|
|
The Dirac equation
Bernd Thaller
[align=justify]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 no relativistic Schrodinger equation have been published, but only very few on the Driac 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 Driac theory. I hope that this text sheds a new light on some aspects of the Driac 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 super symmetric derivation of solutions 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 single particle. The main difficulty with a quantum mechanical on-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 interpretation will serve as a guiding principle for the mathemathical 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 basis knowledge of Lie groups is required. Since a detailed exposition (even of the definitions 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 play a fundamental role. Nevertheless, I 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 Driac equation following the physically motivated approach of Dirac. The Hamiltonian of the system is the Driac 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 Driac 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 implement 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-1/2 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.
The Dirac operator in an external field – as well as the free Dirac operator – can be written in 2*2 block-matrix form. This feature is best described in the framework of supesymmetric 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 +-mc^2 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 Driac 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 eigenfucnctions 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 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 consequeces 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 Universit鋞 Berlin and at the Karl-Franzens Universit鋞 Graz in 1986-1988. 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. Beiglb6ck, 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. Wrist, 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. 7.1.
Graz, October 1991
狄拉克方程
Bernd Thaller
序言
Dirac方程从1929年创立至今，已经在现代物理和数学各个领域扮演着基本原理的角色。其应用是如此广泛，因此在仅仅一本书之中，无法对其各方面的应用都加以深入阐述。本书着眼于Dirac方程在1/2自旋粒子的相对论量子力学的作用，覆盖了从单个自由粒子的相对论量子力学到量子电动力学的外加场问题。
相对论量子力学是Dirac方程及其发展为理论物理教育一个确定分支的历史渊源。有不少著名的教科书论及这一领域。遵循这些标准课本不少关于非相对论Schrodinger方程（包括数学和物理）的书籍均已出版，但关于Dirac方程的书籍却非常少见。我之所以写这本书，是因为感觉到数学和物理依然缺乏一种关于满足数学精确性和严密性要求的Dirac电子理论的现代的系统化的描述。
Dirac方程的丰富的数学结构已在近些年引起了广泛的兴趣，由此获得了很多令人惊异的结果应当纳入Dirac理论的体系。我希望这本书在Dirac理论的尚未发现其一些新的知识与结果纳入教科书的有效方法等方面发挥抛砖引玉的作用，如非相对论极限的严密处理方法，库仑场问题及异常磁矩效应的超对解，相对论散射态的渐近分析，一些关于磁场的结果，以及孤粒子方程的超对称解等。
或许极少有关于Dirac方程的专门书籍的原因是明确的量子力学性解释的缺乏。Dirac电子理论似乎留给了我们并无明确界定其有效适用范围的一部理论，其特性和应用尺度并不是十分明朗。事实上，人们是否应该将Dirac方程解释为量子力学的进展方程也不是清楚的，像单粒子的Schrodinge方程那样。描写粒子的量子力学性主要困难是负（动能）能量状态的出现。相互作用有可能引起粒子态转化为负能态，因此在其框架内人们不能够指望物质是稳定的。鉴于这一困难，R.Jost表明，“非量子化Dirac场因而无有意义的物理解释”（[Jo 65], P.39）。尽管如此，我们将注重实效地处理这些问题。作为数学理论发展的指导原则，试验性的量子力学描述发挥着重要作用。这导致负能量预示着反物质的重大发现，然而同时对粒子和反粒子的描述人们不得不扩充过去量子力学的形式。因此Dirac理论可以被认为是理解量子场论的一个重要步骤。（参考第10章）
另一方面，我感觉电子的相对论量子力学有着在其它数学物理中意味深长的地方。概略地，我们刻画其有效范围是速度如此之高而相对论性运动可以测量，但其能量是足够小，典型的例子是其概率极小的正负电子偶的产生。氢原子成功描述清楚表明这一范围并不是空想的。在电子的描述中用Dirac方程描述主要优势有：（1）Diarc方程同相对论理论是一致的；（2）它很自然地描写了电子自旋和自旋磁矩。因此我将Dirac方程作为进一步描述一个粒子以上的Schrodinger方程的一个步骤。然而我们不得不面临着一个残酷的事实，将Dirac方程推向过深入过广泛的范围，那么量子力学就将导致矛盾。因此我们讨论包含了对那些预示着理论局限性的困难和矛盾处理，尤其是局部问题和Klein矛盾。这些问题还没有明确的解释，即使在量子电动力学中。
当着手写这本书稿的时候，我考虑到读者对象主要是理论物理学家和数学家们，我希望两个领域的专家将由此而发现人们有兴趣的新奇的东西。涵盖本书的主题的许多数学工具和物理概念在过去十余年里已经有了很大的发展。而时下一部理论的发展，一种数学语言是不可缺少的，只要人们试图潜心地思考研究物理学问题及物理现象。我想我不脱离Dirac的思想太远，“……关于新物理的一本书，如果不纯粹论述实验工作，本质上就是数学。”（[Di 76], 序言）（物理学家们在遇到物理悖论问题的时候常常误以为自己跳出了数学的魔圈而把握了深奥莫测的真理，因而回避悖论的数学和物理严密逻辑，实际上极大的阻碍了理论物理的发展——Sunroom注）。然而，我从不企图表现出于个人兴趣的数学。我所做的是仅仅使用这些适合准确表达所面临问题的答案的工具，虽然有时候一些算术的普遍结果见诸文献。偶尔读者甚至将发现出自相关文献中缺少证明的一个冠名定理。
为了使读者清楚理解本书中的内容，通晓线性泛函分析——作为量子力学分析的数学基础——将是非常有用的而且有时候也是必备的。其主要定理就是关于自伴随矩阵的谱定理和关于幺正演化群的斯通定理（亦即Hille Yoshida定理的特例）。不熟悉这些结果的读者可以参考某本书引用的关于Hilbert空间线性算子的定理。一些章节涉及Lorentz-Poincare群和Lie群的一些基本知识都是必须的。由于详细的说明（甚至专门的解释）需要费较多的笔墨，对其数学背景感兴趣的读者可以参考很多有关的优秀著作。
本书的取材大体上属于个人的初步尝试和见解，很多方面没有涉及到应有的详细论述。例如，我感到遗憾的是，我一直都没有时间探讨极短寿命的不稳定基本粒子，磁单极子，指数的意义和QED（量子电动力学）中不规则性等问题，或重力场中的Dirac方程。在此省略掉数学多种题材，其中之一是与作为最基本工具Dirac算子相联系的自旋结构的多种几何形式。但我给了较多的注解和参考文献，因此有兴趣的读者可通过这些文献进一步理解。
