The Scope and History of Commutative and Noncommutative Harmonic Analysis George W. Mackey HISTORY OF MATHEMATICS Volume 5 AMERICAN MATHEMATICAL SOCIETY LONDON MATHEMATICAL SOCIETY The Scope and History of Commutative and Noncommutative Harmonic Analysis Titles in This Series Volume 5 George W. Mackey The scope and history of commutative and noncommutative harmonic analysis 1992 4 Charles W. McArthur Operations analysis in the U.S. Army Eighth Air Force in World War II 1990 3 Peter Duren, editor, et al. A century of mathematics in America, part III 1989 2 Peter Duren, editor, et al. A century of mathematics in America, part II 1989 1 Peter Duren, editor, et al. A century of mathematics in America, part I 1988 https://doi.org/10.1090/hmath/005 The Scope and History of Commutative and Noncommutative Harmonic Analysis George W. Mackey HISTORY OF MATHEMATICS Volume5 AMERICAN MATHEMATICAL SOCIETY LONDON MATHEMATICAL SOCIETY 2000 Mathematics Subject Classification. Primary OOB60; Secondary 22D30, 01-02, 11---02, 81Q99. Library of Congress Cataloging-in-Publication Data Mackey, George Whitelaw, 1916- The scope and history of commutative and noncommutative harmonic analysis / George W. Mackey. p. cm. - (History of mathematics, ISSN 0899-2428 ; v. 5) Consists of six reprinted articles originally written as expanded versions of talks given at various conferences and published between 1978 and 1990. Includes bibliographical references. ISBN 0-8218-9903-1 (acid-free paper) ISBN 0-8218-3790-7 (soft cover) 1. Harmonic analysis-History. I. Title. II. Series. QA403.M28 1992 92-12857 515'.2433-dc20 CIP Harmonic Analysis as the Exploitation of Symmetry-A Historical Survey, Rice Uni versity Studies, volume 64, numbers 2 and 3, pages 73-228. Copyright © 1978. Reprinted by permission of Rice Uhiversity Press. Herman Weyland the Application of Group Theory to Quantum Mechanics, Exact Sci ences and their Philosophical Foundations/Exakte Wissenschaften und ihre philosophische Grundlegung, Vortrage des Internationalen Hermann-Weyl-Kongresses, Kiel, 1985, edited by Wolfgang Deppert, Kurt Hubner, Arnold Oberschelp, and Volker Weidemann, pages 131-159. Copyright© 1985. Reprinted by permission of Verlag Peter Lang GmbH. The Significance of Invariant Measures for Harmonic Analysis, Colloquia Mathematica Societatis Janos Bolyai, volume 49, pages 551-609. Copyright © 1985. Reprinted by permission of Janos Bolyai Mathematical Society. Weyl's Program and Modem Physics, Differential Geometrical Methods in Theoretical Physics, edited by K. Bleuleer and M. Werner, pages 11-36. Copyright© 1988. Reprinted by permission of Kluwer Academic Publishers. Induced Representations and the Applications of Harmonic Analysis, Springer Lecture Notes in Mathematics, volume 1359, edited by P. Eymand and J. P. Pier, pages 16-51. Copyright© 1988. Reprinted by permission of Springer Verlag. Von Neumann and the Early Days of Ergodic Theory, Proceedings of Symposia in Pure Mathematics, volume 50, edited by James Glimm, John lmpagliazzo, Isadore Singer, pages 25-38. Copyright © 1990 American Mathematical Society. © 1992 by the American Mathematical Society. All rights reserved. Reprinted by the American Mathematical Society, 2005 Printed in the United States of America. The American Mathematical Society retains all rights except those granted to the United States Government. § The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://wvw.ams.org/ 10 9 8 7 6 5 4 3 2 1 10 09 08 07 06 05 Contents Introduction vii Harmonic Analysis as the Exploitation of Symmetry: A Historical Survey 1 Herman Weyland the Application of Group Theory to Quantum Mechanics 159 The Significance of Invariant Measures for Harmonic Analysis 189 Weyl's Program and Modern Physics 249 Induced Representations and the Applications of Harmonic Analysis 275 Von Neumann and the Early Days of Ergodic Theory 311 Final Remarks 325 GEORGE W. MACKEY v This page intentionally left blank Introduction This book consists mainly of reprints of six articles of mine, all origi nally written as expanded versions of talks given at various conferences held between 1977 and 1988 and first published in the relevant conference pro ceedings. All of them include a fair amount of historical material and are primarily, but not exclusively, expository in character. They are intimately related to one another and may be thought of as overlapping presentations of various aspects of a single theme. In addition, it contains a lengthy section entitled "Final Remarks" whose nature is explained in its own introduction. The first and earliest is by far the longest and most comprehensive. To some extent, the others may be regarded as fuller treatments or updatings of parts of the first with the special purposes of the corresponding conference in mind. For instance, three of the conferences were organized to do honor to the memories of Hermann Weyl, Alfred Haar, and John von Neumann, respectively, and my talks naturally put emphasis on their contributions to my central theme. However, in each case I had one or more new things to say as a result of insights and points of view gained since the first was written. Before discussing the six articles individually, it will be useful to comment on the background of the first. In the summer of 1965, Jack Feldman invited me to spend six weeks at the University of California in Berkeley and to give some lectures on my work. For some time I had become increasingly impressed by the extent and interrelatedness of the applications of the theory of unitary group representations to physics, probability, and number theory and I seized the opportunity to organize my thoughts on these matters. I gave twelve 90 minute lectures and wrote extensive lecture notes which were typed and mimeographed. The first paragraph of the introduction to these notes reads as follows: In these lectures I am going to attempt something a little un usual. Instead of giving a detailed account with proofs of a rela tively small body of mathematical theory, I propose to give a series of interrelated survey lectures. These will be designed to show the extent to which the theory of infinite dimensional group represen tations is a universal tool with significant applications in subjects as GEORGE