Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal and proceedings of the Royal Society of New South Wales

The Moyal-Dirac controversy revisited

Basil J. Hiley

Published Dec 1, 2021• 1 citation
Publisher

Abstract

In this paper we revisit the controversy that arose between Paul Dirac and Joe Moyal.Our motivation is provided by the fact that the algebraic aspects of both their approaches are becoming more appreciated as interest in the development of non-commutative geometry's attempt to "geometrise" quantum mechanics grows.Both were seeking to understand the role of non-commutativity, Moyal from a consideration of the differences between classical and quantum statistics, while Dirac was exploring its implications for the dynamics.The disagreement arose essentially over which should be given priority, the dynamics or the statistics.We will provide the background to show that both are essential aspects of the same overall mathematical structure. IntroductionN this paper I want to focus on the controversy that arose between Paul Dirac and Joe Moyal as a consequence of their different proposals for developing a phase space approach to quantum theory.This controversy gets to the heart of the fundamental problems in the description of quantum phenomena.Interestingly Dirac and Moyal were both "outsiders" in the sense that they started out as engineers, more so Moyal because of the way he entered academia, a story beautifully told in a biography penned by his wife Ann Moyal (2006).! We pick up the story when Moyal was working in the Meteorological branch of the Ministére de |'Air in Paris at the outbreak of World War IJ.In the ensuing chaos, he managed to escape to England and was eventually given a job at the de Havilland Aircraft Company where he became the Assistant Director of the wartime Vibrations Department.After the war Moyal wanted to work in quantum physics when fortunately an opportunity arose for him to enter academia by becoming an assistant lecturer in the Department of Mathematical Physics at Queen's University Belfast.It was there in 1949 that he published his classic paper which is the focus of his disagreement with Dirac.This disagreement involves the question as how best to develop a (p, x,t) phase space formulation for quantum phenomena, the well-recognised difficulty arising from the non-commutativity of the quantum observables for momentum and position.In other words, if we replace the classical variables, (p,x), by operators, or rather 1. See also Moyal (2017).[Ed.]

DOI
10.5962/p.361972

Related papers

Quantum Trajectories: Dirac, Moyal and Bohm

Basil J. Hiley, Maurice A. de Gosson, Glen Dennis

We recall Dirac's early proposals to develop a description of quantum phenomena in terms of a non-commutative algebra in which he suggested a way to construct what he called quantum trajectories. Generalising these ideas, we show how they are related to weak values and explore their use in the experimental construction of quantum trajectories. We discuss covering spaces which play an essential role in accounting for the wave properties of quantum particles. We briefly point out how new mathematical techniques take us beyond Hilbert space and into a deeper structure which connects with the algebras originally introduced by Born, Heisenberg and Jordan. This enables us to bring out the geometric aspects of quantum phenomena. Quanta 2019; 8: 11–23.

arXiv:1610.07130•Quanta•Jun 5, 2019•3 citations

A unifying perspective on the Moyal and Voros products and their physical meanings

Prasad Basu, Biswajit Chakraborty, Frederik G. Scholtz

The Moyal and Voros formulations of non-commutative quantum field theory has been a point of controversy in the recent past. Here we address this issue in the context of non-commutative non-relativistic quantum mechanics. In particular we show that the two formulations simply correspond to two different representations associated with two different choices of basis on the quantum Hilbert space. From a mathematical perspective the two formulations are therefore completely equivalent, but we also argue that only the Voros formulaton admits a consistent physical interpretation. These considerations are elucidated by considering the free particle transition amplitude in the two representations.

arXiv:1101.2495•Journal of Physics A Mathematical and Theoretical•Jun 10, 2011•30 citations

