Monographs and Surveys in

116

Pure and Applied Mathematics

SUPERSYMMETRY IN

QUANTUM AND

CLASSICAL

MECHANICS

© 2001 by Chapman & Hall/CRC

CHAPMAN & HALL/CRC

Monographs and Surveys in Pure and Applied Mathematics

Main Editors

H. Brezis, Universit© de Paris

R.G. Douglas, Texas A&M University

A. Jeffrey, University of Newcastle upon Tyne (Founding Editor)

Editorial Board

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R. Aris, University of Minnesota

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© 2001 by Chapman & Hall/CRC

CHAPMAN & HALL/CRC

Monographs and Surveys in

116

Pure and Applied Mathematics

SUPERSYMMETRY IN

QUANTUM AND

CLASSICAL

MECHANICS

BIJAN KUMAR BAGCHI

CHAPMAN & HALL/CRC

Boca Raton London New York Washington, D.C.

© 2001 by Chapman & Hall/CRC

Library of Congress Cataloging-in-Publication Data

Bagchi, B. (Bijan Kumar)

Supersymmetry in quantum and classical mechanics / B. Bagchi.

p. cm.-- (Chapman & Hall/CRC monographs and surveys in pure and applied mathematics)

Includes bibliographical references and index.

ISBN 1-58488-197-6 (alk. paper)

1. Supersymmetry. I. Title. II. Series.

QC174.17.S9 2000

539.7′25 --dc21 00-059602

This book contains information obtained from authentic and highly regarded sources. Reprinted material

is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable

efforts have been made to publish reliable data and information, but the author and the publisher cannot

assume responsibility for the validity of all materials or for the consequences of their use.

Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic

or mechanical, including photocopying, micro¬lming, and recording, or by any information storage or

retrieval system, without prior permission in writing from the publisher.

The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for

creating new works, or for resale. Speci¬c permission must be obtained in writing from CRC Press LLC

for such copying.

Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are

used only for identi¬cation and explanation, without intent to infringe.

© 2001 by Chapman & Hall/CRC

No claim to original U.S. Government works

International Standard Book Number 1-58488-197-6

Library of Congress Card Number 00-059602

Printed in the United States of America 1 2 3 4 5 6 7 8 9 0

Printed on acid-free paper

© 2001 by Chapman & Hall/CRC

For Basabi and Minakshi

© 2001 by Chapman & Hall/CRC

Contents

Preface

Acknowledgments

1 General Remarks on Supersymmetry

1.1 Background

1.2 References

2 Basic Principles of SUSYQM

2.1 SUSY and the Oscillator Problem

2.2 Superpotential and Setting Up a Supersymmetric Hamil-

tonian

2.3 Physical Interpretation of Hs

2.4 Properties of the Partner Hamiltonians

2.5 Applications

2.6 Superspace Formalism

2.7 Other Schemes of SUSY

2.8 References

3 Supersymmetric Classical Mechanics

3.1 Classical Poisson Bracket, its Generalizations

3.2 Some Algebraic Properties of the Generalized Poisson

Bracket

3.3 A Classical Supersymmetric Model

3.4 References

4 SUSY Breaking, Witten Index, and Index Condition

4.1 SUSY Breaking

4.2 Witten Index

© 2001 by Chapman & Hall/CRC

4.3 Finite Temperature SUSY

4.4 Regulated Witten Index

4.5 Index Condition

4.6 q-deformation and Index Condition

4.7 Parabosons

4.8 Deformed Parabose States and Index Condition

4.9 Witten™s Index and Higher-Derivative SUSY

4.10 Explicit SUSY Breaking and Singular Superpotentials

4.11 References

5 Factorization Method, Shape Invariance

5.1 Preliminary Remarks

5.2 Factorization Method of Infeld and Hull

5.3 Shape Invariance Condition

5.4 Self-similar Potentials

5.5 A Note On the Generalized Quantum Condition

5.6 Nonuniqueness of the Factorizability

5.7 Phase Equivalent Potentials

5.8 Generation of Exactly Solvable Potentials in SUSYQM

5.9 Conditionally Solvable Potentials and SUSY

5.10 References

6 Radial Problems and Spin-orbit Coupling

6.1 SUSY and the Radial Problems

6.2 Radial Problems Using Ladder Operator Techniques

in SUSYQM

6.3 Isotropic Oscillator and Spin-orbit Coupling

6.4 SUSY in D Dimensions

6.5 References

7 Supersymmetry in Nonlinear Systems

7.1 The KdV Equation

7.2 Conservation Laws in Nonlinear Systems

7.3 Lax Equations

7.4 SUSY and Conservation Laws in the KdV-MKdV

Systems

7.5 Darboux™s Method

7.6 SUSY and Conservation Laws in the KdV-SG Systems

7.7 Supersymmetric KdV

© 2001 by Chapman & Hall/CRC

7.8 Conclusion

7.9 References

8 Parasupersymmetry

8.1 Introduction

8.2 Models of PSUSYQM

8.3 PSUSY of Arbitrary Order p

8.4 Truncated Oscillator and PSUSYQM

8.5 Multidimensional Parasuperalgebras

8.6 References

Appendix A

Appendix B

© 2001 by Chapman & Hall/CRC

Preface

This monograph summarizes the major developments that have taken

place in supersymmetric quantum and classical mechanics over the

past 15 years or so. Following Witten™s construction of a quantum

mechanical scheme in which all the key ingredients of supersymme-

try are present, supersymmetric quantum mechanics has become a

discipline of research in its own right. Indeed a glance at the litera-

ture on this subject will reveal that the progress has been dramatic.

The purpose of this book is to set out the basic methods of super-

symmetric quantum mechanics in a manner that will give the reader

a reasonable understanding of the subject and its applications. We

have also tried to give an up-to-date account of the latest trends in

this ¬eld. The book is written for students majoring in mathemati-

cal science and practitioners of applied mathematics and theoretical

physics.

I would like to take this opportunity to thank my colleagues

in the Department of Applied Mathematics, University of Calcutta

and members of the faculty of PNTPM, Universite Libre de Brux-

eles, especially Prof. Christiane Quesne, for their kind cooperation.

Among others I am particularly grateful to Profs. Jules Beckers,

Debajyoti Bhaumik, Subhas Chandra Bose, Jayprokas Chakrabarti,

Mithil Ranjan Gupta, Birendranath Mandal, Rabindranath Sen, and

Nandadulal Sengupta for their interest and encouragement. It also

gives me great pleasure to thank Prof. Rajkumar Roychoudhury and

Drs. Nathalie Debergh, Anuradha Lahiri, Samir Kumar Paul, and

Prodyot Kumar Roy for fruitful collaborations. I am indebted to my

students Ashish Ganguly and Sumita Mallik for diligently reading

the manuscript and pointing out corrections. I also appreciate the

help of Miss Tanima Bagchi, Mr. Dibyendu Bose, and Dr. Mridula

© 2001 by Chapman & Hall/CRC

Kanoria in preparing the manuscript with utmost care. Finally, I

must thank the editors at Chapman & Hall/CRC for their assistance

during the preparation of the manuscript. Any suggestions for im-

provement of this book would be greatly appreciated.

I dedicate this book to the memory of my parents.

Bijan Kumar Bagchi

© 2001 by Chapman & Hall/CRC

Acknowledgments

This title was initiated by the International Society for the Inter-

action of Mechanics and Mathematics (ISIMM). ISIMM was estab-