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    Lie Groups for Pedestrians
    Lie Groups for Pedestrians
    Lie Groups for Pedestrians
    Ebook294 pages2 hours

    Lie Groups for Pedestrians

    By Harry J. Lipkin

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    According to the author of this concise, high-level study, physicists often shy away from group theory, perhaps because they are unsure which parts of the subject belong to the physicist and which belong to the mathematician. However, it is possible for physicists to understand and use many techniques which have a group theoretical basis without necessarily understanding all of group theory. This book is designed to familiarize physicists with those techniques. Specifically, the author aims to show how the well-known methods of angular momentum algebra can be extended to treat other Lie groups, with examples illustrating the application of the method.
    Chapters cover such topics as a simple example of isospin; the group SU3 and its application to elementary particles; the three-dimensional harmonic oscillator; algebras of operators which change the number of particles; permutations; bookkeeping and Young diagrams; and the groups SU4, SU6, and SU12, an introduction to groups of higher rank. Four appendices provide additional valuable data.

    LanguageEnglish
    PublisherDover Publications
    Release dateJun 8, 2012
    ISBN9780486137889
    Lie Groups for Pedestrians

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      Lie Groups for Pedestrians - Harry J. Lipkin

      Copyright

      Copyright © 1965, 1966 by Harry J. Lipkin All rights reserved.

      Bibliographical Note

      This Dover edition, first published in 2002, is an unabridged republication of the second edition (1966) of the work originally published by North-Holland Publishing Company, Amsterdam, in 1965.

      Library of Congress Cataloging-in-Publication Data

      Lipkin, Harry J.

      Lie groups for pedestrians / Harry J. Lipkin.

      p. cm.

      This Dover edition . . . is an unabridged republication of the second edition (1966) of the work originally published by North-Holland Publishing Company, Amsterdam, in 1965—T.p. verso.

      Includes bibliographical references and index.

      9780486137889

      1. Particles (Nuclear physics) 2. Lie groups. I. Title.

      QC782 .L56 2002

      539.7’2’0151255—dc21

      2002022117

      Manufactured in the United States by Courier Corporation

      42185604

      www.doverpublications.com

      PREFACE TO FIRST EDITION

      As a graduate student in experimental physics, I found the study of group theory considered to be a useless ‘high-brow’ luxury. Furthermore all attempts to follow a lecture course resulted in a losing battle with characters, cosets, classes, invariant subgroups, normal divisors and assorted lemmas. It was impossible to learn all the definitions of new terms defined in one lecture and remember them until the next lecture. The result was complete chaos.

      It was a great surprise to find later on that (1) techniques based on group theory can be useful; (2) they can be learned and used without memorizing the large number of definitions and lemmas which frighten the uninitiated. Angular momentum is presented in elementary quantum mechanics courses without a detailed analysis of the Lie group of continuous rotations in three dimensions. The student learns about angular momentum multiplets and coupling of angular momenta without realizing that these are the irreducible representations of the rotation group. He also does not realize that the algebraic properties of other Lie groups can be applied to physical problems in the same way as he has used angular momentum algebra, with no need for characters, classes, cosets, etc.

      This book began as a short article with the aim of presenting the ‘group theoretical’ methods used in nuclear structure in a simple way. Another short article was begun to point out that bilinear products of creation and annihilation operators lead to Lie algebras, and to classify the algebras obtained in this way. These were then combined with a discussion of ‘quasispin’ operators acting like fictitious angular momenta which arise in various areas in physics. This material, now in Chapters 4 and 5, was presented in a series of lectures at Argonne National Laboratory in the summer of 1961, discussing simple models of many-particle systems and the application of group theory. The article thus became a set of lecture notes.

      The Argonne lecture notes were still unfinished when unitary symmetry appeared and created a demand from high energy physicists for intelligible lectures on group theory. They wanted to understand and use unitary symmetry without learning about characters and cosets. A series of lectures was given at the University of Illinois and the lecture notes had a different emphasis from the Argonne notes. The audience was interested in unitary symmetry and elementary particles, not in nuclear structure and many-body problems. After several revisions and additions the lecture notes from Illinois and Argonne were combined and extended to form this book.

      The aim of the book is to show how the well-known techniques of angular momentum algebra can be extended to treat other Lie groups, and to give several examples illustrating the application of the method. Because of the present interest in symmetries of elementary particles, this particular application is stressed. Chapter 1 presents the essential features of the method by analogy with angular momentum and points out that bilinear products of creation and annihilation operators lead to Lie algebras. Chapter 2 presents isospin as the first example of the method. Chapter 3 presents the group SU3 and its application to elementary particles. Chapter 4 gives the treatment of the three-dimensional harmonic oscillator using SU3 and discusses its application to nuclear structure. Chapter 5 considers the classification of Lie algebras of bilinear products of creation and annihilation operators, symplectic groups, and the applications to pairing correlations and seniority in many-particle systems. Chapter 6 discusses permutation symmetry and gives a simplified version of Young diagrams as a guide to their use.