最后，我想给出本书内容一个简短的介绍。开始三章阐述了自由粒子的相对论量子力学的多个方面。自由电子的运动学由自由狄拉克方程——一个四维系统的偏微分方程描述。第一章我们介绍基于狄拉克之自然逼近的狄拉克方程。系统的哈密顿函数是作为矩阵微分算子的狄拉克算子，它不是半有界的，还有负能谱的存在暴露了一些仅能在微粒论中才得以克服的概念上的困难。然而，在二次量子化中，负能量导致反粒子（正电子）存在的预言而被看作狄拉克方程的最伟大成就之一(第10章)。第一章我们在量子力学的基础上讨论了相对论运动学。除了狄拉克算子的数学工具之外,我们研究诸如位置，速度，动量等看得见的客观属性，描述了颤振，并阐述一些局部问题。
第二章我们论述了相对论量子力学的必备条件并阐述了狄拉克方程在希尔伯特空间里庞加莱群的处理方法，特别地我们强调覆盖群（关于量子力学对称变换表述的“旋量表象”）。为什么狄拉克方程的波函数有四个分量和狄拉克矩阵如何产生于表象理论就应该变得明朗了。第三章我们从庞加莱群着手构造合适的希尔伯特空间的多种幺正表象。这里狄拉克方程得以充分证明其映射到一个“1/2自旋表象”的不可约子空间的群理论。
第四章介绍外场和决定于其具体变换方法的分类。我们讨论一些必要的限制（狄拉克算子明确地存在着伴随势能表达式奇点的发散性，当核电荷Z>137时库仑场中奇异性是确认无疑的(因为那时候基态能量都是虚能量，显然荒唐，但至今理论物理学家们几乎还没有发现带来这种荒唐推论的根本原因是因为求解复杂波动方程缺乏了正确的数学知识，特别嗜好解释人都不是真正物理学家——sunroom注)），根据光谱理论描述一些有趣的结果，分解球对称问题的波函数。一个令人惊奇的现象是电谐势关不住粒子。这一事实涉及到后面简单讨论到的Klein佯谬（通常错误的东西都能够被理论物理学家们解释为佯谬，原因是实验数据可以拥有两方面的奥妙：杜撰与牵强附会的解释。如果有人说太阳是三角形的，总有人能够发挥其聪明才智用实验来证明三角形太阳理论的极多推论。——sunroom注）。
连同自由粒子的狄拉克算子在内，外场中狄拉克算子能够写成2*2矩阵形式。这种特性最好地被用于描写超对称量子力学的框架。在第５章，我们就这些作为几乎整部书进阶基础的数学概念给出了一个导言。例如，我们获得尤其简单（同时也是最普遍的）对Foldy Wouthuysen变换的描述，这变换就是对超对称狄拉克算子的对角化。对角化形式清楚表现出“带超对称的狄拉克算子”光谱的正负粒子的对称性。放弃这“光谱超对称”仅在＋-mc^2的极端情形才有可能，并得助于作为重要的拓扑变量的狄拉克算子指数而进一步发展。我们引入几个数学工具计算狄拉克算子指数并讨论相对论量子力学中的具体例子的应用。
第六章我们计算狄拉克方程的非相对论极限和一级相对论修正，然后我们利用超对称结果获得了一个简单，严密而又普遍适用的算法。这种处理手段可能似乎是非传统的，因为它不用Foldy-Wouthuysen变换，取而代之的是基于解析混沌理论的解决方法。
第七章致力于研究一些拓宽视野的特殊体系，可由超对称方法得到。第一部分论述引起非常有趣的现象和狄拉克算子奇异光谱的磁场。第二部分我们确定了库仑场中（相对论氢原子）的本征值和本征函数，这几乎全是代数。同时我们也考虑到由单一势项描述但实际上具有诸如库仑—狄拉克算子变为明确定义为核电荷值的有规则影响的异常磁矩。
第8章的内容是散射理论。对于电磁场和标量场情形，我们给出一个渐近完备性的和描述波以及散射算子的几何学的含时的证明。因散射理论起见，磁场在庞加莱度规——即使是长程作用亦致使其表现为短程作用——中（散射算子保持其不变性，即使经典运动无渐近线）得到最好的描述。一维含时标量场的狄拉克算子的散射理论在孤子理论中有着有趣的应用。在一维Schr鰀inger方程关联Korteweg-de Vries方程完全相同的情形，狄拉克方程涉及到非线性波动方程（即改进的Korteweg-de Vries方程）。超对称能够被作为一种理解（和“转化”）关联KdV和mKdV解的Miura变换的一种工具。对这些联系的解释构成第9章的内容。
最后，第10章提供了处理多粒子结构负能量问题的一个统一框架。我们描述一种（非量子化的）强外场中“二次量子化”狄拉克理论。这一系统的希尔伯特空间实际上就是包含由任意数目可变的粒子和反粒子组成的状态的Fock空间。然而，Fock空间的动力学本质上是由根据狄拉克方程实现幺正时间演化所描述的。我们研究幺正和自伴随矩阵的实现，粒子产生和湮灭的因果关系，以及光作为真空电荷的联系，指标理论，和自然电子偶的产生。
至于出现在书中论述光的附加内容，读者应该参考书末所列注释中的文献。这些注释描述其来源并包含一些有关物理应用及进一步数学发展的参考文献。
本书产生于1986-1988年期间我在柏林Freie大学和格拉茨Karl-Franzens大学的几个讲义。原稿的部分章节已由几位同行认真阅读，并且我已经收到一些有价值的注释。我尤其感激W. Beiglb6ck, 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. Wrist，他们提出了有价值的建议并指出了原稿中的几处错误。
谨以此书献给我的妻子西格丽德，和我十岁的儿子沃尔夫冈，他帮助我写了描绘插图7.1的计算机程序。
1991年10月于格拉茨
Bernd Thaller |
|
发表于 2007-5-19 14:27:39
