1.1 Motivation and Summary
5
(infinite) complete tensor product representation on a nonseparable Hilbert space
H . The representation U (G) describing an elementary subsystem determines a
unitary (discontinuous) representation U (G) on H which, in turn, determines an
automorphism group σ G of the algebra A
of quasilocal observables of the infinite
system. A natural definition of a classical subsystem of the large quantal system
(A
; σ G ) appearing in this case can be extended to the case of arbitrary systems
(A; σ G ), as it is shown in Sect. 5.2. The arising classical (macroscopic) subsystem
(M; σ G ) is naturally mapped into the classical Poisson system (G
∗
; Ad
∗
(G)), or to
its generalizations.
1.1.8 Chapter 6 is devoted to an application of Sects. 5.1 and 5.2:
It is shown that ‘mean-field’ type time evolutions can be determined on a large
quantal system (A; σ G ) by specification of a Hamiltonian dynamics of a classical
(macroscopic) Poisson system—the macroscopic limit of (A; σ G ).
This is a perhaps simplest example of (infinite-)long-range interactions in many
body systems. The correspondence between classical and quantum descriptions of
systems appear there ‘selfconsistently’: The quantum theory of the entering ‘elementary subsystems’ is built ‘on the background’ of the classical ‘environment’ what is
compound of the infinite collection of those ‘elementary subsystems’.
6 The dynamics of a general class of such systems is described in Sect. 6.3, and the statistical
thermodynamics of equilibrium states is introduced in Sect. 6.4.
A slightly alternative approach to these quantum mean-field theories is described
in the papers [40, 41].
1.1.9 Finally, the Chap. 7 contains four exactly solved models of interaction of
a microscopic quantal system with a ‘macroscopic’ one. Due to this interaction
the macroscopic quantal system changes its classical state to a different one. Such
a change of a macroscopic (classical) state can be interpreted as a change of a
‘pointer position’, hence these models could be considered as models of ‘quantum
measurements’ in the sense of Klaus Hepp [153]. The change of the macroscopic
state is reached in the limit t → ∞ of infinite time, and the convergence in the first
three models is very slow.
In the last of the described models (in Sect. 7.6) the ‘macroscopic’ quantal system is described as a finite collection of ‘small’ quantal systems. This leads to
problems with an unambiguous definition of ‘macroscopic states’, since it is possible (formally, in this abstract theory) to observe interference effects between such
different ‘macroscopic states’. To make clear the correspondence of quantum theory with observations, it would be necessary to introduce also quantum models of
observation apparatuses used for detection of states of such a large but finite ‘macroscopic system’. Some discussion on this problem (including reports of observations
of ‘macroscopic interference phenomena’) appeared in literature in last decades, cf.
e.g. [55, 56, 190–192, 195, 196]. In the model of Sect. 7.6, the (large but finite)
6 Ideas of this kind could, perhaps, reconcile the basic idea of Niels Bohr [26, 27] on fundamental
role of a “classical background” in formulations of QM with the postulate that QM is the basic
theory.
5
(infinite) complete tensor product representation on a nonseparable Hilbert space
H . The representation U (G) describing an elementary subsystem determines a
unitary (discontinuous) representation U (G) on H which, in turn, determines an
automorphism group σ G of the algebra A
of quasilocal observables of the infinite
system. A natural definition of a classical subsystem of the large quantal system
(A
; σ G ) appearing in this case can be extended to the case of arbitrary systems
(A; σ G ), as it is shown in Sect. 5.2. The arising classical (macroscopic) subsystem
(M; σ G ) is naturally mapped into the classical Poisson system (G
∗
; Ad
∗
(G)), or to
its generalizations.
1.1.8 Chapter 6 is devoted to an application of Sects. 5.1 and 5.2:
It is shown that ‘mean-field’ type time evolutions can be determined on a large
quantal system (A; σ G ) by specification of a Hamiltonian dynamics of a classical
(macroscopic) Poisson system—the macroscopic limit of (A; σ G ).
This is a perhaps simplest example of (infinite-)long-range interactions in many
body systems. The correspondence between classical and quantum descriptions of
systems appear there ‘selfconsistently’: The quantum theory of the entering ‘elementary subsystems’ is built ‘on the background’ of the classical ‘environment’ what is
compound of the infinite collection of those ‘elementary subsystems’.
6 The dynamics of a general class of such systems is described in Sect. 6.3, and the statistical
thermodynamics of equilibrium states is introduced in Sect. 6.4.
A slightly alternative approach to these quantum mean-field theories is described
in the papers [40, 41].
1.1.9 Finally, the Chap. 7 contains four exactly solved models of interaction of
a microscopic quantal system with a ‘macroscopic’ one. Due to this interaction
the macroscopic quantal system changes its classical state to a different one. Such
a change of a macroscopic (classical) state can be interpreted as a change of a
‘pointer position’, hence these models could be considered as models of ‘quantum
measurements’ in the sense of Klaus Hepp [153]. The change of the macroscopic
state is reached in the limit t → ∞ of infinite time, and the convergence in the first
three models is very slow.
In the last of the described models (in Sect. 7.6) the ‘macroscopic’ quantal system is described as a finite collection of ‘small’ quantal systems. This leads to
problems with an unambiguous definition of ‘macroscopic states’, since it is possible (formally, in this abstract theory) to observe interference effects between such
different ‘macroscopic states’. To make clear the correspondence of quantum theory with observations, it would be necessary to introduce also quantum models of
observation apparatuses used for detection of states of such a large but finite ‘macroscopic system’. Some discussion on this problem (including reports of observations
of ‘macroscopic interference phenomena’) appeared in literature in last decades, cf.
e.g. [55, 56, 190–192, 195, 196]. In the model of Sect. 7.6, the (large but finite)
6 Ideas of this kind could, perhaps, reconcile the basic idea of Niels Bohr [26, 27] on fundamental
role of a “classical background” in formulations of QM with the postulate that QM is the basic
theory.