Supersymmetry in quantum and classical mechanics

Bijan Kumar Bagchi1584881976, 9781584881971

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, and in recent years has made dramatic progress. This Monograph summarizes the major developments of the last decade, provides an up-to-date review of the literature, and offers a detailed exposition of the underlying SCQM principles.

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Table of contents :
SUPERSYMMETRY IN QUANTUM AND CLASSICAL MECHANICS……Page 1
SUPERSYMMETRY IN QUANTUM AND CLASSICAL MECHANICS……Page 3
Table of Contents……Page 6
Preface……Page 9
Acknowledgments……Page 11
1.1 Background……Page 12
Table of Contents……Page 0
1.2 References……Page 16
2.1 SUSY and the Oscillator Problem……Page 19
2.2 Superpotential and Setting Up a Supersymmetric Hamiltonian……Page 24
2.3 Physical Interpretation of Hs……Page 28
2.4 Properties of the Partner Hamiltonians……Page 30
2.5 Applications……Page 32
2.6 Superspace Formalism……Page 41
2.7 Other Schemes of SUSY……Page 46
2.8 References……Page 48
3.1 Classical Poisson Bracket, its Generalizations……Page 55
3.2 Some Algebraic Properties of the Generalized Poisson Bracket……Page 59
3.3 A Classical Supersymmetric Model……Page 62
3.4 References……Page 64
4.1 SUSY Breaking……Page 66
4.2 Witten Index……Page 68
4.3 Finite Temperature SUSY……Page 70
4.4 Regulated Witten Index……Page 72
4.5 Index Condition……Page 77
4.6 q-deformation and Index Condition……Page 78
4.7 Parabosons……Page 81
4.8 Deformed Parabose States and Index Condition……Page 83
4.9 Witten’s Index and Higher-Derivative SUSY……Page 86
4.10 Explicit SUSY Breaking and Singular Superpotentials……Page 91
4.11 References……Page 94
5.1 Preliminary Remarks……Page 99
5.2 Factorization Method of Infeld and Hull……Page 100
5.3 Shape Invariance Condition……Page 104
5.4 Self-similar Potentials……Page 115
5.5 A Note On the Generalized Quantum Condition……Page 116
5.6 Nonuniqueness of the Factorizability……Page 118
5.7 Phase Equivalent Potentials……Page 120
5.8 Generation of Exactly Solvable Potentials in SUSYQM……Page 125
5.9 Conditionally Solvable Potentials and SUSY……Page 130
5.10 References……Page 136
6.1 SUSY and the Radial Problems……Page 142
6.2 Radial Problems Using Ladder Operator Techniques in SUSYQM……Page 147
6.3 Isotropic Oscillator and Spin-orbit Coupling……Page 152
6.4 SUSY in D Dimensions……Page 156
6.5 References……Page 158
7.1 The KdV Equation……Page 160
7.2 Conservation Laws in Nonlinear Systems……Page 162
7.3 Lax Equations……Page 168
7.4 SUSY and Conservation Laws in the KdV-MKdV Systems……Page 170
7.5 Darboux’s Method……Page 171
7.6 SUSY and Conservation Laws in the KdVSG Systems……Page 174
7.7 Supersymmetric KdV……Page 176
7.8 Conclusion……Page 181
7.9 References……Page 182
8.1 Introduction……Page 186
8.2 Models of PSUSYQM……Page 188
8.3 PSUSY of Arbitrary Order p……Page 195
8.4 Truncated Oscillator and PSUSYQM……Page 198
8.5 Multidimensional Parasuperalgebras……Page 205
8.6 References……Page 209
The D-dimensional Schroedinger Equation in
a Spherically Symmetric Potential V (r)……Page 212
Derivation of the Results (A19) and (A21)……Page 221