Quantum mechanics in phase space

Philip Watson

The objective of this thesis is to describe the fundamental concepts relating to the reformulation of quantum mechanics in phase space. It is assumed that the interested reader is familiar with the principles and techniques of ordinary quantum mechanics. In classical statistical mechanics the expectation values of physical quantities are calculated as averages over phase space distribution functions. It is possible to obtain a similar procedure in quantum mechanics. However, phase space is a classical concept and thus has no quantum mechanical equivalent, owing to the non-commutability of the operators p and q. A sensible basic requirement for the formulation of quantum mechanics in phase space is a linear one-one mapping between quantum operators and classical functions. This was achieved by the pioneering work of Weyl, Wigner and Moyal. The phase space that they devised we shall call "pseudo phase space". It is completely quantum. The pseudo phase space variables are represented by commuting c-numbers of momentum, p, and position, q; they are the result of applying the Weyl correspondence rule to the operators p and q, respectively. In some ways the pseudo phase space formulation of quantum mechanics appears to return us to an "almost" classical arena, since we dispense with the calculus of non-commuting operators for purely algebraic methods. The pseudo phase space formulation of quantum mechanics is self-contained. That is, there is no need, in principle, to switch to the Schroedinger or Heisenberg pictures when solving physical problems, although it may often be convenient to do so. The classical appearance of the pseudo phase space scheme is especially useful when considering the semi-classical limit of quantum mechanics. In fact, for potentials up to and including the harmonic oscillator the equations represent classical motion. The structure of the thesis is such that it takes the reader steadily through the major concepts of pseudo phase space theory. The mathematical techniques that shall be needed are developed early in Chapter 1. These are used throughout the thesis. This presents the advantage that the reader is not faced with learning new mathematical methods as the thesis proceeds. The pseudo phase space formulation of quantum mechanics as found applications in all areas of mathematical physics. Hence, there is a considerable amount of specialist literature available on the subject. Itis hoped that this thesis serves has an introduction.

•Jan 1, 1991•0 citations

From Noncommutative Phase Space to Hilbert Space

B. Muthukumar, Aalok Misra

In the Hamiltonian formulation, classical mechanics employs the commutative algebra of functions that are defined on phase space a point of which could be represented using Dirac delta distributions. In the absence of such a concrete existence of the notion of point in the quantum domain, we show how the quantum mechanical formalism emerges by replacing the commutative algebra by a noncommutative algebra of functions and introducing the quantum condition. The noncommutativity is achieved by deforming the product of two functions and by introducing the Moyal brackets, the basic Moyal brackets between two spatial coordinates and between position and momentum coordinates being noncommutative. The only other input we make use of is the quantum condition which is motivated from its classical analogue—the square of Dirac delta distributions.

AIP conference proceedings•Jan 1, 2007•2 citations

The Algebraic Way

Basil J. Hiley

In this paper we examine in detail the non-commutative symplectic algebra underlying quantum dynamics. We show that this algebra contains both the Weyl-von Neumann algebra and the Moyal algebra. The latter contains the Wigner distribution as the kernel of the density matrix. The underlying non-commutative geometry can be projected into either of two Abelian spaces, so-called `shadow phase spaces'. One of these is the phase space of Bohmian mechanics, showing that it is a fragment of the basic underlying algebra. The algebraic approach is much richer, giving rise to two fundamental dynamical time development equations which reduce to the Liouville equation and the Hamilton-Jacobi equation in the classical limit. They also include the Schrödinger equation and its wave function, showing that these features are a partial aspect of the more general non-commutative structure. We discuss briefly the properties of this more general mathematical background from which the non-commutative symplectic algebra emerges.

arXiv:1602.06071•IMPERIAL COLLEGE PRESS eBooks•Apr 4, 2016•3 citations

On the Relationship between the Moyal Algebra and the Quantum Operator Algebra of von Neumann

