For a large (infinite) quantal system, an automorphic group action of G on the
C
à -algebra A of its bounded observables enables us to define a macroscopic subsystem being a classical Hamiltonian system of the same type as we obtained in the
case of a finite number of degrees of freedom. There is a difference, however, between
the interpretations of ‘classical projections’ and of these ‘macroscopic limits’: The
classical (mechanical) projection describes classical mechanics of expectation values
of quantal observables whereas the macroscopic limit describes a quantal subsystem
with classical properties—its observables are elements of a subalgebra M G of the
center Z of the double dual A
ÃÃ of A. Any state ! on A has a unique ‘macroscopic
limit’ p M ! which is represented by a probability measure on the corresponding
(generalized) classical phase space. This offers us a possibility of deriving a classical
(macroscopic) time evolution (which is, in general, in a certain sense stochastic, cf.
[29, Sect. III.G]) from the underlying reversible quantal dynamics.
A scheme of ‘macroscopic quantization’ is outlined, according to which a
(nonunique) reconstruction of the infinite quantal system ðA; r G Þ from its macroscopic limit is possible. By determining a classical Hamiltonian function in the
macroscopic limit of ðA; r G Þ we can define a ‘mean-field’ time evolution in the
infinite system ðA; r G Þ. Our definition of the ‘mean-field’ evolutions extends the
usual ones. The schemes and results developed in the work are applicable to models
in the statistical mechanics as well as in gauge-theories (in the ‘large N limit’). They
might be relevant also in general considerations of ‘quantizations’ and of foundations of quantum theory.
The last Chapter of this work is devoted to the description of several models of
interacting ‘microsystems’ with ‘macrosystems’, mimicking a description of the
‘process of measurement in QM’. In these models, certain ‘quantal properties’
of the system, namely a (coherent) superposition of specific vector states (eigenstates of a ‘measured’ observable), transform by the unitary continuous time evolutions (for t ! 1) into the corresponding ‘proper mixtures’ of macroscopically
different states of the ‘macrosystems’ occurring in the models.
In this connection we shall shortly discuss the old ‘quantum measurement
problem’, which however, in the light of certain experiments performed in the past
decades and suggesting the possibilities of quantum-mechanical interference of
several macroscopically different states of a macroscopic system, need not be at all
a fundamental theoretical problem; this might mean that the often discussed
‘measurement process’ can be included into the presently widely accepted model of
quantum theory.
Acknowledgements
This work is a revised and completed version of the unpublished text: “Classical
Projections and Macroscopic Limits of Quantum Mechanical Systems” written
roughly in the years 1985–1986. The author is indebted to Klaus Hepp for his
stimulations and the kind help with correcting many formulations of the original
vi
Preface
C
à -algebra A of its bounded observables enables us to define a macroscopic subsystem being a classical Hamiltonian system of the same type as we obtained in the
case of a finite number of degrees of freedom. There is a difference, however, between
the interpretations of ‘classical projections’ and of these ‘macroscopic limits’: The
classical (mechanical) projection describes classical mechanics of expectation values
of quantal observables whereas the macroscopic limit describes a quantal subsystem
with classical properties—its observables are elements of a subalgebra M G of the
center Z of the double dual A
ÃÃ of A. Any state ! on A has a unique ‘macroscopic
limit’ p M ! which is represented by a probability measure on the corresponding
(generalized) classical phase space. This offers us a possibility of deriving a classical
(macroscopic) time evolution (which is, in general, in a certain sense stochastic, cf.
[29, Sect. III.G]) from the underlying reversible quantal dynamics.
A scheme of ‘macroscopic quantization’ is outlined, according to which a
(nonunique) reconstruction of the infinite quantal system ðA; r G Þ from its macroscopic limit is possible. By determining a classical Hamiltonian function in the
macroscopic limit of ðA; r G Þ we can define a ‘mean-field’ time evolution in the
infinite system ðA; r G Þ. Our definition of the ‘mean-field’ evolutions extends the
usual ones. The schemes and results developed in the work are applicable to models
in the statistical mechanics as well as in gauge-theories (in the ‘large N limit’). They
might be relevant also in general considerations of ‘quantizations’ and of foundations of quantum theory.
The last Chapter of this work is devoted to the description of several models of
interacting ‘microsystems’ with ‘macrosystems’, mimicking a description of the
‘process of measurement in QM’. In these models, certain ‘quantal properties’
of the system, namely a (coherent) superposition of specific vector states (eigenstates of a ‘measured’ observable), transform by the unitary continuous time evolutions (for t ! 1) into the corresponding ‘proper mixtures’ of macroscopically
different states of the ‘macrosystems’ occurring in the models.
In this connection we shall shortly discuss the old ‘quantum measurement
problem’, which however, in the light of certain experiments performed in the past
decades and suggesting the possibilities of quantum-mechanical interference of
several macroscopically different states of a macroscopic system, need not be at all
a fundamental theoretical problem; this might mean that the often discussed
‘measurement process’ can be included into the presently widely accepted model of
quantum theory.
Acknowledgements
This work is a revised and completed version of the unpublished text: “Classical
Projections and Macroscopic Limits of Quantum Mechanical Systems” written
roughly in the years 1985–1986. The author is indebted to Klaus Hepp for his
stimulations and the kind help with correcting many formulations of the original
vi
Preface