Texts and Monographs in Physics. Series Editors: W. Beiglbock E. H. Lieb W. Thirring

Transcription

1 Texts and Monographs in Physics Series Editors: W. Beiglbock E. H. Lieb W. Thirring

2 Bernd Thaller The Dirac Equation Springer-Verlag Berlin Heidelberg GmbH

3 Dr. Bernd Thaller Institut fur Mathematik, Karl-Franzens-Universităt Graz Heinrichstr. 36, A-SOlO Graz, Austria Editors Wolf BeiglbOck Institut fur Angewandte Mathematik Universităt Heidelberg Im Neuenheimer Feld 294 W-6900 Heidelberg 1, FRG Walter Thirring Institut fur Theoretische Physik der Universităt Wien Boltzmanngasse 5 A-I090 Wien, Austria Elliott H. Lieb Jadwin Hali Princeton University P. O. Box 708 Princeton, NJ , USA ISBN Library ofcongress Cataloging-in-Publication Data. Thaller. Bernd, 1956~ The Dirac equation / Bernd Thaller. p. cm, ~ (Texts and monographs in physics) Includes bibliographical references and index, ISBN ISBN (ebook) DOI / Dirac equation. 2. Relativistic quantum theory. 3. Quantum electrodynamics. I. Title. II. Series. QC W28T '24~dc This work is subject to copyright. AII rights are reserved, whether the whole ar part of the material is concerned, specifically the rights oftranslation, reprinting, reuse of illustrations. recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication ar 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 far prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 1992 Urspriinglich erschienen bei Springer-Verlag Berlin Heidelberg New York 1992 Softcover reprint of the hardcover 1 st edition 1992 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 therefare frec for general use. Typesetting: Camera ready by authar 55/ Printed on acid-free paper

4 Preface Ever since its invention in 1929 the Dirac equation has played a fundamental role in various areas of modern physics and mathematics. Its applications are so widespread that a description of all aspects cannot be done with sufficient depth within a single volume. In this book the emphasis is on the role of the Dirac equation in the relativistic quantum mechanics of spin-1/2 particles. We cover the range from the description of a single free particle to the external field problem in quantum electrodynamics. Relativistic quantum mechanics is the historical origin of the Dirac equation and has become a fixed part of the education of theoretical physicists. There are some famous textbooks covering this area. Since the appearance of these standard texts many books (both physical and mathematical) on the nonrelativistic Schrodinger equation have been published, but only very few on the Dirac equation. I wrote this book because I felt that a modern, comprehensive presentation of Dirac's electron theory satisfying some basic requirements of mathematical rigor was still missing. The rich mathematical structure of the Dirac equation has attracted a lot of interest in recent years. Many surprising results were obtained which deserve to be included in a systematic exposition of the Dirac theory. I hope that this text sheds a new light on some aspects of the Dirac theory which to my knowledge have not yet found their way into textbooks, for example, a rigorous treatment of the nonrelativistic limit, the supersymmetric solution of the Coulomb problem and the effect of an anomalous magnetic moment, the asymptotic analysis of relativistic observables on scattering states, some results on magnetic fields, or the supersymmetric derivation of solitons of the mkdv equation. Perhaps one reason that there are comparatively few books on the Dirac equation is the lack of an unambiguous quantum mechanical interpretation. Dirac's electron theory seems to remain a theory with no clearly defined range of validity, with peculiarities at its limits which are not completely understood. Indeed, it is not clear whether one should interpret the Dirac equation as a quantum mechanical evolution equation, like the Schrodinger equation for a single particle. The main difficulty with a quantum mechanical one-particle interpretation is the occurrence of states with negative (kinetic) energy. Interaction may cause transitions to negative energy states, so that there is no hope for a stability of matter within that framework. In view of these difficulties R. Jost stated, "The unquantized Dirac field has therefore no useful physical interpretation" ([Jo 65], p. 39). Despite this verdict we are going to approach these questions in a pragmatic way. A tentative quantum mechanical interpre-

