2
1 Introduction
Quantal models of many systems, on the other hand, might be constructed from
classical models of the same systems (which are adequate in a certain range of conditions, e.g. classical gases in some intervals of temperature and density) by a more
or less standard procedure of ‘quantization’, compare, e.g. [19, 93, 117], [225, 228,
314], and works quoted therein. The (vaguely stated) question arising from these
considerations is: What is a ‘physically justified way’ of correct determination
of quantal models from their classical approximates ?
1.1.3 One of the remarkable features of QM is the occurence of the universal (Planck)
constant , which might be used to measure mutual ‘deviation’ of quantal and classical descriptions of a given physical system (we shall not discuss here the nontrivial methodological question: how to determine a ‘physical system’ and what is its
dependence on theoretical concepts used in the process of the determination). Consequently, an approximate description of processes in the framework of QM that are
characterized by some quantities S large compared to the Planck constant (S being
of the same physical dimension as ) is often reached in the limit of large values
of S
−1 (‘short wave asymptotics’). If, however, the system described by QM has
some features (‘variables’ etc.) which are adequately described by CM too, then the
description of this ‘classical subsystem’ has to be contained in QM with the fixed
value of Planck constant (i.e. the classical description should be exact consequence
of QM without any approximation procedure, which is often formally performed by
the limit → 0
1 ). We shall introduce a standard procedure of obtaining classical
systems from quantal ones. Such a classical system is called here a ‘classical projection’ of the quantal system (contrary to the ‘classical limit’ obtained in some way
by → 0).
1.1.4 This work is considered as a conceptually and intuitively (however, not always
technically) simple way to give some insight into the indicated questions. Much
more complete and extensive overview of these and related technical topics is given
in the recent book [192] by Landsman. Many relevant questions are discussed in
the author’s work [37], containing also a detailed discussion of possible extensions
of the QM formalism to its nonlinear versions; these nonlinear quantum motions
are closely connected with the theory presented in our Chap. 6, corresponding to
the motions of a single “microsystem” moving in the “mean-field” acting on it by
interaction with infinite number of similar microsystems; the dynamics of the whole
infinite collection of “microsystems” is, however, linear. Such a nonlinear quantum
dynamics is also discussed by S.Weinberg in [328], whose work is also discussed
and reformulated in [37, Sect. 3.6].
The mentioned work of S. Weinberg is not intrinsically consistent in the case of
nonlinear motions of nontrivial density matrices, resp. “mixtures”. To obtain successful picture of nonlinear quantum dynamics of “mixed states” together with their
physically satisfactory quantal interpretation, one has to introduce two kinds of
“mixed states”: The usual one used in (linear) QM are described in the standard way
1 Consider here macroscopic quantal effects (e.g. superconductivity, superfluidity) vanishing for
→0.
1 Introduction
Quantal models of many systems, on the other hand, might be constructed from
classical models of the same systems (which are adequate in a certain range of conditions, e.g. classical gases in some intervals of temperature and density) by a more
or less standard procedure of ‘quantization’, compare, e.g. [19, 93, 117], [225, 228,
314], and works quoted therein. The (vaguely stated) question arising from these
considerations is: What is a ‘physically justified way’ of correct determination
of quantal models from their classical approximates ?
1.1.3 One of the remarkable features of QM is the occurence of the universal (Planck)
constant , which might be used to measure mutual ‘deviation’ of quantal and classical descriptions of a given physical system (we shall not discuss here the nontrivial methodological question: how to determine a ‘physical system’ and what is its
dependence on theoretical concepts used in the process of the determination). Consequently, an approximate description of processes in the framework of QM that are
characterized by some quantities S large compared to the Planck constant (S being
of the same physical dimension as ) is often reached in the limit of large values
of S
−1 (‘short wave asymptotics’). If, however, the system described by QM has
some features (‘variables’ etc.) which are adequately described by CM too, then the
description of this ‘classical subsystem’ has to be contained in QM with the fixed
value of Planck constant (i.e. the classical description should be exact consequence
of QM without any approximation procedure, which is often formally performed by
the limit → 0
1 ). We shall introduce a standard procedure of obtaining classical
systems from quantal ones. Such a classical system is called here a ‘classical projection’ of the quantal system (contrary to the ‘classical limit’ obtained in some way
by → 0).
1.1.4 This work is considered as a conceptually and intuitively (however, not always
technically) simple way to give some insight into the indicated questions. Much
more complete and extensive overview of these and related technical topics is given
in the recent book [192] by Landsman. Many relevant questions are discussed in
the author’s work [37], containing also a detailed discussion of possible extensions
of the QM formalism to its nonlinear versions; these nonlinear quantum motions
are closely connected with the theory presented in our Chap. 6, corresponding to
the motions of a single “microsystem” moving in the “mean-field” acting on it by
interaction with infinite number of similar microsystems; the dynamics of the whole
infinite collection of “microsystems” is, however, linear. Such a nonlinear quantum
dynamics is also discussed by S.Weinberg in [328], whose work is also discussed
and reformulated in [37, Sect. 3.6].
The mentioned work of S. Weinberg is not intrinsically consistent in the case of
nonlinear motions of nontrivial density matrices, resp. “mixtures”. To obtain successful picture of nonlinear quantum dynamics of “mixed states” together with their
physically satisfactory quantal interpretation, one has to introduce two kinds of
“mixed states”: The usual one used in (linear) QM are described in the standard way
1 Consider here macroscopic quantal effects (e.g. superconductivity, superfluidity) vanishing for
→0.