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CHAPMAN & HALL/CRC Monographs and Surveys in 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 116 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) 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, microfilming, 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 Specific 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 identification 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 Printed on acid-free paper © 2001 by Chapman & Hall/CRC For Basabi and Minakshi © 2001 by Chapman & Hall/CRC Contents Preface Acknowledgments General Remarks on Supersymmetry 1.1 Background 1.2 References Basic Principles of SUSYQM 2.1 SUSY and the Oscillator Problem 2.2 Superpotential and Setting Up a Supersymmetric Hamiltonian 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 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 SUSY Breaking, Witten Index, and Index Condition 4.1 SUSY Breaking 4.2 Witten Index © 2001 by Chapman & Hall/CRC 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 Finite Temperature SUSY Regulated Witten Index Index Condition q-deformation and Index Condition Parabosons Deformed Parabose States and Index Condition Witten’s Index and Higher-Derivative SUSY Explicit SUSY Breaking and Singular Superpotentials References 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 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 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 7.9 Conclusion References 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 supersymmetry are present, supersymmetric quantum mechanics has become a discipline of research in its own right Indeed a glance at the literature on this subject will reveal that the progress has been dramatic The purpose of this book is to set out the basic methods of supersymmetric 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 mathematical 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 Bruxeles, 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 improvement 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 Interaction of Mechanics and Mathematics (ISIMM) ISIMM was established in 1975 for the genuine interaction between mechanics and mathematics New phenomena in mechanics require the development of fundamentally new mathematical ideas leading to mutual enrichment of the two disciplines The society fosters the interests of its members, elected from countries worldwide, by a series of biannual international meetings (STAMM) and by specialist symposia held frequently in collaboration with other bodies © 2001 by Chapman & Hall/CRC CHAPTER General Remarks on Supersymmetry 1.1 Background It is about three quarters of a century now since modern quantum mechanics came into existence under the leadership of such names as Born, de Broglie, Dirac, Heisenberg, Jordan, Pauli, and Schroedinger At its very roots the conceptual foundations of quantum theory involve notions of discreteness and uncertainty Schroedinger and Heisenberg, respectively, gave two distinct but equivalent formulations: the configuration space approach which deals with wave functions and the phase space approach which focuses on the role of observables Dirac noticed a connection between commutators and classical Poisson brackets and it was chiefly he who gave the commutator form of the Poisson bracket in quantum mechanics on the basis of Bohr’s correspondence principle Quantum mechanics continues to attract the mathematicians and physicists alike who are asked to come to terms with new ideas and concepts which the tweory exposes from time to time [1-2] Supersymmetric quantum mechanics (SUSYQM) is one such area which has received much attention of late This is evidenced by the frequent appearances of research papers emphasizing different aspects of SUSYQM [3-9] Indeed the boson-fermion manifestation in soluble models has considerably enriched our understanding of degeneracies © 2001 by Chapman & Hall/CRC Appendix A The D-dimensional Schroedinger Equation in a Spherically Symmetric Potential V (r) In Cartesian coordinates, the Schroedinger equation under the influence of a potential V (r) reads − where h ¯2 → → → ∇ ψ( r ) + V (r)ψ( r ) = Eψ( r ) 2m D → r (A1) → = (x1 , x2 , xD ), r = | r | (A2) ∂ ∂ = ∂xi ∂xi Our task is to transform (A1) to D-dimensional polar coordinates The latter are related to the Cartesian coordinates by ∇2 D x1 = r cos θ1 sin θ2 sin θ3 sin θD−1 x2 = r sin θ1 sin θ2 sin θ3 sin θD−1 x3 = r cos θ2 sin θ3 sin θ4 sin θD−1 x4 = r cos θ3 sin θ4 sin θ5 sin θD−1 xj = r cos θj−1 sin θj sin θj+1 sin θD−1 = r cos θD−1 sin θD−1 xD−1 xD = r cos θD−1 © 2001 by Chapman & Hall/CRC (A3) where D 0 = < ≤ ≤ 3, 4, r