      The appendices constitute a large portion of the book and present a detailed study of the application of SU3 algebra to unitary symmetry of elementary particles. Appendix A builds up the structure of the SU3 multiplets by combining fundamental triplets. Appendix B develops the U-spin method for calculating experimental predictions from unitary symmetry. Appendix C presents many detailed examples of experimental predictions from unitary symmetry. Appendix D is a short discussion on the phases which plague all investigators.

      I should like to express my appreciation to many colleagues at the University of Illinois and Argonne National Laboratory who forced me to explain this material to them in a series of constantly interrupted lectures, and to the secretarial staff, particularly M. Runkel andE. Kinstle who performed the incredible job of getting the notes out almost before the lectures were given. It is a pleasure to thank Y. Ne’eman for introducing me to unitary symmetry and C. A. Levinson and S. Meshkov for showing me how the techniques they developed for nuclear structure could be used for elementary particles. I should also like to thank G. Racah for many stimulating discussions and to acknowledge having learned a great deal from a series of his seminar lectures which showed how many useful results could be obtained with the use of simple but powerful algebraic methods. Finally I should like to thank all my colleagues at the Weizmann Institute who helped in the preparation of this book, and particularly L. Mirvish, who typed the manuscript, R. Cohen, who prepared the figures, and H. Harari for criticism of the manuscript.

      PREFACE TO SECOND EDITION

      At the time when the first edition of this book was going to press new symmetries of elementary particles appeared which were based on the Lie groups SU4, SU6 and SU12. These groups were not discussed in any detail in the first edition. However, the general approach is easily extended to treat such groups of rank three and higher. Thus at the time of a second printing of this book, it appears that the addition of material on these groups would be useful to the reader, even though only a short time has elapsed since the first edition went to press.

      This second edition was prepared with the aim of incorporating useful material on groups of higher rank with a minimum of modification of the book in order to allow it to appear as soon as possible. Chapter 7 has been added to present material on the groups SU4, SU6 and SU12. This additional chapter was written in a style as close as possible to that of the preceding work. There has been no revision of the previous text except for the correction of a few minor errors.

      It is dangerous to attempt to keep a book like this exactly up to date with current research developments. However, it seems as if groups of rank higher than two should be of interest to physicists for some time in the future. The purpose of Chapter 7 is not to present an up-to-date picture of the present status of elementary particle symmetries, but rather to show how groups of higher rank can be treated with the same ‘pedestrian’ approach presented earlier for groups of rank two. The material on elementary particles may be only of historical interest by the time the book appears. However, the general algebraic techniques should still be useful. For this reason no attempt has been made to include detailed comparisons of higher symmetry predictions with experimental data. The Sakata model has been retained for the fundamental triplet of SU3, to keep the same style as in the earlier chapters without rewriting them.

      Another development since the appearance of the first edition of this book has been the sudden interest in non-compact groups and their infinite-dimensional representations. The possibility of expanding the very brief treatment of this subject in § 5.6 was considered, but is not feasible at this time.

      At the time of writing of this preface the book ‘High Energy Physics and Elementary Particles’ published by the International Atomic Energy Agency, Vienna 1965, has recently appeared. The reader is referred to this book for more detail comparisons of higher symmetries with experimental data as well as discussions of mathematical theory of non-compact algebras and their applications to elementary particles. However, the characteristic difficulty in the preparation of such material is also evident in this book despite its rapid preparation. New experimental data have already made some of the material obsolete.

      I should like to express my appreciation to H. Harari whose Ph.D. thesis (Hebrew University, Jerusalem–in Hebrew) contains many useful tables which were an aid in the preparation of Chapter 7, and to D. Agassi and C. Robinson for pointing out errors in the first edition.

