Supersymmetry In Quantum and Classical Mechanics / Edition 1

by Bijan Kumar Bagchi

Overview

Following Witten's remarkable discovery of the quantum mechanical scheme in which all the salient features of supersymmetry are embedded, SCQM (supersymmetric classical and quantum mechanics) has become a separate area of research . In recent years, progress in this field has been dramatic and the literature continues to grow. Until now, no book has offered an overview of the subject with enough detail to allow readers to become rapidly familiar with its key ideas and methods.

Supersymmetry in Classical and Quantum Mechanics offers that overview and summarizes the major developments of the last 15 years. It provides both an up-to-date review of the literature and a detailed exposition of the underlying SCQM principles. For those just beginning in the field, the author presents step-by-step details of most of the computations. For more experienced readers, the treatment includes systematic analyses of more advanced topics, such as quasi- and conditional solvability and the role of supersymmetry in nonlinear systems.

Product Details

ISBN-13:	9780367398125
Publisher:	Taylor & Francis
Publication date:	09/05/2019
Series:	Monographs and Surveys in Pure and Applied Mathematics
Pages:	240
Product dimensions:	6.12(w) x 9.19(h) x (d)

About the Author

Bijan Kumar Bagchi is a Reader in the Department of Applied Mathematics at the University of Calcutta. His research interests include quantum mechanics and nonlinearity, and he has authored or co-authored more than 80 articles in journals of international repute.

Table of Contents

Preface xi

Acknowledgments xiii

1 General Remarks on Supersymmetry 1

1.1 Background 1

1.2 References 5

2 Basic Principles of SUSYQM 9

2.1 SUSY and the Oscillator Problem 9

2.2 Superpotential and Setting Up a Supersymmetric Hamiltonian 14

2.3 Physical Interpretation of H_s 18

2.4 Properties of the Partner Hamiltonians 20

2.5 Applications 22

2.6 Superspace Formalism 31

2.7 Other Schemes of SUSY 36

2.8 References 38

3 Supersymmetric Classical Mechanics 45

3.1 Classical Poisson Bracket, its Generalizations 45

3.2 Some Algebraic Properties of the Generalized Poisson Bracket 49

3.3 A Classical Supersymmetric Model 52

3.4 References 54

4 SUSY Breaking, Witten Index, and Index Condition 57

4.1 SUSY Breaking 57

4.2 Witten Index 59

4.3 Finite Temperature SUSY 61

4.4 Regulated Witten Index 63

4.5 Index Condition 68

4.6 q-deformation and Index Condition 69

4.7 Parabosons 72

4.8 Deformed Parabose States and Index Condition 74

4.9 Witten's Index and Higher-Derivative SUSY 77

4.10 Explicit SUSY Breaking and Singular Super potentials 82

4.11 References 85

5 Factorization Method, Shape Invariance 91

5.1 Preliminary Remarks 91

5.2 Factorization Method of Infeld and Hull 92

5.3 Shape Invariance Condition 96

5.4 Self-similar Potentials 107

5.5 A Note On the Generalized Quantum Condition 108

5.6 Nonuniqueness of the Factorizability 110

5.7 Phase Equivalent Potentials 112

5.8 Generation of Exactly Solvable Potentials in SUSYQM 117

5.9 Conditionally Solvable Potentials and SUSY 122

5.10 References 128

6 Radial Problems and Spin-orbit Coupling 135

6.1 SUSY and the Radial Problems 135

6.2 Radial Problems Using Ladder Operator Techniques in SUSYQM 140

6.3 Isotropic Oscillator and Spin-orbit Coupling 145

6.4 SUSY in D Dimensions 149

6.5 References 151

7 Super symmetry in Nonlinear Systems 153

7.1 The KdV Equation 153

7.2 Conservation Laws in Nonlinear Systems 155

7.3 Lax Equations 161

7.4 SUSY and Conservation Laws in the KdV-MKdV Systems 163

7.5 Darboux's Method 164

7.6 SUSY and Conservation Laws in the KdV-SG Systems 167

7.7 Supersymmetric KdV 169

7.8 Conclusion 174

7.9 References 175

8 Parasupersymmetry 179

8.1 Introduction 179

8.2 Models of PSUSYQM 181

8.3 PSUSY of Arbitrary Order p 188

8.4 Truncated Oscillator and PSUSYQM 191

8.5 Multidimensional Parasuperalgebras 198

8.6 References 202

Appendix A 205

Appendix B 215

Index 221