Logical reformulation of quantum mechanics. III. Classical limit and irreversibility

Abstract

This paper deals with two questions: (1) It contains a proof of the fact that consistent quantum representations of logic tend to the classical representation of logic when Planck's constant tends to zero. This result is obtained by using the microlocal analysis of partial differential equations and the Weyl calculus, which turn out to be the proper mathematical framework for this type of problems. (2) The analysis of the limitations of this proof turn out to be of physical significance, in particular when one considers quantum systems having for their classical version a KolmogorovK-system. These limitations are used to show the existence of a “best” classical description for such a system leading to an objective definition of entropy. It is shown that in such a description the approach to equilibrium is strictly reduced to a Markov process.