The Phase Space Model of Nonrelativistic Quantum Mechanics.

Jaromir Tosiek,Maciej Przanowski

doi:10.3390/e23050581

Abstract

We focus on several questions arising during the modelling of quantum systems on a phase space. First, we discuss the choice of phase space and its structure. We include an interesting case of discrete phase space. Then, we introduce the respective algebras of functions containing quantum observables. We also consider the possibility of performing strict calculations and indicate cases where only formal considerations can be performed. We analyse alternative realisations of strict and formal calculi, which are determined by different kernels. Finally, two classes of Wigner functions as representations of states are investigated.

Highlights

In contrast to the Hilbert space approach to quantum physics, in classical mechanics, we deal with constraints or with curvature without problems and the mathematical apparatus of differential geometry used for that purpose is well developed
Every fibre in the Weyl bundle admits several ◦-products leading to different ∗-products. Considerations devoted to this aspect of phase space quantum mechanics can be found in [58,65]
Since the Wigner function is defined on the phase space and it represents the state of the system, it should be similar to the density of probability

Summary

Introduction

It seems to be natural that an alternative to the Hilbert space version of quantum mechanics, compatible with classical physics, should exist. Phase space quantum mechanics has remained present in the scientific world. It has developed in parallel as a part of physics and independently as a subdiscipline of mathematics called deformation quantisation. Every quantum observable from that class is related to its classical limit by a relation called ordering This statement implies that there exist several admissible ∗-products related to different orderings. In the case of discrete phase space, we construct a Wigner function in an alternative manner with the use of the trace of a density operator multiplied by a generalised Stratonovich–Weyl quantiser

Some Comments on the Hilbert Space Formulation of Quantum Theory

The Structure of Quantum Phase Space

Algebra of Quantum Observables

Physically Motivated Orderings

Representation of States on a Quantum Phase Space

Conclusions