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The Weyl Operator and Its Generalization

Leon Cohen
Publication Date: 
Number of Pages: 
Pseudo-Differential Operators Theory and Applications 9
[Reviewed by
Michael Berg
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This compact book addresses an exceedingly important theme that straddles mathematics and physics. The exemplar to bear in mind in this connection is that of quantization, i.e. the manoeuvre of formally accounting for non-commutativity in a theory of interactions (and other “measurable” phenomena) at the level of the Planck scale, by positing a correspondence of non-commuting operators with classical commuting functions or variables.

We accordingly encounter first and foremost that watershed in the history and philosophy of science wrought by the Knabenfysik of the 1920s, involving Heisenberg, Born, Dirac, and Schrödinger, viz. matrix mechanics and wave mechanics, soon gelling into quantum mechanics. Chronologically, the first great break with classical physical conceptions came in the form of denying the commutation of measurements of position and momentum of a particle, discovered by the very young Werner Heisenberg. This monumental discovery was soon crafted into matrix language by Max Born.

Erwin Schrödinger, only some six months down the line, along lines prophesied by none other that David Hilbert whom Born and Heisenberg consulted on the question of interpreting their matrix mechanics (see p. 182 of Reid’s biography Hilbert), presented his wave mechanics with partial differential equations taking center stage: witness the famous and ubiquitous Schrödinger wave equation. Paul Dirac, not long afterward, went at the same theme using Poisson brackets, evolving an exceedingly fecund language and formalism for quantum mechanics that still holds sway: his classic and gorgeously written Principles of Quantum Mechanics is still a fabulous place to learn the subject.

Dirac and, independently, Schrödinger, established the equivalence of matrix mechanics and wave mechanics, and when the dust had properly settled, the new and radical theory was put into a proper overarching functional analytic framework by John von Neumann (cf. his Mathematical Foundations of Quantum Mechanics).

Against this historical backdrop one naturally considers the aforementioned mathematical question of how precisely one should associate operators (e.g. unitary operators on a Hilbert space of states for a quantum mechanical system) to classical variables, taking into account that in the process commutativity takes it on the chin. While Born and Pasqual Jordan took up this matter in the context of the new physics, the mathematician who, more than any one else, addressed this matter was Hermann Weyl, whose approach led to what is known as Weyl quantization, a theme that is ubiquitous in this area of mathematical physics. It constitutes a particularly powerful way to effect quantization with a very broad mathematical sweep. For one thing, there are immediate Lie algebraic connections to be had (not surprisingly, of course, given that Weyl is at the helm), as well as an according predisposition to representation theory.

This latter phenomenon is responsible for marvelous connections with analytic number theory, for instance, with the general theory of the metaplectic group i.e. the double cover of the symplectic group in the presence of the projective representation introduced by André Weil in his seminal 1964 paper Sur certains groupes d’opérateurs unitaires. This projective representation’s pre-history is rooted in work by David Shale concerning particles satisfying Bose-Einstein statistics, illustrating the marvel of this part of unitary representation theory as being central to both quantum theory and analytic number theory in essentially the same way.

Returning to quantization as such, however, the book under review is concerned with this process explicitly as a theme in operator theory. It constitutes a terse but very effective exposition of the process covering essentially all aspects of not only its pivotal relation to quantum mechanics, but also its relation to other areas such as time-frequency analysis, image processing, and acoustics.

This is not an “applications” book, however: it’s all hard-core analysis with a no apologies. The opening chapters are concerned with the background material in operator algebra, the Weyl operator being featured in Chapter 3 and general operators in Chapter 4. Then we get to phase space, a number of special cases including Born-Jordan ordering and Weyl ordering, and a discussion of unitary transformations.

Cohen includes, next, an explicit (if terse) discussion of the path integral approach, thus giving Feynman’s famous approach to quantum mechanics (going back to his doctoral thesis) its due. After this a number of specific topics are dealt with, including time-frequency operators, the eigenvalue problem in phase space, and the uncertainty principle “for arbitrary operators” where it arises as a consequence of the interplay of the standard deviations associated to these operators. This is indeed both a beautiful bit of mathematics and famously, or notoriously, when these operators are assigned the indicated meanings in quantum mechanics, something of huge physical importance.

Again, the book is tersely written, and so is not a textbook in the ordinary sense — indeed, there are no exercises, and Cohen isn’t given to expansive essays on motivation and context. On the other hand, the fact that he gets to the point quickly is a virtue for readers who wish to learn this material quickly, modulo their possessing a decent measure of mathematical experience and maturity. With this proviso in mind, the book succeeds very well.



Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Introduction.- The Fundamental Idea, Terminology, and Operator Algebra.- The Weyl Operator.- The Algebra of the Weyl Operator.- Product of Operators, Commutators, and the Moyal Sin Bracket.- Some Other Ordering Rules.- Generalized Operator Association.- The Fourier, Monomial, and Delta Function Associations.- Transformation Between Associations.- Path Integral Approach.- The Distribution of a Symbol and Operator.- The Uncertainty Principle.- Phase-Space Distributions.- Amplitude, Phase, Instantaneous Frequency, and the Hilbert Transform.- Time - Frequency Analysis.- The Transformation of Differential Equations into Phase Space.- The Representation of Functions.- The N Operator Case.