W. MACKEY vii viii G. MACKEY diverse as number theory, ergodic theory, quantum physics, prob ability theory, and the theory of automorphic functions. In fact, to speak of significant applications is to understate the case. Large sections of some of these subjects may be looked upon as nearly identical with certain branches of the theory of group representa tions. Moreover, one obtains a clearer view of the many known relationships between the subjects in question by looking at them in this way. While I was in Berkeley, I received a formal invitation (with Michael Atiyah as sponsor) to go to Oxford for the academic year 1966-1967 and give a course of lectures as George Eastman Visiting Professor. By this time, my enthusiasm for my new program had increased, and I lost no time in deciding to treat the same theme in my Oxford lectures, taking advantage of a several-fold increase in lecturing time, to do a more complete and thorough job. Once again mimeographed lecture notes were produced and distributed on a small scale. More than a decade later (in 1978), these lecture notes, slightly revised and supplemented by some 9000 words of "Notes and Ref erences," were published in book form [5]. The first paper in this collection may be regarded as being a presenta tion of the point of view of the Berkeley and Oxford lectures, but with less detailed development, much more emphasis on the period before 1940, espe cially before 1896, and organized as a history of the applications of harmonic analysis. Recall that group representations were not invented until 1896 and that it was only in 1927 that Peter and Weyl pointed out and emphasized the (still insufficiently appreciated) fact that classical Fourier analysis can be illuminatingly regarded as a chapter in the representation theory of compact commutative Lie groups. When I was invited to speak at the conference on the history of analysis given at Rice University on March 12-13, 1977, I decided that it might be interesting to review the hist~ry of mathematics and physics in the last three hundred years or so, with heavy emphasis on those parts in which harmonic analysis had played a decisive or, at least, a major role. I was pleased and somewhat astonished to find how much of both subjects could be included under this rubric. To put it slightly differently, I decided to sketch the history of harmonic analysis-not as an interesting self-contained branch of mathematics, but as a widely applicable method with considerable unifying power. My subject turned out to be much too vast to be treated adequately in two lectures. Thus, the talks actually given were short on concrete details and long on vague "handwaving." In writing my talks up for publication, I decided to fill in some of these gaps, and found myself drawn into a much more extensive project than I had anticipated. It took several months of full-time work but turned out to be an extremely rewarding experience. The picture that gradually emerged as the various details fell into place was one viii COMMUTATIVE AND NONCOMMUTATIVE HARMONIC ANALYSIS INTRODUCTION ix that I found very beautiful, and the process of seeing it do so left me in an almost constant state of euphoria. I would like to believe that others can be led to see this picture by reading my paper; to facilitate this, I have included a large number of short expositions of topics which are not widely understood by non-specialists. However, I fear that there is little hope of achieving my goal for those not willing to take the time to go through the paper rather slowly and read each exposition with care. The paper is written for the mathematical public at large and, moreover, can be read selectively. However, its full message is only available to those who are willing to read the whole paper. The papers given at the Kiel conference honoring the one hundredth birth day of Hermann Weyl and at the Como conference on Differential Geomet rical Methods in Theoretical Physics deal exclusively with the applications of unitary group representations to quantum mechanics. These began in 1927 with independent papers by Weyland Wigner published in volumes 46 and 43, respectively, of Zeitschrift fur Physik. While Wigner was concerned with the use of group representations in the solution of concrete physical problems, Weyl was interested in using the same subject to clarify the foundations. In these two papers we discuss only Weyl's program. The first 40% of the Weyl centenary paper is devoted to a detailed account of the historical background for Weyl's fundamental work of 1924-1925 on group representation theory, its application to quantum mechanics, and its unification with Fourier analysis. The middle 25% is a review of the contents of his 1927 paper in the Zeitschrift far Physik and the now classic book (13] which appeared a year later. The final 35% first explains how this paper stimulated a fundamental paper by M. H. Stone, which in turn (almost two decades later) stimulated the present author to prove a general theorem about group representations known as the imprimitivity theorem. It then goes on to explain how the imprimitivity theorem makes it possible to give a much more satisfactory answer to the fundamental foundational question posed by Weyl in the Zeitschrift paper of 1927 than Weyl could give with the tools available at the time. The paper prepared for the Como conference overlaps the Weyl centenary paper in that the first half of the former is a review of parts of the latter. However, the author believes that the exposition of section IV of the Como paper is a substantial improvement on the corresponding exposition in the centenary paper. In the second half of the Como paper, it is explained how one can go on from the account of free particle quantum statics, contained in the first part, to include dynamics, particle interactions, isotopic spin, etc., and to make contact with such relatively recent developments in elementary particle quantum mechanics as gauge fields and supersymmetry. The Haar centenary paper is, in a sense, a shortened version of the Rice article. However it is organized differently. The role of Haar measure is underlined and certain topics not treated at all in the Rice article are included. GEORGE W. MACKEY ix