Supersymmetric Methods in Quantum and Statistical Physics

The idea of supersymmetry was originally introduced in relativistic quantum field theories as a generalization of Poincare symmetry. The purpose of this book is to give an introduction to supersymmet­ ric quantum mechanics and review some of the recent developments of vari­ ous supersymmetric methods in quantum and statistical physics. This book gives an introduction to supersymmetric quantum mechanics and a comprehensive review of its applications in quantum and statistical physics. The author discusses the classical and quantum versions of Witten's model and exact spectral properties of the model for the so-called shape invariant potentials. The quasi-classical quantization rules are derived and other topics include the supersymmetric structure of a classical stochastic dynamical system obeying the Langevin or the Fokker-Planck equation, Pauli's Hamiltonian and its application to the paragmagnetism of a non-interacting electron gas in two and three dimensions, and supersymmetry of Dirac's Hamiltonian. The book addresses graduate students as well as scientists. The idea of supersymmetry was originally introduced in relativistic quantum field theories as a generalization of Poincare symmetry. In 1976 Nicolai sug­ gested an analogous generalization for non-relativistic quantum mechanics. With the one-dimensional model introduced by Witten in 1981, supersym­ metry became a major tool in quantum mechanics and mathematical, sta­ tistical, and condensed-IIll;l. tter physics. Supersymmetry is also a successful concept in nuclear and atomic physics. An underlying supersymmetry of a given quantum-mechanical system can be utilized to analyze the properties of the system in an elegant and effective way. It is even possible to obtain exact results thanks to supersymmetry. The purpose of this book is to give an introduction to supersymmet­ ric quantum mechanics and review some of the recent developments of vari­ ous supersymmetric methods in quantum and statistical physics. Thereby we will touch upon some topics related to mathematical and condensed-matter physics. A discussion of supersymmetry in atomic and nuclear physics is omit­ ted. However, the reader will find some references in Chap. 9. Similarly, super­ symmetric field theories and supergravity are not considered in this book. In fact, there exist already many excellent textbooks and monographs on these topics. A list may be found in Chap. 9. Yet, it is hoped that this book may be useful in preparing a footing for a study of supersymmetric theories in atomic, nuclear, and particle physics. The plan of the book is as follows.

Giving an introduction to supersymmetric quantum mechanics and a review of its applications in quantum and statistical physics, this work examines the classical version and the quantum version of Witten's model. Exact spectral properties for the so-called shape invariant potentials are discussed. The quasi-classical quantization rukes are derived. The topics covered also include the supersymmetric structure of a classical stochastic dynamical system obeying the Langevin of the Fokker-Planck equation, Pauli's Hamiltonian and its application to the paramagnetism of a non-interacting electron gas in two and three dimensions, and supersymmetry of Dirac's Hamiltonian.