4
1 Introduction
ρ ∈ P(H) (for the orbit O ) and can be made arbitrarily small (in the sense of weak
convergence of distributions to the distribution V ).
For a general time evolution, the orbits O are not invariant with respect to the
quantal time evolution, and also on various orbits of the same quantal system the
projected classical evolutions are mutually different. This brings in mind an idea of
some stochastic time evolution on a classical phase space reflecting the underlying
quantal evolution.
Such an idea is not, however, realizable for systems with finite number degrees
of freedom (briefly: finite systems) because their density matrices have not unique
decomposition into convex combinations of pure states ∈ P(H). This is just a
crude intuition which was not clearly formulated and realized in the following text.
2
1.1.6 Quantal systems with infinite number of degrees of freedom (briefly: infinite
systems) are considered in the Chap. 5. A physical motivation for such a consideration connected with our investigation of the relations between QM and CM consists
in the fact, that ‘macroscopicality’ and ‘classicality’ are almost synonyma: most of
physical systems containing an operationally well defined classical subsystem are
compound of a large number (say: of the order 10
20 and more) of microscopic constituents (like atoms) and vice versa.
3 Described approximately as infinite quantal
systems, these systems have some characteristic properties distinguishing them from
finite ones: the existence of nontrivial sets of ‘classical observables’ in given representations of observable algebra (this fact is a consequence of the existence of
various inequivalent unitary representations), the existence of quite a rich simplexes
(in the sense of Choquet) in the state space of the system allowing (in the presence of
some additional assumptions) unique decomposition of their elements into extremal
elements etc. This enables us to describe their ‘classical subsystems’ directly in terms
of the quantal description—hence the name ‘macroscopic limit’. This means that,
contrary to the case of finite systems,
4 in the case of infinite systems quantal and
classical interpretations of the ‘macroscopic observables’ coincide (at least on a Ginvariant subset of states): classical, resp. macroscopic quantities are represented by
operators belonging to the center of the weak closure of the algebra of observables
in some representations.
5
1.1.7 The Chap. 5 is divided into two sections. In the first one we consider the system
consisting of denumerably infinite number of quantal subsystems, each of which is
described by a G-covariant representation of its algebra of bounded observables. To
be more specific, we consider a sequence of copies of the same finite system in the
2 Some more specific hints on this possible classical stochastic evolutions from quantal time development could be found perhaps in [29].
3 The macroscopic quantal effects like superfluidity and superconductivity are additional effects
observed in these ‘classical subsystems’ of the large quantal systems.
4 where the quantal interpretation of classical quantities (i.e. expectation values of generators of
U (G) in corresponding states) was different from the classical interpretation (i.e. sharp values of
corresponding classical generators).
5 The center Z(A) of a C ∗ -algebra A is the commutative C ∗ -subalgebra of A consisting of all
elements of A, each commuting with all elements of A: Z(A) := {z ∈ A : z ·x − x ·z = 0, ∀x ∈ A}.
1 Introduction
ρ ∈ P(H) (for the orbit O ) and can be made arbitrarily small (in the sense of weak
convergence of distributions to the distribution V ).
For a general time evolution, the orbits O are not invariant with respect to the
quantal time evolution, and also on various orbits of the same quantal system the
projected classical evolutions are mutually different. This brings in mind an idea of
some stochastic time evolution on a classical phase space reflecting the underlying
quantal evolution.
Such an idea is not, however, realizable for systems with finite number degrees
of freedom (briefly: finite systems) because their density matrices have not unique
decomposition into convex combinations of pure states ∈ P(H). This is just a
crude intuition which was not clearly formulated and realized in the following text.
2
1.1.6 Quantal systems with infinite number of degrees of freedom (briefly: infinite
systems) are considered in the Chap. 5. A physical motivation for such a consideration connected with our investigation of the relations between QM and CM consists
in the fact, that ‘macroscopicality’ and ‘classicality’ are almost synonyma: most of
physical systems containing an operationally well defined classical subsystem are
compound of a large number (say: of the order 10
20 and more) of microscopic constituents (like atoms) and vice versa.
3 Described approximately as infinite quantal
systems, these systems have some characteristic properties distinguishing them from
finite ones: the existence of nontrivial sets of ‘classical observables’ in given representations of observable algebra (this fact is a consequence of the existence of
various inequivalent unitary representations), the existence of quite a rich simplexes
(in the sense of Choquet) in the state space of the system allowing (in the presence of
some additional assumptions) unique decomposition of their elements into extremal
elements etc. This enables us to describe their ‘classical subsystems’ directly in terms
of the quantal description—hence the name ‘macroscopic limit’. This means that,
contrary to the case of finite systems,
4 in the case of infinite systems quantal and
classical interpretations of the ‘macroscopic observables’ coincide (at least on a Ginvariant subset of states): classical, resp. macroscopic quantities are represented by
operators belonging to the center of the weak closure of the algebra of observables
in some representations.
5
1.1.7 The Chap. 5 is divided into two sections. In the first one we consider the system
consisting of denumerably infinite number of quantal subsystems, each of which is
described by a G-covariant representation of its algebra of bounded observables. To
be more specific, we consider a sequence of copies of the same finite system in the
2 Some more specific hints on this possible classical stochastic evolutions from quantal time development could be found perhaps in [29].
3 The macroscopic quantal effects like superfluidity and superconductivity are additional effects
observed in these ‘classical subsystems’ of the large quantal systems.
4 where the quantal interpretation of classical quantities (i.e. expectation values of generators of
U (G) in corresponding states) was different from the classical interpretation (i.e. sharp values of
corresponding classical generators).
5 The center Z(A) of a C ∗ -algebra A is the commutative C ∗ -subalgebra of A consisting of all
elements of A, each commuting with all elements of A: Z(A) := {z ∈ A : z ·x − x ·z = 0, ∀x ∈ A}.