《狄拉克方程》——翻译连载（５）

sunroom

[align=justify]《狄拉克方程》序言——翻译对吗？（1）[align=justify]
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.
[align=justify]　　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，以至于关于各个方面问题的深入全面的描述均不能缺少这一方程。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。[align=justify]
[align=justify][align=justify]《狄拉克方程》序言——翻译对吗？（２）
[align=justify][align=justify][align=justify][align=justify]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). [align=justify]
　　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）

《狄拉克方程》序言——翻译对吗？（３）
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.
[align=justify]　　另一方面，我感觉电子的相对论量子力学有其数学物理的意味深长的地方。概略地说，我们刻画了其有效范围是速度如此之高而相对论性运动效应竟可以测量，但物质的能量又是如此之小，典型的例子是其产生的概率极小的正负电子偶。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。
[align=justify][align=justify]《狄拉克方程》序言——翻译对吗？（4）
[align=justify]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.[align=justify]
　　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和研究物理学问题及物理现象。我想我不应该脱离Dirac观点太远，“……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。”（[Di76], 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，我从不企图表现那些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。

sunroom

不是很懂第一段"sufficient depth within a single volume"的意思!

mmdzz

Its applications are so widespread that a description of all aspects cannot be done with sufficient depth within a single volume.

其应用极其广泛，因此在仅仅一本书之中（指容量有限），无法对其各方面的应用都加以深入阐述。：）

volume指一本、一册书。

老先生不妨装个金山词霸，有生词随时对照查。

sunroom

引用第2楼mmdzz于2007-04-29 20:34发表的 :
Its applications are so widespread that a description of all aspects cannot be done with sufficient depth within a single volume.

其应用极其广泛，因此在仅仅一本书之中（指容量有限），无法对其各方面的应用都加以深入阐述。：）

volume指一本、一册书。
.......

兄弟实聪明之至，＂volume指一本、一册书＂，我怎么也琢磨不出来．而今老矣，尚能食粥！非常感谢了．这本书我要读一年吧？ 不是很精通电脑，装金山词霸老是死机，今天请个内行帮我看看．再谢小兄弟！

doctorliubo

希望你把对《狄拉克》的翻译继续下去，最好开一个专帖，便于管理~~~

sunroom

引用第4楼doctorliubo于2007-04-30 08:28发表的 :
希望你把对《狄拉克》的翻译继续下去，最好开一个专帖，便于管理~~~

    

我努力吧，争取一天一段。如何开专贴，我还不懂。

doctorliubo

你把《狄拉克》序言——翻译对吗？（2）也拷贝过来，我把这个帖子给你置顶，然后以后所有的翻译都发到这个帖子里~~~

doctorliubo

建议你多下载一些这方面的外文文献，然后多看看人间是怎么翻译这方面的内容的~~~

sunroom

只要努力，就会有收获！活到老，我也努力到老！

sunroom

今天翻译的是（２），把昨天的（２）改为（３）了。

sunroom

（２）的翻译有无原则性错误或极不妥的地方，请批评，先谢了！

sunroom

（４）的译文也贴上去了。节日快乐！

sunroom

5月1日，这一天，火箭赢了，依然赢得那样惊心动魄，不像爵士，赢得那样从容不迫！

sunroom

[align=justify]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.[align=justify]
　　为了使读者清楚理解本书中的内容，通晓线性函数分析——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佯随矩阵的谱定理和关于酉群的斯通定理（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。一些章节所涉及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相关的优秀著作。

sunroom

序言还没有译完，后面说是每章节的主要内容。我觉得还是把文章写完后再译的好，因为那时候具体的逻辑可能可以翻译得更准确一些。

zhengfei_dlut

linear functional analysis 是 线性泛函分析

sunroom

引用第15楼zhengfei_dlut于2007-05-04 07:46发表的 :
linear functional analysis 是 线性泛函分析

对了。谢！

sunroom

[align=justify]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.

　　本书的取材大体上属于个人初步的尝试和见解，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。

sunroom

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.

　　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。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，负能量导致了存在反粒子（正电子）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。

sunroom

[align=justify]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". [align=justify]
　　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”）。为什么狄拉克的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酉表示。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。