Table Of ContentTexts and Monographs in Physics
Series Editors: R. Balian W. Beiglbock .. H. Grosse E. H. Lieb
N. Reshetikhin H. Spohn W. Thirring
Springer
Berlin
Heidelberg
New York
Barcelona
Budapest
Hong Kong
London
Milan
Paris
Santa Clara
Singapore
Tokyo
Texts and Monographs in Physics
Series Editors: R. Balian W Beiglbock H. Grosse E. H. Lieb
N. Reshetikhin H. Spohn W Thirring
From Microphysics to Macrophysics Scattering Theory of Waves
I + II Methods and Applications of and Particles 2nd edition
Statistical Physics By R. Balian By R. G. Newton
Variational Methods in Mathematical Quantum Entropy and Its Use
Physics A Unified Approach By M. Ohya and D. Petz
By P. Blanchard and E. Bruning
Generalized Coherent States
Quantum Mechanics: and Their Applications
Foundations and Applications By A. Perelomov
3rd enlarged edition By A. Bohm
Essential Relativity Special, General,
The Early Universe and Cosmological Revised 2nd edition
Facts and Fiction 3rd corrected By W. Rindler
and enlarged edition By G. Borner
Path Integral Approach
Operates Algebras and Quantum to Quantum Physics An Introduction
Statistical Mechanics I + II 2nd edition 2nd printing By G. Roepstorff
By O. Bratteli and D. W. Robison
Advanced Quantum Theory
Geometry of the Standard Model and Its Applications Through Feynman
of Elementary Particles Diagrams 2nd edition
By A. Derdzinski By M. D. Scadron
Random Walks, Critical Phenomena, Finite Quantum Electrodynamics
and Triviality in Quantum Field The Causal Approach 2nd edition
Theory By R. Fernandez, J. Frohlich ByG. Scharf
and A. D. Sokal
From Electrostatics to Optics
Quantum Relativity A Concise Electrodynamics Course
A Synthesis of the Ideas of Einstein ByG. Scharf
and Heisenberg
Large Scale Dynamics of Interacting
By D. R. Finkelstein
Particles By H. Spohn
Quantum Mechanics I + II
General Relativity and Relativistic
By A. Galindo and P. Pascual
Astrophysics By N. Straumann
The Elements of Mechanics
The Mechanics and Thermodynamics
By G. Gallavotti
of Continuous Media By M. Silhary
Local Quantum Physics
The Dirac Equation By B. Thaller
Fields, Particles, Algebras
2nd revised and enlarged edition The Theory of Quark and Gluon
ByR. Haag Interactions 2nd completely revised
and enlarged edition By F. J. Yndurmn
Elementary Particle Physics
Concepts and Phenomena Relativistic Quantum Mechanics and
By O. Nachtmann Introduction to Field Theory
By F. J. Yndurmn
Inverse SchrOdinger Scattering
in Three Dimensions Supersymmetric Methods in Quantum
By R. G. Newton and Statistical Physics By G. Junker
Georg Junker
Supersymmetric
Methods
in Quantum
and Statistical
Physics
With 36 Figures and 13 Tables
Springer
Priv.-Doz. Dr. Georg Junker
Institut fur Theoretische Physik
Universitiit Erlangen-Nurnberg
Staudtstrasse 7
0-91058 Erlangen, Germany
Editors
Roger Balian Nicolai Reshetikhin
CEA Oepartment of Mathematics
Service de Physique Theorique de Saclay University of California
F-91191 Gif-sur-Yvette, France Berkeley, CA 94720-3840, USA
Wolf Beiglbock Herbert Spohn
Institut fUr Angewandte Mathematik Theoretische Physik
Universitiit Heidelberg Ludwig-Maximilians-Universitiit Munchen
1m Neuenheimer Feld 294 Theresienstrasse 37
0-69120 Heidelberg, Germany 0-80333 Munchen, Germany
Harald Grosse Walter Thirring
Institut fUr Theoretische Physik Institut fUr Theoretische Physik
Universitiit Wien Universitiit Wien
Boltzmanngasse 5 Boltzmanngasse 5
A-I090 Wien, Austria A-I090 Wien, Austria
Elliott H. Lieb
Jadwin Hall
Princeton University, P. O. Box 708
Princeton, NJ 08544-0708, USA
Library of Congress Cataloging-in-Publication Data applied for.
