Table Of ContentThe 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
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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