Basil J. Hiley

The primary motivation for Moyal's approach to quantum mechanics was to develop a phase space formalism for quantum phenomena by generalising the techniques of classical probability theory. To this end, Moyal introduced a quantum version of the characteristic function which immediately provides a probability distribution. The approach is sometimes perceived negatively merely as an attempt to return to classical notions, but the mathematics Moyal develops is simply a re-expression of what is at the heart of quantum mechanics, namely the non-commutative algebraic structure first introduced by von Neumann in 1931. In this paper we will establish this relation and show that the "distribution function", F(P,X,t) is simply the quantum mechanical density matrix for a single particle. The coordinates, X and P, are not the coordinates of the particle but the mean co-ordinates of a cell structure (a `blob') in phase space, giving an intrinsically non-local description of each individual particle, which becomes a point in the limit to order $\hbar^2$. We discuss the significance of this non-commutative structure on the symplectic geometry of the phase space for quantum processes.

arXiv:1211.2098•Nov 9, 2012•5 citations
In this paper we revisit the controversy that arose between Paul Dirac and Joe Moyal.Our motivation is provided by the fact that the algebraic aspects of both their approaches are becoming more appreciated as interest in the development of non-commutative geometry's attempt to "geometrise" quantum mechanics grows.Both were seeking to understand the role of non-commutativity, Moyal from a consideration of the differences between classical and quantum statistics, while Dirac was exploring its implications for the dynamics.The disagreement arose essentially over which should be given priority, the dynamics or the statistics.We will provide the background to show that both are essential aspects of the same overall mathematical structure. IntroductionN this paper I want to focus on the controversy that arose between Paul Dirac and Joe Moyal as a consequence of their different proposals for developing a phase space approach to quantum theory.This controversy gets to the heart of the fundamental problems in the description of quantum phenomena.Interestingly Dirac and Moyal were both "outsiders" in the sense that they started out as engineers, more so Moyal because of the way he entered academia, a story beautifully told in a biography penned by his wife Ann Moyal (2006).! We pick up the story when Moyal was working in the Meteorological branch of the Ministére de |'Air in Paris at the outbreak of World War IJ.In the ensuing chaos, he managed to escape to England and was eventually given a job at the de Havilland Aircraft Company where he became the Assistant Director of the wartime Vibrations Department.After the war Moyal wanted to work in quantum physics when fortunately an opportunity arose for him to enter academia by becoming an assistant lecturer in the Department of Mathematical Physics at Queen's University Belfast.It was there in 1949 that he published his classic paper which is the focus of his disagreement with Dirac.This disagreement involves the question as how best to develop a (p, x,t) phase space formulation for quantum phenomena, the well-recognised difficulty arising from the non-commutativity of the quantum observables for momentum and position.In other words, if we replace the classical variables, (p,x), by operators, or rather 1. See also Moyal (2017).[Ed.]
We recall Dirac's early proposals to develop a description of quantum phenomena in terms of a non-commutative algebra in which he suggested a way to construct what he called quantum trajectories. Generalising these ideas, we show how they are related to weak values and explore their use in the experimental construction of quantum trajectories. We discuss covering spaces which play an essential role in accounting for the wave properties of quantum particles. We briefly point out how new mathematical techniques take us beyond Hilbert space and into a deeper structure which connects with the algebras originally introduced by Born, Heisenberg and Jordan. This enables us to bring out the geometric aspects of quantum phenomena. Quanta 2019; 8: 11–23.
The Moyal and Voros formulations of non-commutative quantum field theory has been a point of controversy in the recent past. Here we address this issue in the context of non-commutative non-relativistic quantum mechanics. In particular we show that the two formulations simply correspond to two different representations associated with two different choices of basis on the quantum Hilbert space. From a mathematical perspective the two formulations are therefore completely equivalent, but we also argue that only the Voros formulaton admits a consistent physical interpretation. These considerations are elucidated by considering the free particle transition amplitude in the two representations.
