Table Of ContentQuaternionic Quantum Mechanics
and Quantum Fields
The International Series
of Monographs
on Physics
GENERAL EDITORS
J. Birman
S. F. Edwards
C. H. Llewellyn-Smith
M. Rees
Ouaternionic Quantum
Mechanics and
Ouanturn Fields
Stephen L. Adler
Institute for Advanct3d Study, Princeton
New York Oxford OXFORD UNIVERSITY PRESS 1995
Oxford University Press
Oxford New York Toronto
Delhi Bombay Calcutta Madras Karachi
Kuala Lumpur Singapore Hong Kong Tokyo
Nairoh1 Dar cs Salaam Cape Town
Melbourne Auckland Madrid
and associated companies in
Berlin lbadan
Copyright (c) 1995 by Oxford University Press, Inc.
Published by Oxford Univcrsrly Press, Inc.,
200 Madison Avenue, New York. New Yor·k 10016
Oxford b a regi~tered trademark of Oxford University Press
All rights reserved. No part of t\m pubhcalion may be reproduced,
stored in a retrieval ~ystcm, or transmitted, in any form or hy any means,
electronic, mechanical, photocopymg, recordmg, or othcnvise.
wrthout the prior permission of Oxford University Press.
Library of Congre;s Cataloging-in-Publicallon Uata
Adler, Stephen L.
Quaternionic quantum mechanics and quantum tlelds , Stephen L. Adler.
p. em. -(International series of monographs on physics; H8)
Includes bibliographical references and index.
ISBN 0-19-506643-X
I. Quantum theory. 2. Quantum lleld theory.
3. Quatcrnions.
4. Mathematrcal physics.
I. Ti tie.
II. Series: International series of monographs on phy~ics
(Oxford, England): 88.
QC17412.A34 1995
530.1'2-dc20 94-6306
13579 8642
Pnnted in the United States of America
on acid-free paper
Dedicated with love to my children
Anthony, Jessica, and Victoria
This page intentionally left blank
Pre1face
This book aims to give a development and exposition of the quaternionic
generalization of standard complex quantum mechanics. The original impetus
for my writing it came from Richard Slansky who, at the Aspen Winter Physics
Conference in January 1988, suggested that I expand my talk there into a
Physics Reports number giving a full-scale review of quaternionic quantum
mechanics. As work on the intended article progressed, however, it became
apparent that what I was writing was actually a research monograph. Therefore
I decided, with encouragement from my colleague John Bah call, to convert the
project into a book. I am convinced that quaternionic quantum mechanics
represents largely uncharted, and potentially very interesting, terrain in theore
tical physics and hope that this work will encourage its further exploration by
others.
In the pursuit of my interest in quaternionic quantum mechanics over the last
14 years, I have benefited from conversations or correspondence with a large
number of people. Specifically, let me mention I. Adler, W. A. Bardeen, I. Bars,
J. S. Bell, L. C. Biedenharn, G. V. Bhanot, J. D. Bjorken, L. S. Brown, C. P.
Burgess, Y. M. Cho, S. Coleman, S. Cotanch, A. Davies, G. Domokos, F. J.
