Supersymmetry in Quantum and Classical Mechanics.pdf

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

--dc21

00-059602

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

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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

′25

Trademark Notice:

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

4.3FiniteTemperatureSUSY

4.4RegulatedWittenIndex

4.5IndexCondition

4.6 q -deformationandIndexCondition

4.7Parabosons

4.8DeformedParaboseStatesandIndexCondition

4.9Witten’sIndexandHigher-DerivativeSUSY

4.10ExplicitSUSYBreakingandSingularSuperpotentials

4.11References

5FactorizationMethod,ShapeInvariance

5.1PreliminaryRemarks

5.2FactorizationMethodofInfeldandHull

5.3ShapeInvarianceCondition

5.4Self-similarPotentials

5.5ANoteOntheGeneralizedQuantumCondition

5.6NonuniquenessoftheFactorizability

5.7PhaseEquivalentPotentials

5.8GenerationofExactlySolvablePotentialsinSUSYQM

5.9ConditionallySolvablePotentialsandSUSY

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

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