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Quaternionic 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