Classical Systems in Quantum Mechanics

TL;DR

This paper explores how classical systems and mechanics can be derived from quantum mechanics through two different mathematical approaches, addressing the emergence of classical behavior from quantum formalism.

Contribution

It presents two novel methods for deriving classical mechanics from quantum mechanics, one via Hamiltonian field theory and the other through macroscopic limits of quantum systems.

Findings

Classical Hamiltonian mechanics can be obtained from quantum formalism.

Macroscopic classical variables emerge from quantum systems with infinite degrees of freedom.

Models of quantum measurement connect microscopic quantum states with macroscopic classical states.

Abstract

If we admit that quantum mechanics (QM) is universal theory, then QM should contain also some description of classical mechanical systems. The presented text contains description of two different ways how the mathematical description of kinematics and dynamics of classical systems emerges from the mathematical formalism of QM. The first of these ways is to obtain an equivalent description of QM (with finite number of degrees of freedom) as a classical Hamiltonian field theory and afterwards restrict it in dependence of specific classical system to obtain the classical Hamiltonian mechanics of that finite system. The second way is transition to QM of systems with infinite number of degrees of freedom - i.e. of macroscopic systems - and extract from it classical mechanics (with finite number of degrees of freedom) of macroscopic variables of this quantum system. The last chapter contains some considerations concerning the "measurement problem in QM", in which a measured quantum "microsystem" has to be dynamically connected with changes of classical states of a macroscopic quantum (sub-)system - the measuring device. Several models of this process are presented.