5 VI Preface tation will serve as a guiding principle for the mathematical development of the theory. It will turn out that the negative energies anticipate the occurrence of antiparticles, but for the simultaneous description of particles and antiparticles one has to extend the formalism of quantum mechanics. Hence the Dirac theory may be considered a step on the way to understanding quantum field theory (see Chapter 10). On the other hand, my feeling is that the relativistic quantum mechanics of electrons has a meaningful place among other theories of mathematical physics. Somewhat vaguely we characterize its range of validity as the range of quantum phenomena where velocities are so high that relativistic kinematical effects are measurable, but where the energies are sufficiently small that pair creation occurs with negligible probability. The successful description of the hydrogen atom is a clear indication that this range is not empty. The main advantages of using the Dirac equation in a description of electrons are the following: (1) The Dirac equation is compatible with the theory of relativity, (2) it describes the spin of the electron and its magnetic moment in a completely natural way. Therefore, I regard the Dirac equation as one step further towards the description of reality than a one-particle Schrodinger theory. Nevertheless, we have to be aware of the fact that a quantum mechanical interpretation leads to inconsistencies if pushed too far. Therefore I have included treatments of the paradoxes and difficulties indicating the limitations of the theory, in particular the localization problem and the Klein paradox. For these problems there is still no clear solution, even in quantum electrodynamics. When writing the manuscript I had in mind a readership consisting of theoretical physicists and mathematicians, and I hope that both will find something interesting or amusing here. For the topics covered by this book a lot of mathematical tools and physical concepts have been developed during the past few decades. At this stage in the development of the theory a mathematical language is indispensable whenever one tries to think seriously about physical problems and phenomena. I hope that I am not too far from Dirac's point of view: "... a book on the new physics, if not purely descriptive of experimental work, must be essentially mathematical" ([Di 76], preface). Nevertheless, I have tried never to present mathematics for its own sake. I have only used the tools appropriate for a clear formulation and solution of the problem at hand, although sometimes there exist mathematically more general results in the literature. Occasionally the reader will even find a theorem stated without a proof, but with a reference to the literature. For a clear understanding of the material presented in this book some familiarity with linear functional analysis - as far as it is needed for quantum mechanics - would be useful and sometimes necessary. The main theorems in this respect are the spectral theorem for self-adjoint operators and Stone's theorem on unitary evolution groups (which is a special case of the Hille-Yoshida theorem). The reader who is not familiar with these results should look up the cited theorems in a book on linear operators in Hilbert spaces. For the sections concerning the Lorentz and Poincare groups some basic knowledge of Lie groups is required. Since a detailed exposition (even of the definitions

6 Preface VII alone) would require too much space, the reader interested in the background mathematics is referred to the many excellent books on these subjects. The selection of the material included in this book is essentially a matter of personal taste and abilities; many areas did not receive the detailed attention they deserved. For example, I regret not having had the time for a treatment of resonances, magnetic monopoles, a discussion of the meaning of indices and anomalies in QED, or the Dirac equation in a gravitational field. Among the mathematical topics omitted here is the geometry of manifolds with a spin structure, for which Dirac operators playa fundamental role. Nevertheless, I have included many comments and references in the notes, so that the interested reader will find his way through the literature. Finally, I want to give a short introduction to the contents of this book. The first three chapters deal with various aspects of the relativistic quantum mechanics of free particles. The kinematics of free electrons is described by the free Dirac equation, a four-dimensional system of partial differential equations. In Chapter 1 we introduce the Dirac equation following the physically motivated approach of Dirac. The Hamiltonian of the system is the Dirac operator which as a matrix differential operator is not semibounded from below. The existence of a negative energy spectrum presents some conceptual problems which can only be overcome in a many particle formalism. In the second quantized theory, however, the negative energies lead to the prediction of antiparticles (positrons) which is regarded as one of the greatest successes of the Dirac equation (Chapter 10). In the first chapter we discuss the relativistic kinematics at a quantum mechanical level. Apart from the mathematical properties of the Dirac operator we investigate the behavior of observables such as position, velocity, momentum, describe the Zitterbewegung, and formulate the localization problem. In the second chapter we formulate the requirement of relativistic invariance and show how the Poincare group is implemented in the Hilbert space of the Dirac equation. In particular we emphasize the role of covering groups ("spinor representations") for the representation of symmetry transformations in quantum mechanics. It should become clear why the Dirac equation has four components and how the Dirac matrices arise in representation theory. In the third chapter we start with the Poincare group and construct various unitary representations in suitable Hilbert spaces. Here the Dirac equation receives its group theoretical justification as a projection onto an irreducible subspace of the "covariant spin-l/2 representation". In Chapter 4 external fields are introduced and classified according to their transformation properties. We discuss some necessary restrictions (Dirac operators are sensible to local singularities of the potential, Coulomb singularities are only admitted for nuclear charges Z < 137), describe some interesting results from spectral theory, and perform the partial wave decomposition for spherically symmetric problems. A very striking phenomenon is the inability of an electric harmonic oscillator potential to bind particles. This fact is related to the Klein paradox which is briefly discussed.