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Junker, Georg:
Supersymmetric methods in quantum and statistical physics / Georg Junker. - Berlin; Heidel
berg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara;
Singapore; Tokyo: Springer, 1996
(Texts and monographs in physics)
Zugl.: Erlangen, Niirnberg, Univ., Habil.-Schr., 1995
ISBN-13:978-3-642-64742-0 e-ISBN-13:978-3-642-61 194-0
DOl: 10.1007/978-3-642-61194-0
ISSN 0172-5998
ISBN-13:978-3-540-61591-0 Springer-Verlag Berlin Heidelberg New York
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication
of this publication or parts thereof is permitted only'under the provisions of the German Copyright
Law of September 9, 1965, in its current version, and permission for use must always be obtained
from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1996
Softcover reprint of the hardcover 1st edition 1996
The.use of general descriptive names, registered names, trademarks, etc. in this publication does not
imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
Typesetting: Camera-ready copy from the author using a Springer TEX macro package
Cover design: design & production GmbH, Heidelberg
SPIN: 10545662 55/3144-543210 -Printed on acid-free paper
Dedicated to the memory of
Asim Orhan Barut
(24 June 1926 - 5 December 1994)
But try to remember that a good man can never die.
The person of a man may go, but the best part of him stays.
It stays forever.
William Saroyan: The Human Comedy
Preface
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. Chapter 1 starts with a brief historical
review of symmetry and supersymmetry.
In Chap. 2 the basic definitions for supersymmetric quantum mechan
ics are given and fundamental properties following from these definitions are
=
discussed. In particular, we study the properties of N 2 supersymmetric
quantum systems. It is shown that the existence of the Witten-parity operator
natural,ly leads to a grading of the Hilbert space. The supersymmetry trans
formation and the Witten index are introduced. Ground-state properties of a
supersymmetric quantum system are also discussed.
\TIll f>reface
Chapter 3 extensively discusses Witten's model, which in essence consists
of a pair of standard one-dimensional Hamiltonians. This model is uniquely
characterized by a scalar function which is often called the supersymmetric
(SUSY) potential. Properties of this SUSY potential and the corresponding
supersymmetric Witten model are studied. Many examples of SUSY poten
tials leading to well-known quantum problems in one dimension are presented.
Supersymmetric classical mechanics is introduced and studied in Chap.
4. Here we consider the pseudoclassical version of Witten's model. A pseu
doclassical system possesses, in addition to the usual degrees of freedom,
Grassmann-valued degrees of freedom. The equations of motion are explicitly
solved and the properties of the solutions are discussed. Here emphasis is put
on the so-called fermionic phase. Quantization of this pseudoclassical system
is achieved via the canonical as well as the path-integral approach.
Exact solutions of quantum mechanical eigenvalue problems are covered
in Chap. 5. An additional property of the SUSYpotential called shape
invariance allows an· explicit derivation of the discrete spectrum of the
Hamiltonian in question and the corresponding eigenstates. The connection
with Schrodinger's factorization method is shown.
As an approximation method we consider in Chap. 6 the quasi-classical
path-integral approach to the Witten model. This approach yields novel quasi
classical quantization conditions having some remarkable properties. These
are discussed in detail and are compared with the usual WKB approximation
method.
An application in statistical physics is studied in Chap. 7. The super
symmetric structure of the Fokker-Planck and that of the Langevin equation
are shown to be closely connected to the quantum mechanical and the pseu
doclassical Witten model, respectively. Implications of supersymmetry are
discussed. Supersymmetric approximation methods for decay rates are also
presented.
In Chap. 8 we consider the supersymmetry of Pauli's Hamiltonian and dis
cuss the paramagnetism of a two- and of a three-dimensional non-interacting
electron gas, and the paramagnetic conjecture. Supersymmetry in Dirac's
equation is briefly discussed and its application to semiconductor heterojunc
tions is also given.
The selection of the topics covered in this book is clearly subjective and
results from my own interests. Chapter 9 is aimed at giving a broad overview
of topics in theoretical physics where supersymmetric methods became and
stitl are important tools.
Preface IX
The overview given in Chap. 9 is not meant to be complete. Similarly, the
list of references, though rather extensive, is certainly incomplete. I apologize
to all those who have been left out due to my ignorance.
This book originated in a course of lectures given at the University of
Erlangen-Niirnberg during the winter term 1993/94. My view of supersym
metric quantum mechanics has been developed during many years of collab
oration with Akira Inomata. Discussions with him, either personally or via
e-mail, are always enjoyable and clarifying. His comments and advice have
been very valuable for this book. Similar thanks go to Hajo Leschke. This
book benefitted a lot from his clarifying comments and suggestions. Also sev
eral other people have read parts of the manuscript and made many valuable
comments. I am especially indebted to Wolf Beiglb6ck, Helmut Fink, Werner
Fischer, Alfred Hiiller, John R. Klauder, Stephan Matthiesen, Peter Miiller,
and last but not least Siegfried Wonneberger for his endurance in reading the
complete manuscript.
The collaboration with Akira Inomata on supersymmetric quantum me
chanics has been supported by the Deutsche Forschungsgemeinschaft. This
support is gratefully acknowledged. I am also grateful to Urda Beiglb6ck and
Frank Holzwarth for their assistance with the Springer 1l\TE]X macro package
CLMonoOl.
Finally I am greatly indebted to my wife Karin and my children Julia and
Michael for their moral support and patience.
