with the whole L(H v )), there
is no possibility of emergence of any disjoint states, hence there is no unambiguously
10 The formulation and main features of the dynamics of this model were presented first time in
[33]. The technical details are described in [39].