7 VIII Preface The Dirac operator in an external field - as well as the free Dirac operator - can be written in 2 x 2 block-matrix form. This feature is best described in the framework of supersymmetric quantum mechanics. In Chapter 5 we give an introduction to these mathematical concepts which are the basis of almost all further developments in this book. For example, we obtain an especially simple (and at the same time most general) description of the famous Foldy-Wouthuysen transformation which diagonalizes a supersymmetric Dirac operator. The diagonal form clearly exhibits a symmetry between the positive and negative parts of the spectrum of a "Dirac operator with supersymmetry". A possible breaking of this "spectral supersymmetry" can only occur at the thresholds ±mc2 and is studied with the help of the "index" of the Dirac operator which is an important topological invariant. We introduce several mathematical tools for calculating the index of Dirac operators and discuss the applications to concrete examples in relativistic quantum mechanics. In Chapter 6 we calculate the nonrelativistic limit of the Dirac equation and the first order relativistic corrections. Again we make use of the supersymmetric structure in order to obtain a simple, rigorous and general procedure. This treatment might seem unconventional because it does not use the Foldy Wouthuysen transformation - instead it is based on analytic perturbation theory for resolvents. Chapter 7 is devoted to a study of some special systems for which additional insight can be obtained by supersymmetric methods. The first part deals with magnetic fields which give rise to very interesting phenomena and strange spectral properties of Dirac operators. In the second part we determine the eigenvalues and eigenfunctions for the Coulomb problem (relativistic hydrogen atom) in an almost algebraic fashion. We also consider the addition of an "anomalous magnetic moment" which is described by a very singular potential term but has in fact a regularizing influence such that the Coulomb-Dirac operator becomes well defined for all values of the nuclear charge. Scattering theory is the subject of Chapter 8; we give a geometric, timedependent proof of asymptotic completeness and describe the properties of wave and scattering operators in the case of electric, scalar and magnetic fields. For the purpose of scattering theory, magnetic fields are best described in the Poincare gauge which makes them look short-range even if they are long-range (there is an unmodified scattering operator even if the classical motion has no asymptotes). The scattering theory of the Dirac equation in one-dimensional time dependent scalar fields has an interesting application to the theory of solitons. The Dirac equation is related to a nonlinear wave equation (the "modified Korteweg-de Vries equation") in quite the same way as the one-dimensional Schrodinger equation is related to the Korteweg-de Vries equation. Supersymmetry can be used as a tool for understanding (and "inverting") the Miura transformation which links the solutions of the KdV and mkdv equations. These connections are explained in Chapter 9. Chapter 10 finally provides a consistent framework for dealing with the negative energies in a many-particle formalism. We describe the "second quantized" Dirac theory in an (unquantized) strong external field. The Hilbert space

8 Preface IX of this system is the Fock space which contains states consisting of an arbitrary and variable number of particles and antiparticles. Nevertheless, the dynamics in the Fock space is essentially described by implementing the unitary time evolution according to the Dirac equation. We investigate the implementation of unitary and self-adjoint operators, the consequences for particle creation and annihilation and the connection with such topics as vacuum charge, index theory, and spontaneous pair creation. For additional information on the topics presented here the reader should consult the literature cited in the notes at the end of the book. The notes describe the sources and contain some references to physical applications as well as to further mathematical developments. This book grew out of several lectures I gave at the Freie Universitat Berlin and at the Karl-Franzens Universitat Graz in Parts of the manuscript have been read carefully by several people and I have received many valuable comments. In particular I am indebted to W. Beiglbock, W. Bulla, V. Enss, F. Gesztesy, H. Grosse, B. Helffer, M. Klaus, E. Lieb, L. Pittner, S. N. M. Ruijsenaars, W. Schweiger, S. Thaller, K. Unterkofler, and R. Wiist, all of whom offered valuable suggestions and pointed out several mistakes in the manuscript. I dedicate this book to my wife Sigrid and to my ten-year-old son Wolfgang, who helped me to write the computer program producing Fig Graz, October 1991 Bernd Thaller