Erlangen, June 1996 Georg Junker
Contents
1. Introduction.............................................. 1
2. Supersymmetric Quantum Mechanics..................... 7
2.1 Definition of SUSY Quantum Mechanics................... 7
=
2.1.1 The Pauli Hamiltonian (N 1) .................... 9
=
2.1.2 Witten's SUSY Quantum Mechanics (N 2) ........ 10
2.1.3 Nicolai's Supersymmetric Harmonic-Oscillator Chain.. 11
=
2.2 Properties of N 2 SUSY Quantum Mechanics ............ 12
2.2.1 The Witten Parity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13
2.2.2 SUSY Transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15
2.2.3 Ground-State Properties for Good SUSY ............ 16
2.2.4 The Witten Index... .. .... ...... ... ... .... .. .. ... 17
2.2.5 SUSY and Gauge Transformations.................. 19
3. The Witten Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 21
3.1 Witten's Model and Its Modification. ........ .... .. .. .. . .. 21
3.2 Witten Parity and SUSY Transformation . . . . . . . . . . . . . . . . .. 22
3.3 The SUSY Potential and Zero-Energy States.. ... . .... .. . .. 23
3.3.1 Ground State for Good SUSY . .. .. .. .. .. .. .. .. .. ... 23
3.3.2 An Additional Symmetry. . . . . . . . . . . . . . . . . . . . . . . . .. 24
3.3.3 Asymptotic Behavior of the SUSY Potential. . . . . . . . .. 25
3.4 Broken Versus Good SUSY ..... . . . . . . . . . . . . . . . . . . . . . . . .. 27
3.5 Examples.............................................. 29
3.5.1 Systems on the Euclidean Line.... ... ... .... .. .. . .. 29
3.5.2 Systems on the Half Line. .. . . .... ...... .... .... ... 32
3.5.3 Systems on a Finite Interval ....................... 34
4. S';1persymmetric Classical Mechanics. . . . . . . . . . . . . . . . . . . . .. 37
4.1 Pseudoclassical Models.... ..... .... . . .... .. .... .... .. . .. 37
4.2 A Supersymmetric Classical Model.... . . .. .... .. . . .. .. ... 38
4.3 The Classical Dynamics .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39
XII Contents
4.4 Discussion of the Fermionic Phase . . . . . . . . . . . . . . . . . . . . . . .. 43
4.5 Quantization........................................... 46
4.5.1 Canonical Approach. . . . . . . . . .. . . . . . . .. . . . . . . . . . .. 47
4.5.2 Path-Integral Approach. . . . . . . .. . . . . . . . . . . . . . . . . . .. 49
5. Exact Solution of Eigenvalue Problems ................... 53
5.1 Supersymmetrization of One-Dimensional Systems. . . . . . . . .. 53
5.2 Shape-Invariance and Exact Solutions. . . . . . . . . . . . . . . . . . . .. 58
5.2.1 An Explicit Example. .. . . . . . . . . . . . . . . . . . . . . . . . . . .. 62
5.2.2 Comparison with the Factorization Method .......... 63
6. Quasi-Classical Path-Integral Approach.. .. ...... .. .... . .. 67
6.1 The Path-Integral Formalism. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 67
6.1.1 The WKB Approximation in the Path Integral. . . . . . .. 68
6.1.2 Quasi-Classical Modification for Witten's Model. . . . .. 70
6.2 Quasi-Classical Quantization Conditions-. .. . . . . . . . . . . . . . . .. 73
6.3 Quasi-Classical Eigenfunctions .. . . . . . . . . . . . . . . . . . . . . . . . .. 78
6.4 Discussion of the Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 79
6.4.1 Exactly Soluble Examples ......................... 79
6.4.2 Numerical Investigations .......................... 81
7. Supersymmetry in Classical Stochastic Dynamics. . ... . ... 93
7.1 Langevin and Fokker-Planck Equation. . . . . . . . . . . . . . . . . . .. 93
7.2 Supersymmetry of the Fokker-Planck Equation.. . . .. . . . . . .. 95
7.3 Supersymmetry of the Langevin Equation.. . . . . . . . . . . . . . . .. 97
7.4 Implications of Supersymmetry ........................... 101
7.4.1 Good SUSY ..................................... 101
7.4.2 Broken SUSY .................................... 105
8. Supersymmetry in the Pauli and Dirac Equation .......... 109
8.1 Pauli's Hamiltonian in Two and Three Dimensions .......... 109
8.2 Pauli Paramagnetism of Non-Interacting Electrons, Revisited. 112
8.2.1 Two-Dimensional Electron Gas ..................... 113
8.2.2 Three-Dimensional Electron Gas ................... 115
8.2.3 The Paramagnetic Conjecture and SUSY ............ 116
8.3 The Dirac Hamiltonian and SUSY ........................ 117
8.3.1 Dirac Hamiltonian with a Scalar Potential ........... 118
9. _ Concluding Remarks and Overview ....................... 121
References .................................................... 125
