168
7 Some Models of “Quantum Measurement”
paper by Klaus Hepp [153], cf. also [253, 254]. According to the previous chapters,
we are able to describe in QM in a mathematically clear way macroscopic systems
(with coordinates undergoing classical behaviour) by models of infinite quantal systems only. Of course, the infinity of the number of degrees of freedom should be
considered as a convenient approximation to large but finite systems. Also infinite
time duration of the processes of changing macroscopic parameters corresponding to
considered microscopic influences is connected with this infinity. In this connection,
it is relevant to be interested in the speed of the corresponding macroscopic changes.
In the ‘infinite models’ presented here the convergence to a macroscopic change is
very slow.
A much larger speed of convergence is reached in the model of finite (arbitrary long) ‘Quantum Domino’—spin chain (cf. 7.1.3 and Sect. 7.3) interacting with
fermion field in such a way that after all the spins in the chain changed their orientations into the opposite ones the chain emits a fermion. In this case the speed of convergence to final stationary state is ‘almost exponential’. The model is described in [39].
Its interpretation as “a model of quantum measurement” is, however, questionable:
Due to its finite dimension a definition of “macroscopic difference” is ambiguous
and it would need probably a longer discussion. Cf. notes on this problem in the
original Hepp’s work [153], and also in our Sect. 7.7.
It should be stressed that we do not intend to present the described models of
micro-macro interaction as a definitive solution of the ‘measurement problem in
QM’, cf. Sect. 7.7. They could be considered rather as an illustration of possibilities
of the standard quantum mechanical formalism to include, by using this specific way
of description of macroscopic observables, some descriptions of possible responses
of large systems (hence changes of their ‘macroscopic variables’) to some of their
interactions with microsystems. It is shown how can various states of a microsystem
interacting with a macrosystem lead in QM to various ‘corresponding changes’ of
values of their macroscopic (resp. ‘classical’) observables.
7.1.3 We present here four models, the second of which is based on the first one, the
“Quantum Domino” (QD), published originally in [36]. The idea of the third model
is similar to that of QD, but it is based on the known X-Y model of the spin chain
[201]. QD is a model of an infinite quantum system—an infinite (or semiinfinite) spin
chain with a short range interaction—in which any local (microscopic) change of a
specific stationary state leads to subsequent evolution (with time t → ∞) to a new,
macroscopically different stationary state. The initial local changes of these stationary
states of this model are realized “by hand”, i.e. a locally perturbed stationary state is
chosen as an initial condition for the forthcoming time-evolved states of that system.
This local perturbation can be realized by a change of quantum state of a single spin
(say the first one in the semiinfinite chain), and this spin can be considered, e.g. as
an additional microsystem (the ‘measured system’) interacting with the infinite rest
of the chain.
1
1 In the case of some different choices of (locally perturbed stationary) initial states in this model,
the subsequent time evolutions of the chain could be different: e.g., an initial segment could move
quasiperiodically and the infinite rest of the chain will converge to a macroscopically different state.
7 Some Models of “Quantum Measurement”
paper by Klaus Hepp [153], cf. also [253, 254]. According to the previous chapters,
we are able to describe in QM in a mathematically clear way macroscopic systems
(with coordinates undergoing classical behaviour) by models of infinite quantal systems only. Of course, the infinity of the number of degrees of freedom should be
considered as a convenient approximation to large but finite systems. Also infinite
time duration of the processes of changing macroscopic parameters corresponding to
considered microscopic influences is connected with this infinity. In this connection,
it is relevant to be interested in the speed of the corresponding macroscopic changes.
In the ‘infinite models’ presented here the convergence to a macroscopic change is
very slow.
A much larger speed of convergence is reached in the model of finite (arbitrary long) ‘Quantum Domino’—spin chain (cf. 7.1.3 and Sect. 7.3) interacting with
fermion field in such a way that after all the spins in the chain changed their orientations into the opposite ones the chain emits a fermion. In this case the speed of convergence to final stationary state is ‘almost exponential’. The model is described in [39].
Its interpretation as “a model of quantum measurement” is, however, questionable:
Due to its finite dimension a definition of “macroscopic difference” is ambiguous
and it would need probably a longer discussion. Cf. notes on this problem in the
original Hepp’s work [153], and also in our Sect. 7.7.
It should be stressed that we do not intend to present the described models of
micro-macro interaction as a definitive solution of the ‘measurement problem in
QM’, cf. Sect. 7.7. They could be considered rather as an illustration of possibilities
of the standard quantum mechanical formalism to include, by using this specific way
of description of macroscopic observables, some descriptions of possible responses
of large systems (hence changes of their ‘macroscopic variables’) to some of their
interactions with microsystems. It is shown how can various states of a microsystem
interacting with a macrosystem lead in QM to various ‘corresponding changes’ of
values of their macroscopic (resp. ‘classical’) observables.
7.1.3 We present here four models, the second of which is based on the first one, the
“Quantum Domino” (QD), published originally in [36]. The idea of the third model
is similar to that of QD, but it is based on the known X-Y model of the spin chain
[201]. QD is a model of an infinite quantum system—an infinite (or semiinfinite) spin
chain with a short range interaction—in which any local (microscopic) change of a
specific stationary state leads to subsequent evolution (with time t → ∞) to a new,
macroscopically different stationary state. The initial local changes of these stationary
states of this model are realized “by hand”, i.e. a locally perturbed stationary state is
chosen as an initial condition for the forthcoming time-evolved states of that system.
This local perturbation can be realized by a change of quantum state of a single spin
(say the first one in the semiinfinite chain), and this spin can be considered, e.g. as
an additional microsystem (the ‘measured system’) interacting with the infinite rest
of the chain.
1
1 In the case of some different choices of (locally perturbed stationary) initial states in this model,
the subsequent time evolutions of the chain could be different: e.g., an initial segment could move
quasiperiodically and the infinite rest of the chain will converge to a macroscopically different state.