      Table of Contents

      Title Page

      Copyright Page

      PREFACE TO FIRST EDITION

      PREFACE TO SECOND EDITION

      CHAPTER 1 - INTRODUCTION

      CHAPTER 2 - ISOSPIN. A SIMPLE EXAMPLE

      CHAPTER 3 - THE GROUP SU3 AND ITS APPLICATION TO ELEMENTARY PARTICLES

      CHAPTER 4 - THE THREE-DIMENSIONAL HARMONIC OSCILLATOR

      CHAPTER 5 - ALGEBRAS OF OPERATORS WHICH CHANGE THE NUMBER OF PARTICLES

      CHAPTER 6 - PERMUTATIONS, BOOKKEEPING AND YOUNG DIAGRAMS

      CHAPTER 7 - THE GROUPS SU4, SU6 AND SU12, AN INTRODUCTION TO GROUPS OF HIGHER RANK

      APPENDIX A - CONSTRUCTION OF THE SU3 MULTIPLETS BY COMBINING SAKATON TRIPLETS

      APPENDIX B - CALCULATIONS OF SU3 USING AN SU2 SUBGROUP: U-SPIN

      APPENDIX C - EXPERIMENTAL PREDICTIONS FROM THE OCTET MODEL OF UNITARY SYMMETRY

      APPENDIX D - PHASES, A PERENNIAL HEADACHE

      BIBLIOGRAPHY

      SUBJECT INDEX

      CHAPTER 1

      INTRODUCTION

      Physicists have not yet learned to live with group theory in the same way as they have learned for other mathematical techniques such as differential equations. When an experimentalist or advanced graduate student encounters a simple differential equation in the course of his work, he does not run away and hide, worry about whether the solution to the equation really exists, or indulge in mathematical exercises of a ‘high-brow’ nature. He either solves the equation or goes to the literature and looks up the solution. On the other hand, many sophisticated theorists who are quite at home in the complex plane seem to be afraid of what might be called elementary exercises in group theory. This is all the more mysterious since many of these so-called group theoretical methods are in principle no different and no more complicated than certain mathematical techniques which every physicist learns in a course in elementary quantum mechanics; namely, the algebra of angular momentum operators.

      The reason for this difficulty may be that physicists have still not made the separation analogous to that made for differential equations between those parts of the subject which belong to the physicist and those which belong to the mathematician. The standard treatment of group theory for physicists begins with complicated definitions, lemmas, and existence proofs which are certainly necessary for a proper understanding of group theory. However, it is possible for physicists to understand and to use many techniques which have a group theoretical basis without necessarily understanding all of group theory, in the same way as he now uses angular momentum algebra without delving deeply into the mysteries of the three-dimensional rotation group.

      The purpose of this treatment is to show how techniques analogous to angular momentum algebra can be extended and applied to other group theoretical problems without requiring a detailed understanding of group theory.

      1.1. REVIEW OF ANGULAR MOMENTUM ALGEBRA

      Consider three angular momentum operators Jx, Jy and Jz which satisfy the well-known commutation rules

      (1.1)

      From these commutation rules it follows that there exists an operator

      which has the property of commuting with all the angular momentum operators:

      (1.2)

      Since J² commutes with all the operators, it commutes with any one of them, and one usually chooses the operator Jz. One can then in any problem find a complete set of states which are simultaneous eigenfunctions of J² and Jz with eigenvalues usually designated by J and M. We use the conventional designation for these states

      (1.3)

      The remaining two operators Jx and Jy do not commute with Jz, but the following simple linear combinations

      (1.4)

      satisfy particularly simple commutation rules. Since J² commutes with all the operators, we have

      (1.5)

      The commutators with Jz are also quite simple,

      (1.6)

      The commutator of each of these operators with Jz is just the same operator again, multiplied by a constant. It then follows that if either of these operators operates on a state which is a simultaneous eigenfunction of J² and Jz with eigenvalues J and M, the result is another state which is an eigenfunction of J² with the same eigenvalue J and which is also an eigenfunction of Jz, but with the eigenvalue M±1.

      (1.7)

      The value of the coefficient appearing on the right-hand side is easily obtained by a little algebra. This result and the trivial

      (1.8)

      define matrix elements for all of the angular momentum operators for all of the complete set of states.

      Beginning with any particular state, |J, M〉, a set of states can be generated by operating successively with the operators (Jx+iJy) and (Jx–iJy). This process cannot be continued indefinitely because M can never be greater than J. Thus one finds restrictions on the possible eigenvalues of J and M, and obtains the well-known result that these may be either integral or half-integral and that for any eigenvalue J there corresponds a set or multiplet of 2J + 1 states all having the same eigenvalue of J and having values of M equal to–J,–J+1, ..., +J. The full set of states in a multiplet can be generated from any one of the states by successive operation with the operators (Jx±iJy).

      Some of these features can be demonstrated simply in diagrams of the type shown in Fig. 1.1. These diagrams are one-dimensional plots of the eigenvalues of Jz. Fig. 1.1a represents the operators (Jx+iJy) and (Jx–iJy) as vectors which change the eigenvalue of Jz by ± 1, respectively. Figure 1.1b illustrates the structure of a typical multiplet, in this case one with J , in which a point is plotted for each value of Jz where a state exists in the multiplet. The operation of any of the operators in Fig. 1.1a on the states in the multiplet of Fig. 1.1 b is represented graphically by taking the appropriate vector of Fig. 1.1a , placing it on Fig. 1.1b and noting which states are connected by this vector.

      Fig. 1.1

      There are also well-known rules for combining multiplets. A system may consist of several parts, each of which is characterized by a multiplet having a particular value of J.) Given the J-values for the multiplets describing parts of the system, there are simple rules for deciding which possible values of J occur for the total system, and there are algebraic techniques involving vector coupling coefficients for expressing the wave functions of the combined system which belong to a given multiplet. There is also one very simple rule which results from the different character of the multiplets having half-integral and integral values of J. If two multiplets having integral values of J are combined, the multiplet describing the overall system must also have integral values of J. If two multiplets having half-integral values of J are

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