Supersymmetry in quantum and classical mechanics

226 Pages·2001·1.631 MB·English

by  Bijan Kumar Bagchi

CHAPMAN & HALL/CRC
Monographs and Surveys in
Pure and Applied Mathematics 116
SUPERSYMMETRY IN
QUANTUM AND
CLASSICAL
MECHANICS
CHAPMAN & HALL/CRC
Monographs and Surveys in Pure and Applied Mathematics
Main Editors
H. Brezis, Université 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 Universität 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. Kirchgässner, Universität 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
CHAPMAN & HALL/CRC
Monographs and Surveys in
Pure and Applied Mathematics 116
SUPERSYMMETRY IN
QUANTUM AND
CLASSICAL
MECHANICS
BIJAN KUMAR BAGCHI
CHAPMAN & HALL/CRC
Boca Raton London New York Washington, D.C.
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
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© 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
For Basabi and Minakshi
Contents
Preface
Acknowledgments
1GeneralRemarksonSupersymmetry
1.1Background
1.2References
2BasicPrinciplesofSUSYQM
2.1SUSYandtheOscillatorProblem
2.2SuperpotentialandSettingUpaSupersymmetricHamil-
tonian
2.3PhysicalInterpretationof H
s
2.4PropertiesofthePartnerHamiltonians
2.5Applications
2.6SuperspaceFormalism
2.7OtherSchemesofSUSY
2.8References
3SupersymmetricClassicalMechanics
3.1ClassicalPoissonBracket,itsGeneralizations
3.2SomeAlgebraicPropertiesoftheGeneralizedPoisson
Bracket
3.3AClassicalSupersymmetricModel
3.4References
4SUSYBreaking,WittenIndex,andIndexCondition
4.1SUSYBreaking
4.2WittenIndex
© 2001 by Chapman & Hall/CRC
4.3FiniteTemperatureSUSY
4.4RegulatedWittenIndex
4.5IndexCondition
4.6 q-deformationandIndexCondition
4.7Parabosons
4.8DeformedParaboseStatesandIndexCondition
4.9Witten’sIndexandHigher-DerivativeSUSY
4.10ExplicitSUSYBreakingandSingularSuperpotentials
4.11References
5FactorizationMethod,ShapeInvariance
5.1Preliminary Remarks
5.2FactorizationMethodofInfeldandHull
5.3ShapeInvarianceCondition
5.4Self-similarPotentials
5.5ANoteOntheGeneralizedQuantumCondition
5.6NonuniquenessoftheFactorizability
5.7PhaseEquivalentPotentials
5.8GenerationofExactly SolvablePotentialsinSUSYQM
5.9Conditionally SolvablePotentialsandSUSY
5.10References
6RadialProblemsandSpin-orbitCoupling
6.1SUSYandtheRadialProblems
6.2RadialProblemsUsingLadderOperatorTechniques
inSUSYQM
6.3IsotropicOscillatorandSpin-orbitCoupling
6.4SUSYin D Dimensions
6.5References
7SupersymmetryinNonlinearSystems
7.1TheKdVEquation
7.2ConservationLawsinNonlinearSystems
7.3LaxEquations
7.4SUSYandConservationLawsintheKdV-MKdV
Systems
7.5Darboux’sMethod
7.6SUSYandConservationLawsintheKdV-SGSystems
7.7SupersymmetricKdV
© 2001 by Chapman & Hall/CRC
7.8Conclusion
7.9References
8Parasupersymmetry
8.1Introduction
8.2ModelsofPSUSYQM
8.3PSUSYofArbitrary Order p
8.4TruncatedOscillatorandPSUSYQM
8.5MultidimensionalParasuperalgebras
8.6References
Appendix A
Appendix
B
© 2001 by Chapman & Hall/CRC
Preface
Thismonographsummarizesthemajordevelopmentsthathavetaken
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 field. 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,
MithilRanjanGupta, BirendranathMandal, RabindranathSen, and
Nandadulal Sengupta for their interest and encouragement. It also
givesmegreatpleasuretothankProf. RajkumarRoychoudhuryand
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
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