9 Contents 1 Free Particles Dirac's Approach The Formalism of Quantum Mechanics Observables and States Time Evolution Interpretation The Dirac Equation and Quantum Mechanics A Hilbert Space for the Dirac Equation Position and Momentum Some Other Observables The Free Dirac Operator The Free Dirac Operator in Fourier Space Spectral Subspaces of Ho The Foldy-Wouthuysen Transformation Self-adjointness and Spectrum of Ho The Spectral Transformation Interpretation of Negative Energies The Free Time Evolution Zitterbewegung The Velocity Operator Time Evolution of the Standard Position Operator Evolution of the Expectation Value Evolution of Angular Momenta The Operators F and G Relativistic Observables Restriction to Positive Energies Operators in the Foldy-Wouthuysen Representation Notions of Localization Localization and Acausality Superluminal Propagation Violation of Einstein Causality Support Properties of Wavefunctions Localization and Positive Energies Approximate Localization The Nonstationary Phase Method Propagation into the Classically Forbidden Region 34 Appendix... 35

10 XII Contents LA 1.B 1.C 1.D I.E 1.F Alternative Representations of Dirac Matrices Basic Properties of Dirac Matrices.... Commutators with Ho.... Distributions Associated with the Evolution Kernel Explicit Form of the Resolvent Kernel.... Free Plane-Wave Solutions The Poincare Group The Lorentz and Poincare Groups The Minkowsky Space Definition of the Lorentz Group Examples of Lorentz Transformations Basic Properties of the Lorentz Group The Poincare Group The Lie Algebra of the Poincare Group Symmetry Transformations in Quantum Mechanics Phases and Rays The Wigner-Bargmann Theorem Projective Representations Representations in a Hilbert Space Lifting of Projective Representations Lie Algebra Representations The Lie Algebra and the Garding Domain The Poincare Lie Algebra Integration of Lie Algebra Representations Integrating the Poincare Lie Algebra Projective Representations Representations of the Covering Group A Criterion for the Lifting The Cohomology of the Poincare Lie Algebra Relativistic Invariance of the Dirac Theory The Covering Group of the Lorentz Group L(2) and Lorentz Group Rotations and Boosts Nonequivalent Representations of 8L(2) Linear Representation of the Space Reflection Gamma Matrices Equivalent Representations Time Reversal and Space-Time Reflections A Covering Group of the Full Poincare Group Appendix A Algebra of Gamma Matrices B Basic Properties of Gamma Matrices C Commutation Formulas D Dirac-Matrix Representation of Lorentz Transformations... 79

11 2.E 2.F Contents Expansion of Products of Gamma Matrices.... Formulas with Traces.... XIII Induced Representations Mackey's Theory ofinduced Representations Induced Representations of Lie Groups A Strategy for Semidirect Products Characters and the Dual Group The Dual Action ofthe Poincare Group Orbits and Isotropy Groups Orbits of the Poincare Group Invariant Measure and Little Group Induced Representations ofthe Poincare Group Wigner's Realization of Induced Representations Wigner States Wigner States for the Poincare Group Irreducibility of the Induced Representation Classification of Irreducible Representations The Defining Representation of the Little Group Covariant Realizations and the Dirac Equation Covariant States Covariant States for the Poincare Group Invariant Subspaces The Scalar Product in the Invariant Subspaces Covariant Dirac Equation The Configuration Space Poincare Transformations in Configuration Space Invariance of the Free Dirac Equation The Foldy-Wouthuysen Transformation Representations of Discrete Transformations Projective Representations of the Poincare Group Antiunitarity of the Time Reversal Operator External Fields Transformation Properties of External Fields The Potential Matrix Poincare Covariance of the Dirac Equation Classification of External Fields Scalar Potential Electromagnetic Vector Potential.... Anomalous Magnetic Moment.... Anomalous Electric Moment.... Pseudovector Potential Pseudoscalar Potential Self-adjointness and Essential Spectrum Local Singularities

