On the classical limit of quantum mechanics I

TL;DR

This paper extends Hepp's method for analyzing the classical limit of quantum mechanics to include unbounded polynomial observables, revealing how quantum expectations relate to classical trajectories and transformations.

Contribution

It introduces a generalized approach to the classical limit that accommodates unbounded polynomial observables, expanding upon Hepp's original bounded observable framework.

Findings

Quantum expectations asymptotically follow classical trajectories.

Next order quantum contributions involve linear canonical transformations.

The method applies to unbounded polynomial observables, broadening previous results.

Abstract

This paper is devoted to the study of the classical limit of quantum mechanics. In more detail we will elaborate on a method introduced by Hepp in 1974 for studying the asymptotic behavior of quantum expectations in the limit as Plank's constant ($\hbar)$ tends to zero. Our goal is to allow for unbounded observables which are (non-commutative) polynomial functions of the position and momentum operators. This is in contrast to Hepp's original paper where the observables were, roughly speaking, required to be bounded functions of the position and momentum operators. As expected the leading order contributions of the quantum expectations come from evaluating the observables along the classical trajectories while the next order contributions are computed by evolving the $\hbar=1$ observables by a linear canonical transformations which is determined by the second order pieces of the quantum mechanical Hamiltonian.