7.6 Radiating Finite Spin Chain
209
behaviour of finite many-particle systems is their complicated mechanical motion
even if they are isolated from any environment. Then we are dealing with such phenomena as various types of “chaos”, and with “recurrences” in their (deterministic
and time-reversible) mechanical motion. We shall not consider here such mechanical explanations of irreversibility, initiated by J. C. Maxwell and L. Boltzmann.
As concerns some study on these topics in the case of classical systems, it might
be interesting to look to nice conference or journal papers like, e.g. [332], but more
elementary and also more complex information could be found in some books on
the “theory of dynamical systems” listed in our Bibliography, e.g. [1, 7, 9, 10, 181,
248, 326]. However long are durations of the Poincaré cycles corresponding to the
above mentioned recurrences in mechanical motions of isolated systems with several degrees of freedom (they are comparable with the lifetime of Universe [332]),
an evolution during which the system approaches some stable stationary state cannot be reached in theoretical description of finite isolated mechanical systems. This
does not exclude, however, effectiveness of the statistical physics, which does not
deal with a unique phase-space trajectory of the considered system; here we have a
certain physical reinterpretation of the mechanics of motions in the system’s phase
space. But full effectiveness of the statistical approach to description of behaviour
of multiparticle systems, e.g. mathematically clear description of thermal equilibria
and phase transitions, is again possible in the ‘thermodynamical limit’ of infinitely
large systems only, e.g. [271].
It is seen that after making the finite quantum spin chain of our model to become
an “open system” by adding to the Hamiltonian of the restricted QD the term corresponding to the radiation of a fermion, the speed of the motion to the limiting state
was enormously increased in comparison with the infinite, but isolated, QD-chain,
cf. (7.3.24) and (7.3.26b), i.e. with respect to (7.3.27). The finite-sized version of the
isolated QD would behave, however, almost-periodically, cf. (7.3.23). The addition
of interaction of the finite QD with Fermi field enabled us to obtain a system’s state
converging for t → ∞ to a new stationary state. But a clear and unambiguous interpretation of some states of a finite system, e.g. the two states appearing in the sum on
the right hand side of (7.6.16), as being approximately ‘mutually macroscopically
different’ (hence their quantum interference being ‘almost impossible’), is still open
to discussion. We shall not further investigate here some other connections of these
phenomena and questions.
7.7 On the “Measurement Problem” in QM
Let us add here several notes to the above mentioned “measurement problem”, considered for a long time to be a fundamental problem of the conceptual structure of
QM, cf. e.g. [63, 236, 237] and [238, Chap. 29]. These notes should be also supplemented by the notes in 7.6.1, esp. by the footnotes 12 and 13.
States of the physical systems are described in the mathematical theory of QM
by mathematical objects like “wave functions”, “density matrices”, or “linear func-
209
behaviour of finite many-particle systems is their complicated mechanical motion
even if they are isolated from any environment. Then we are dealing with such phenomena as various types of “chaos”, and with “recurrences” in their (deterministic
and time-reversible) mechanical motion. We shall not consider here such mechanical explanations of irreversibility, initiated by J. C. Maxwell and L. Boltzmann.
As concerns some study on these topics in the case of classical systems, it might
be interesting to look to nice conference or journal papers like, e.g. [332], but more
elementary and also more complex information could be found in some books on
the “theory of dynamical systems” listed in our Bibliography, e.g. [1, 7, 9, 10, 181,
248, 326]. However long are durations of the Poincaré cycles corresponding to the
above mentioned recurrences in mechanical motions of isolated systems with several degrees of freedom (they are comparable with the lifetime of Universe [332]),
an evolution during which the system approaches some stable stationary state cannot be reached in theoretical description of finite isolated mechanical systems. This
does not exclude, however, effectiveness of the statistical physics, which does not
deal with a unique phase-space trajectory of the considered system; here we have a
certain physical reinterpretation of the mechanics of motions in the system’s phase
space. But full effectiveness of the statistical approach to description of behaviour
of multiparticle systems, e.g. mathematically clear description of thermal equilibria
and phase transitions, is again possible in the ‘thermodynamical limit’ of infinitely
large systems only, e.g. [271].
It is seen that after making the finite quantum spin chain of our model to become
an “open system” by adding to the Hamiltonian of the restricted QD the term corresponding to the radiation of a fermion, the speed of the motion to the limiting state
was enormously increased in comparison with the infinite, but isolated, QD-chain,
cf. (7.3.24) and (7.3.26b), i.e. with respect to (7.3.27). The finite-sized version of the
isolated QD would behave, however, almost-periodically, cf. (7.3.23). The addition
of interaction of the finite QD with Fermi field enabled us to obtain a system’s state
converging for t → ∞ to a new stationary state. But a clear and unambiguous interpretation of some states of a finite system, e.g. the two states appearing in the sum on
the right hand side of (7.6.16), as being approximately ‘mutually macroscopically
different’ (hence their quantum interference being ‘almost impossible’), is still open
to discussion. We shall not further investigate here some other connections of these
phenomena and questions.
7.7 On the “Measurement Problem” in QM
Let us add here several notes to the above mentioned “measurement problem”, considered for a long time to be a fundamental problem of the conceptual structure of
QM, cf. e.g. [63, 236, 237] and [238, Chap. 29]. These notes should be also supplemented by the notes in 7.6.1, esp. by the footnotes 12 and 13.
States of the physical systems are described in the mathematical theory of QM
by mathematical objects like “wave functions”, “density matrices”, or “linear func-