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General approach to quantum mechanics as a statistical theory

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journal contribution
posted on 05.02.2019 by Russell P. Rundle, Todd Tilma, John Samson, Vincent Dwyer, R.F. Bishop, Mark Everitt
Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics as a statistical theory. This is equivalent to describing quantum states and ensembles together with their dynamics entirely in terms of phase-space distributions. Finite dimensional systems have historically been an issue. In recent works [Phys. Rev. Lett. 117, 180401 (2016) and Phys. Rev. A 96, 022117 (2017)] we presented a framework for representing any quantum state as a complete continuous Wigner function. Here we extend this work to its partner function—the Weyl function. In doing so we complete the phase-space formulation of quantum mechanics—extending work by Wigner, Weyl, Moyal, and others to any quantum system. This work is structured in three parts. First we provide a brief modernized discussion of the general framework of phase-space quantum mechanics. We extend previous work and show how this leads to a framework that can describe any system in phase space—putting it for the first time on a truly equal footing to Schrödinger's and Heisenberg's formulation of quantum mechanics. Importantly, we do this in a way that respects the unifying principles of “parity” and “displacement” in a natural broadening of previously developed phase-space concepts and methods. Secondly we consider how this framework is realized for different quantum systems; in particular we consider the proper construction of Weyl functions for some example finite dimensional systems. Finally we relate the Wigner and Weyl distributions to statistical properties of any quantum system or set of systems.

Funding

T.T. notes that this work was supported in part by JSPS KAKENHI (C) Grant No. JP17K05569. R.P.R. is funded by the EPSRC (Grant No. EP/N509516/1).

History

School

  • Science

Department

  • Physics

Published in

Physical Review A

Volume

99

Issue

1

Citation

RUNDLE, R.P. ... et al, 2019. General approach to quantum mechanics as a statistical theory. Physical Review A, 99 (1), 012115.

Publisher

© American Physical Society (APS)

Version

VoR (Version of Record)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Acceptance date

04/12/2018

Publication date

2019-01-16

Notes

This paper was published in the journal Physical Review A and is also available at https://doi.org/10.1103/physreva.99.012115.

ISSN

2469-9926

eISSN

2469-9934

Language

en

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