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Fundamental Theories of Physics 180

Lawrence P. Horwitz

Relativistic Quantum Mechanics

Fundamental Theories of Physics

Volume 180

Series editors

Henk van Beijeren Philippe Blanchard Paul Busch Bob Coecke Dennis Dieks Detlef Dürr Roman Frigg Christopher Fuchs Giancarlo Ghirardi Domenico J.W. Giulini Gregg Jaeger Claus Kiefer Nicolaas P. Landsman Christian Maes Hermann Nicolai Vesselin Petkov Alwyn van der Merwe Rainer Verch R.F. Werner Christian Wuthrich

The international monograph series “Fundamental Theories of Physics” aims to stretch the boundaries of mainstream physics by clarifying and developing the theoretical and conceptual framework of physics and by applying it to a wide range of interdisciplinary scientific fields. Original contributions in well-established fields such as Quantum Physics, Relativity Theory, Cosmology, Quantum Field Theory, Statistical Mechanics and Nonlinear Dynamics are welcome. The series also provides a forum for non-conventional approaches to these fields. Publications should present new and promising ideas, with prospects for their further development, and carefully show how they connect to conventional views of the topic. Although the aim of this series is to go beyond established mainstream physics, a high profile and open-minded Editorial Board will evaluate all contributions carefully to ensure a high scientific standard.

More information about this series at http://www.springer.com/series/6001

Lawrence P. Horwitz

Relativistic Quantum Mechanics

123

Lawrence P. Horwitz School of Physics Raymond and Beverly Sackler Faculty of Exact Sciences

Tel Aviv University Ramat Aviv Israel

and

Department of Physics Bar Ilan University Ramat Gan Israel

and

Department of Physics Ariel University Samaria Israel

Fundamental Theories of Physics ISBN 978-94-017-7260-0 ISBN 978-94-017-7261-7 (eBook) DOI 10.1007/978-94-017-7261-7

Library of Congress Control Number: 2015944499

Springer Dordrecht Heidelberg New York London © Springer Science+Business Media Dordrecht 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.

Printed on acid-free paper

Springer Science+Business Media B.V. Dordrecht is part of Springer Science+Business Media (www.springer.com)

Acknowledgments

I would like to thank Constantin Piron for his collaboration in developing the main ideas of the theory discussed in this book during visits to the University of Geneva, hosted initially by J.M. Jauch, and for many discussions and collaborations which followed. I am grateful to Stephen L. Adler for his warm hospitality at the Institute for Advanced Study in Princeton over a period of several visits during which, among other things, much of the research on this subject was done. I would also like to thank William C. Schieve for his collaborations over many years, particu- larly in the development of relativistic statistical mechanics, and for the hospitality he and his wife, Florence, offered me at the University of Texas at Austin during many visits. I would also like to thank Fritz Rohrlich for his hospitality at Syracuse University and for many discussions on relativistic quantum theory and for his collaboration in our study of the constraint formalism.

I am also particularly indebted to Rafael I. Arshansky for his collaboration on developing many of the basic ideas, such as the solution of the two-body bound state problem, the Landau-Peierls relation and many aspects of scattering theory, as well as Ishi Lavie (Z”L), who fell in the first Lebanon-Israel war; he played an essential role in the development of many of the basic ideas in scattering theory as well. I am grateful to Y. Rabin who worked with me on the first analyses of the phenomenon of interference in time in 1976, Ori Oron for his contributions to the application of the eikonal approximations to the relativistic equations and to the relativistic generalization of Nelson’s approach to the quantum theory through the use of Brownian motion, and Nadav Shnerb for the development of the second quantized formalism with the help and guidance of Kurt Haller (Z”L), to Igal Aharonovich for his intensive work on the classical electromagnetic properties of the relativistically charged particle and the associated five-dimensional fields, to Martin Land for his many invaluable contributions to the theory and many of its applications, to Avi Gershon for his invaluable contribution to the application of the theory to the TeVeS program (a geometrical approach to the dark matter problem), to Andrew Bennett, who found the way to compute the anomalous magnetic moment of the electron in this framework, as well as many discussions, Meir Zeilig-Hess for his contribution to the black body problem and tensor representa- tions, and to many other students and colleagues who have contributed to the effort to understand the structure and consequences of the theory discussed here.

v

I also wish to thank Kirsten Theunissen of Springer, for her patience and assistence during the preeparation of the manuscript, and Alwyn van der Merwe for his encouragement at the beginning of this project.

Finally, I wish to thank my wife, Ruth, for her patience and support in the many years during which the research was done in developing and exploring the con- sequences of the approach to relativistic quantum mechanics that is discussed in this book.

vi Acknowledgments

Contents

1 Introduction and Some Problems Encountered in the Construction of a Relativistic Quantum Theory . . . . . . . . . 1 1.1 States in Relativistic Quantum and Classical Mechanics . . . . . . 1 1.2 The Problem of Localization for the Solutions of Relativistic

Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Relativistic Classical and Quantum Mechanics . . . . . . . . . . . . . . . 9 2.1 The Einstein Notion of Time . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 The Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 The Newton-Wigner Problem. . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 The Landau-Peierls Problem . . . . . . . . . . . . . . . . . . . . . . . . . 24 Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Spin, Statistics and Correlations . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1 Relativistic Spin and the Dirac Representation . . . . . . . . . . . . . 33 3.2 The Many Body Problem with Spin, and Spin-Statistics . . . . . . 42 3.3 Construction of the Fock Space and Quantum Field Theory . . . . 44 3.4 Induced Representation for Tensor Operators . . . . . . . . . . . . . . 47 Appendix B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4 Gauge Fields and Flavor Oscillations . . . . . . . . . . . . . . . . . . . . . . 51 4.1 Abelian Gauge Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2 Nonabelian Gauge Fields and Neutrino Oscillations . . . . . . . . . 59 4.3 The Hamiltonian for the Spin 12 Neutrinos . . . . . . . . . . . . . . . . 65 4.4 CP and T Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5 The Relativistic Action at a Distance Two Body Problem . . . . . . . 71 5.1 The Two Body Bound State for Scalar Particles . . . . . . . . . . . . 72 5.2 Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.3 The Induced Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.4 The Stueckelberg String . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

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