Dyson, D. Finkelstein, B. Grossman, M. Giinaydin, J. B. Hartle, G. Hegerfeldt,
T. J. Higgins, L. P. Horwitz, R. Jackiw, T. Kicu, G. Kilcup, J. R. Klauder,
A. Klein, R. Langlands, S.-C. Lee, T. D. Lee, A. J. Leggett, G. W. Mackey,
A. Mcintosh, B. H. J. McKellar, A. Millard, R. L. Mills, C. Moreira,
M. Mueller, H. C. Myung, Y. Nambu, R. Narayanan, C. Nash, Y. J. Ng,
V. Novikov, S. Okubo, G. I. Opat, B. Ovrut, A. Pais, E. A. Paschos, S. G.
Rajeev, P. Ramond, H. Rees. M. Sachs, J. Sandweiss, N. Seiberg, A. Shapere,
P. Shaw, R. Slansky, A. Soffer, D. Speiser, A. Strominger, C. Teitelboim, S. B.
Trciman, R. Wald, J. D. Weckel, S. Weinberg, K. Westerberg, E. P. Wigner,
D. J. Wineland, B. Winstein, E. Witten, C. Wolf, Y.-S. Wu, C. N. Yang, A.
Zee, and B. Zumino. In particular, my decision to embark on a detailed inves
tigation of quaternionic quantum mechanics arose both from a question posed
to me by Frank Yang and from my study of a preliminary version of the 1984
paper by Larry Biedenharn and Larry Horwitz sent to me by the authors. The
demonstration in Chapter 6 that the S-matrix in quaternionic scattering is
<C( I. i) was motivated by questions posed by Geoffrey Opat and Anthony Klein;
the analysis of second quantization in Sees. 7.4 and I 0.1 was strongly influenced
by remarks made by Larry Horwitz and John Klauder; and the field theory
discussion of Sees. 13.4-13.7 owes much to pertinent questions posed by Lowell
viii PREFACE
Brown and Edward Witten. I am grateful to Murat Giinaydin, Robert Lang
lands, and Susumu Okubo for their critical comments on Chapter I, to James
Hartle and Anthony Leggett for a critical reading of the initial draft of Sec.
14.2, to Steven Weinberg for comments on the discussion of nonlinear quantum
mechanics, and to Larry Biedenharn for his useful remarks on several issues. I
especially wish to thank Larry Horwitz and John Klauder for their many
perceptive comments on large portions of the manuscript. Larry Horwitz's
thorough and insightful critical rereading of the revised draft led to many
improvements in the manuscript, as did John Klauder's critique of the first draft
of Chapters 1--9 and the final two chapters. I am deeply indebted to Karl
Westerberg, who faithfully attended my 1991 -1992 Princeton University
lectures based on Chapters 1-12 of this book and whose many probing ques
tions led to significant improvements in the manuscript.
I also want to thank Joseph Birman for directing me toward Oxford
University Press as publisher, and my editor there, Jeffrey Robbins, for his
patience and assistance. I am deeply grateful to Sarah Brett-Smith for her
encouragement during the final stages of research and writing. I have appre
ciated the hospit~lity of the Aspen Center for Physics during several summers
when portions of this work were done, and of course for the past 14 years (and
more) have enjoyed the marvelous environment for theoretical physics provided
by the Institute for Advanced Study. I am grateful to the State of New Jersey,
and in the 1992-1993 academic year, the Robert E. Brennan Foundation, for
funding the Albert Einstein chair at the Institute, which I have held since 1979,
and to the Department of Energy for its continuing support of my research
under Grant No. DE-FG02-90ER40542. The School of Natural Sciences
computing staff, and in particular Judith Nuskey, provided valuable technical
assistance, and Margaret Best and Paula Bozzay assisted with the TEX
composition. Proofreading was facilitated by attentive help from Gise!e Murphy
and Michelle Sage. Finally, I wish to thank my long-time secretary, Valerie
Nowak, for her patience and her beautiful work in the overall TEX composition
of the manuscript.
Princeton S. L.A.
March 1994
Contents
I INTRODUCTION AND GENERAL FORMALISM, 1
1 Introduction, 3
1.1. Classical Versus Quantum Mechanics, 4
1.2. Number Systems Used for Probability Amplitudes, 5
1.3. Alternative Formulations of Quantum Mechanics, I 0
1.4. Notation and Introduction to Quaternionic Arithmetic, II
2 General Framework of Quaternionic Quantum Mechanics, 19
2.1. States, Operators, Wave Functions, and Inner Products, 20
2.2. Observables and Self-adjoint Operators, 27
2.3. Symmetry Transformations and Anti-self-adjoint Operators, 29
2.4. Time Development, 36
2.5. Relationships Between Quaternionic, Complex, and Real Quantum
Mechanics, 40
2.6. Energy Eigenstates in Quaternionic, Complex, and Real Quantum
Mechanics, and the Complex Embedding of Real Quantum
Mechanics, 45
2.7. Nonextendability to Octonionic Quantum Mechanics, 49
3 Further General Results in Quaternionic Quantum Mechanics, 53
3.1. Space Translations and Momentum, 53
3.2. Rotations and Angular Momentum, 64
3.3. Time Translations, Evolution of Expectation Values, and the
Heisenberg Picture, 68
3.4. The Uncertainty Principle in Quaternionic Quantum
Mechanics, 70
3.5. Representation of Symmetnies of if, 74
3 .6. Simultaneous Diagonalization of Mutually Commuting Self-adjoint
and Anti-self-adjoint Operators, 76
3.7. Spin Angular Momentum and Hamiltonian Structure, 84
