论文题目：
作者简介：戴柬
，男,1973年08月出生，1999年09月师从于北京大学宋行长教授，于2002年02月获博士学位。
摘
要
本文意在给出孔内斯学派非交换几何一个较完整的介绍，在孔内斯几何不适用的数理结构上建立非交换几何，并考虑相应的经典和量子场论。本文使用的数学方法主要是结合代数及其模理论和有限维欧氏空间上的数学分析。对于孔内斯几何，我们认为一个较全面的适用于物理学家的综述是很有必要的，因此文中以非交换拓扑，非交换向量丛，非交换微分及量子化表示，至孔内斯非交换几何的中心结构K-循环为线索，通过大量实例对其加以阐明。交换的至多可列维结合代数无法以较自然的方式纳入孔内斯几何体系，这类对象上的非交换几何构造是我们工作的主要贡献。文中介绍的线索仍然是从微分拓扑到微分几何，所不同的是由于存在更丰富的代数结构，如群结构，相应微分结构具有新的特征，如不变形式等。在这类具有非交换几何结构的空间上，我们推广定义标量场，费米场和规范场等物理理论并尝试把他们应用于强耦合标量四次相互作用，点阵狄拉克算子和规范场，夸克质量等现象学场合。文中新的数学结果有：离散点集向量场分类，使得联络表达更直观；相应在有限群上有导子系数；有限群微分的对合符号法则及相容性，流形形变的星李超代数描述；交换群上的正则
一阶微分，交错表象，高阶微分反对称实现；有限群厄密结构分类，拉普拉斯算子定义和谱定理；一维孔内斯距离。新的物理结果有：一般标量理论和守恒流的定义，强耦合标量四次相互作用理论的初等讨论；半直积群上的狄拉克算子和流，三阶改进的狄拉克－威尔逊算子；半直积有限群上的质量矩阵理论；点阵上非么正的规范场论，循环点阵规范理论的经典解。
以下摘引本文第一章《导言》和第五章《结语》作为导读。
本文的主题是离散非交换微分几何的数学基础及离散非交换空间上的场论构造。在这个导言中，我们首先做一个非常简短的非技术性的历史综述，以显示非交换几何与近代数学物理和理论物理的成长的关系，以及非交换几何的一个最粗略的特征，然后介绍本文的组织，同时对本文的论题加以明确限定。回顾科学史至少在方法论意义上常常是有益的。经验表明，近代理论物理的发展与微分几何学的发展是紧密联系，相互促进的。这里我们侧重关怀的是几何的代数表达。几何对象：在常识的意义上，几何学与几何体的代数表达至少可以上溯至笛卡儿；对欧氏空间中几何体的坐标化描述，使得对其局部的精密分析成为可能，从而为牛顿力学及对应的几何光学，以及晚两个世纪出现的曲线论提供了可以操作的形式体系。事实上尽管牛顿力图以古典的公理化形式表述力学理论，但对无穷小进行完整的公理化甚至在今天也未得到广泛认同。坐标几何的逻辑延伸抽象出微分流形的概念，而矢量力学在复杂系统的应用发展成位形空间上的拉格朗日力学和相空间上的哈密顿力学。几何结构：尽管用三维欧氏空间中经典场论的语言，波动光学和经典电动力学在十九世纪业已完美表述，但其背后作为向量丛上规范联络的几何内涵却经过近一百年才被揭示出来。与此相反，高斯发现二维曲面的内蕴度量属性早爱因斯坦对引力的几何化描述近一百年。而联络这样的几何结构可以用光滑函数及其模上的微分算子代数表出。概言之，以上经验显示相互作用的媒介的运动学常常可以表征为微分流形上某种几何结构的代数形式。而非交换几何方法拓展了微分流形的概念，相应地推广了有关的几何结构。简单讲，非交换几何方法从考虑流形上函数代数的角度研讨流形本身，这一角度的变化使得圈出比微分流形更广泛的一类对象并以代数语言加以描述成为可能，如果几何结构能够在其上加以定义，则这些对象成为推广意义上的流形。狭义而言, 非交换几何体系专指以孔内斯为领袖的集中在法国巴黎和马赛的一个数学学派创制的业已公理化的形式体系。经过二十年的发展，这一体系作为一个数学分支已趋成熟，其标志是一系列的研究专著的涌现，从孔内斯早期的总结性文献《非交换微分几何》（1985）到这一领域的权威之著孔内斯的《非交换几何》（1994）等。我们了解的这一体系物理应用的最早尝试是威腾的立方开弦场论，而除近几年来在弦理论中的运用外，它给出粒子物理标准模型中西格斯机制一个诠释，并描述了整数量子霍耳效应。就物理而言，与体系逻辑的恰当性同样重要的是其是否在自然界存在物理实现，一个抽象得很完备的数理体系未必恰当地描述观测现象。在此意义上，非交换几何方法较非交换几何体系有更宽泛的外延。事实上，存在一类在物理上广为使用的几何模型，即点阵模型无法纳入孔内斯的公理系统。而这正是本文发轫之处。下面的三章完成本文的三个任务。第二章给出非交换几何体系一个简要但完整的综述，鉴于国内在这方面的参考文献较少，我们认为这一服务于物理学家的综述是很有必要的。对点阵的抽象是一类离散对象，第三章旨在在这类对象上构造完整的非交换几何学，而第四章讨论相应的场论建造。需要指出的是在离散对象上推广标量场，矢量场及至张量场的概念是直接的，但费米场（旋量场）的推广并没有一个“正则”方向，这里只是给出其中一种可能和尝试。这一事实恰是自旋－统计定理的反面。
