Supersymmetry in Classical and Quantum Mechanics

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.

GENERAL REMARKS ON SUPERSYMMETRY
Background
BASIC PRINCIPLES OF SUPERSYMMETRIC QUANTUM MECHANICS
SUSY and the Oscillator Problem
Superpotential and Setting Up a Supersymmetric Hamiltonian
Physical Interpretation of Hs
Properties of the Partner Hamiltonians
Applications
Superspace Formalism
Other Schemes of SUSY
SUPERSYMMETRIC CLASSICAL MECHANICS
Classical Poisson Bracket, Its Generalizations
Some Algebraic Properties of the Generalized Poisson Bracket
A Classical Supersymmetric Model
SUSY Breaking, Witten Index and Index Condition
SUSY Breaking
Witten Index
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
FACTORIZATION METHOD, SHAPE INVARIANCE AND GENERATION OF SOLVABLE PROBLEMS
Preliminary Remarks
Factorization Method of Infeld and Hull
Shape Invariance Condition
Self-Similar Potentials
A Note on the Generalized Quantum Condition
Non-Uniqueness of the Factorizability
Phase Equivalent Potentials
Generation of Exactly Solvable Potentials in SUSYQM
Conditionally Solvable Potentials and SUSY
RADIAL PROBLEMS AND SPIN-ORBIT COUPLING
SUSY and the Radial Problems
Radial Problems Using Ladder Operator Techniques in SUSYQM
Isotropic Oscillator and Spin-Orbit Coupling
SUSY in D Dimensions
SUPERSYMMETRY IN NONLINEAR SYSTEMS
The KdV Equation
Conservation Laws in Nonlinear Systems
Lax Equations
SUSY and Conservation Laws in the KdV - MKdV Systems
Darboux's Method
SUSY and Conservation Laws in the KdV-SG Systems
Supersymmetric KdV
Conclusion
Parasupersymmetry
Introduction
Models of PSUSYQM
PSUSY of Arbitrary Order p
Truncated Oscillator and PSUSYQM
Multidimensional Parasuperalgebras
APPENDIX A