12 XIV Contents Behavior at Infinity The Coulomb Potential Invariance of the Essential Spectrum and Local Compactness Time Dependent Potentials Propagators Time Dependence Generated by Unitary Operators Gauge Transformations Klein's Paradox Spherical Symmetry Assumptions on the Potential Transition to Polar Coordinates Operators Commuting with the Dirac Operator Angular Momentum Eigenfunctions The Partial Wave Subspaces The Radial Dirac Operator Selected Topics from Spectral Theory Potentials Increasing at Infinity The Virial Theorem Number of Eigenvalues Supersymmetry Supersymmetric Quantum Mechanics The Unitary Involution T The Abstract Dirac Operator Associated Supercharges The Standard Representation Some Notation Nelson's Trick Polar Decomposition of Closed Operators Commutation Formulas Self-adjointness Problems Essential Self-adjointness of Abstract Dirac Operators Particles with an Anomalous Magnetic Moment Dirac Operators with Supersymmetry Basic Definitions and Properties Standard Representation Examples of Supersymmetric Dirac Operators Dirac Operator with a Scalar Field Supersymmetry in Electromagnetic Fields The Klein-Gordon Equation Normal Forms of Supersymmetric Dirac Operators The Abstract Foldy-Wouthuysen Transformation The Abstract Cini-Touschek Transformation Connection with the Cayley Transform

13 Contents XV 5.7 The Index of the Dirac Operator The Fredholm Index Regularized Indices Spectral Shift and Witten Index Krein's Spectral Shift Function The Witten Index and the Axial Anomaly The Spectral Asymmetry Topological Invariance of the Witten Index Perturbations Preserving Supersymmetry Invariance of the Index Under Perturbations Fredholm Determinants Regularized Indices in Exactly Soluble Models Scalar Potential in One Dimension Magnetic Field in Two Dimensions Callias Index Formula The N onrelativistic Limit c-dependence of Dirac Operators General Setup Supersymmetric Dirac Operators Analyticity of the Resolvent Nonrelativistic Limit with Anomalous Moments c-dependence of Eigenvalues Browsing Analytic Perturbation Theory The Reduced Dirac Operator Analyticity of Eigenvalues and Eigenfunctions First Order Corrections of Nonrelativistic Eigenvalues Interpretation of the First Order Correction Example: Separable Potential Special Systems Magnetic Fields Introduction Dirac and Pauli Operators Homogeneous Magnetic Field The Ground State in a Magnetic Field Two Dimensions Three Dimensions Index Calculations Magnetic Fields and the Essential Spectrum Infinitely Degenerate Threshold Eigenvalues A Characterization of the Essential Spectrum Cylindrical Symmetry The Coulomb Problem The Hidden Supersymmetry The Ground State

14 XVI Contents Excited States The BJL Operator Discussion Stationary Coulomb Scattering Scattering States Preliminaries Scattering Theory Wave Operators The Scattering Operator Asymptotic Observables Introduction Invariant Domains RAGE Asymptotics of Zitterbewegung Asymptotic Observables Propagation in Phase Space Asymptotic Completeness Short-Range Potentials Coulomb Potentials Supersymmetric Scattering and Magnetic Fields Existence of Wave Operators Scattering in Magnetic Fields Time-Dependent Fields Solitons Time-Dependent Scalar Potential Scalar Potentials in One Dimension Generation of Time Dependence The Miura Transformation The Korteweg-de Vries Equation Construction of an Operator B Differential Equations for the Potentials Inversion of the Miura Transformation Scattering in One Dimension The Scattering Matrix Relative Scattering and the Regularized Index Soliton Solutions Solitons of the KdV Equation mkdv Solitons in the Critical Case mkdv Solitons in the Sub critical Case Quantum Electrodynamics in External Fields Quantization of the Dirac Field The Fock Space Creation and Annihilation Operators

15 Contents The Algebra of Field Operators Irreducibility of the Fock Representation Operators in Fock Space Implementation of Unitary Operators The Time Evolution Number and Charge Operators One-Particle Operators Bogoliubov Transformations Unitary Implementation in the General Case Even and Odd Parts of Unitary Operators The Shale-Stinespring Criterion Unitary Groups, Schwinger Terms, and Indices Example: The Shift Operator Particle Creation and Scattering Theory The S-Matrix in Fock Space Spontaneous Pair Creation.... Appendix.... Notes Books Articles Symbols Index XVII