The objective of this thesis is to describe the fundamental concepts relating to the reformulation of quantum mechanics in phase space. It is assumed that the interested reader is familiar with the principles and techniques of ordinary quantum mechanics. In classical statistical mechanics the expectation values of physical quantities are calculated as averages over phase space distribution functions. It is possible to obtain a similar procedure in quantum mechanics. However, phase space is a classical concept and thus has no quantum mechanical equivalent, owing to the non-commutability of the operators p and q. A sensible basic requirement for the formulation of quantum mechanics in phase space is a linear one-one mapping between quantum operators and classical functions. This was achieved by the pioneering work of Weyl, Wigner and Moyal. The phase space that they devised we shall call "pseudo phase space". It is completely quantum. The pseudo phase space variables are represented by commuting c-numbers of momentum, p, and position, q; they are the result of applying the Weyl correspondence rule to the operators p and q, respectively. In some ways the pseudo phase space formulation of quantum mechanics appears to return us to an "almost" classical arena, since we dispense with the calculus of non-commuting operators for purely algebraic methods. The pseudo phase space formulation of quantum mechanics is self-contained. That is, there is no need, in principle, to switch to the Schroedinger or Heisenberg pictures when solving physical problems, although it may often be convenient to do so. The classical appearance of the pseudo phase space scheme is especially useful when considering the semi-classical limit of quantum mechanics. In fact, for potentials up to and including the harmonic oscillator the equations represent classical motion. The structure of the thesis is such that it takes the reader steadily through the major concepts of pseudo phase space theory. The mathematical techniques that shall be needed are developed early in Chapter 1. These are used throughout the thesis. This presents the advantage that the reader is not faced with learning new mathematical methods as the thesis proceeds. The pseudo phase space formulation of quantum mechanics as found applications in all areas of mathematical physics. Hence, there is a considerable amount of specialist literature available on the subject. Itis hoped that this thesis serves has an introduction.
In the Hamiltonian formulation, classical mechanics employs the commutative algebra of functions that are defined on phase space a point of which could be represented using Dirac delta distributions. In the absence of such a concrete existence of the notion of point in the quantum domain, we show how the quantum mechanical formalism emerges by replacing the commutative algebra by a noncommutative algebra of functions and introducing the quantum condition. The noncommutativity is achieved by deforming the product of two functions and by introducing the Moyal brackets, the basic Moyal brackets between two spatial coordinates and between position and momentum coordinates being noncommutative. The only other input we make use of is the quantum condition which is motivated from its classical analogue—the square of Dirac delta distributions.
In this paper we examine in detail the non-commutative symplectic algebra underlying quantum dynamics. We show that this algebra contains both the Weyl-von Neumann algebra and the Moyal algebra. The latter contains the Wigner distribution as the kernel of the density matrix. The underlying non-commutative geometry can be projected into either of two Abelian spaces, so-called `shadow phase spaces'. One of these is the phase space of Bohmian mechanics, showing that it is a fragment of the basic underlying algebra. The algebraic approach is much richer, giving rise to two fundamental dynamical time development equations which reduce to the Liouville equation and the Hamilton-Jacobi equation in the classical limit. They also include the Schrödinger equation and its wave function, showing that these features are a partial aspect of the more general non-commutative structure. We discuss briefly the properties of this more general mathematical background from which the non-commutative symplectic algebra emerges.
The primary motivation for Moyal's approach to quantum mechanics was to develop a phase space formalism for quantum phenomena by generalising the techniques of classical probability theory. To this end, Moyal introduced a quantum version of the characteristic function which immediately provides a probability distribution. The approach is sometimes perceived negatively merely as an attempt to return to classical notions, but the mathematics Moyal develops is simply a re-expression of what is at the heart of quantum mechanics, namely the non-commutative algebraic structure first introduced by von Neumann in 1931. In this paper we will establish this relation and show that the "distribution function", F(P,X,t) is simply the quantum mechanical density matrix for a single particle. The coordinates, X and P, are not the coordinates of the particle but the mean co-ordinates of a cell structure (a `blob') in phase space, giving an intrinsically non-local description of each individual particle, which becomes a point in the limit to order ℏ2\hbar^2ℏ2. We discuss the significance of this non-commutative structure on the symplectic geometry of the phase space for quantum processes.