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CHAPMAN & HALL/CRC
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
H. Amann, University of Zürich
R. Aris, University of Minnesota
G.I. Barenblatt, University of Cambridge
H. Begehr, Freie Universit¤t Berlin
P. Bullen, University of British Columbia
R.J. Elliott, University of Alberta
R.P. Gilbert, University of Delaware
R. Glowinski, University of Houston
D. Jerison, Massachusetts Institute of Technology
K. Kirchg¤ssner, Universit¤t Stuttgart
B. Lawson, State University of New York
B. Moodie, University of Alberta
S. Mori, Kyoto University
L.E. Payne, Cornell University
D.B. Pearson, University of Hull
I. Raeburn, University of Newcastle
G.F. Roach, University of Strathclyde
I. Stakgold, University of Delaware
W.A. Strauss, Brown University
J. van der Hoek, University of Adelaide




© 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
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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-

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