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arXiv:1211.2098

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

Basil J. Hiley

Published Nov 9, 2012• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1211.2098

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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.
In the Wigner-Moyal approach to quantum mechanics, we show that Moyal's starting point, the characteristic function M(τ,θ)=∫ψ∗(x)ei(τp^+θx^)ψ(x)dxM(τ,θ)=\int ψ^{*}(x)e^{i(τ{\hat p}+θ{\hat x})}ψ(x)dxM(τ,θ)=∫ψ∗(x)ei(τp^​+θx^)ψ(x)dx, is essentially the primitive idempotent used by von Neumann in his classic paper "Die Eindeutigkeit der Schrödingerschen Operatoren". This paper provides the original proof of the Stone-von Neumann equation. Thus the mathematical structure Moyal develops is simply a re-expression of what is at the heart of quantum mechanics and reproduces exactly the results of the quantum formalism. The "distribution function" F(X,P,t)F(X,P,t)F(X,P,t) is simply the quantum mechanical density matrix expressed in an (X,P)( X,P)(X,P)-representation, where XXX and PPP are the mean co-ordinates of a cell structure in phase space. The whole approach therefore clearly has little to do with classical statistical theories but is a consequence of a non-commutative nature of the theory.
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.
In this paper we discuss the notion of a ‘particle’ and its ‘trajectory’ in the Bohm approach to quantum theory in the context of a non-commutative dynamics which has classical mechanics as a commutative sub-algebra. We study the von Neumann-Moyal algebra and show that the key equations used in the Bohm approach are already contained within this structure. They appear as projections from the non-commutative phase space into space-time. This structure leads us to a more radical view of quantum processes involving individual particles.
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.
We show that the theory of quantum statistical mechanics is a special model in the framework of the quantum probability theory developed by mathematicians, by extending the characteristic function in the classical probability theory to the quantum probability theory. As dynamical variables of a quantum system must respect certain commutation relations, we take the group generated by a Lie algebra constructed with these commutation relations as the bridge, so that the classical characteristic function defined in a Euclidean space is transformed to a normalized, non-negative definite function defined in this group. Indeed, on the quantum side, this group-theoretical characteristic function is equivalent to the density matrix; hence, it can be adopted to represent the state of a quantum ensemble. It is also found that this new representation may have significant advantages in applications. As two examples, we show its effectiveness and convenience in solving the quantum-optical master equation for a harmonic oscillator coupled with its thermal environment, and in simulating the quantum cat map, a paradigmatic model for quantum chaos. Other related issues are reviewed and discussed as well.
We start from Wootter's construction of discrete phase spaces and Wigner functions for qubits and more generally for finite-dimensional Hilbert spaces. We look at this framework from a non-commutative space perspective and we focus on the Moyal product and the differential calculus on these discrete phase spaces. In particular, the qubit phase space provides the simplest example of a four-point non-commutative phase space. We give an explicit expression of the Moyal bracket as a differential operator. We then compare the quantum dynamics encoded by the Moyal bracket to the classical dynamics: we show that the classical Poisson bracket does not satisfy the Jacobi identity thus leaving the Moyal bracket as the only consistent symplectic structure. We finally generalize our analysis to Hilbert spaces of prime dimensions d and their associated d × d phase spaces.