是否关怀存在终极的物理规律把物理学家，不论实验家抑或理论家，分成两类－实用的和形而上的。非交换几何这一类型的课题容易使人倾向关怀终极理论。自1985年始，相当一批物理学家开始相信弦理论是统一物质世界的理论。而1994年以来弦理论内部的逻辑突破使得弦的物理图象，即世界叶上的共形场论，变得含混不清了，以至于什么是弦理论的基本自由度成为一个本质问题。这时弦论专家开始诉诸非交换几何，因此如果这些专家的直觉正确，则我们可以期望在普朗克能标上下的物理，特别是量子引力这样的深刻问题，有可能通过这一新的几何学得到更好的了解。从实用的角度讲，对于作为迄今人类对物质结构与运动了解的最深层经验的汇总的标准模型，希格斯粒子是最后一片缺失的拼图。非交换几何在电弱能标上模型构造的共同特征是希格斯场不可或缺。反过来，这使得非交换几何是否是在这个能标近旁描述自然的恰当工具可以得到证伪，从而我们有可能回答在数理逻辑范畴内为真的非交换几何在物理实证下是否亦为真。在大约二千五百年前发现的不可通约量很好地描述着宏观低速的物理世界，而对波函数对称性的分析需要我们在纳米以下或较低温度环境中依赖复数域，至今没有人知道在一千亿电子伏与普朗克能量间何处我们需要不可交换的数。
Abstracti
This dissertation aims to review Connes' noncommutative geometry ina thorough manner, to device noncommutative geometry over thestructures on which Connes' geometry stops to make sense, and toconstruct field theories over these structures at both classical and quantum levels. Associative algebra and correspondent module theory, as well as finite dimensional analysis are the main athematical tools of this contribution. In our understanding, a omprehensive review to Connes' geometry for physicist in China is ncessary. Therefore, we follow the clue from noncommutative opology, noncommutative vector bundles, noncommutative dfferential together with its quantized representations, to the key concept of Connes' geometry --- K-cycle, and we give a lot of detailed examples to clarify this mathematical framework. Then
noncommutative geometry is endowed upon countable dimensional commutative associative algebras, upon which Connes geometry has to be implemented unnaturally. This work is our main efforts. Still we go from differential topology to differential geometry, but due to the existence of more algebraic structures, like a group, here the differential possesses additional characters, as invariant forms. Based upon this geometric construction, scalar field, fermionic field, and gauge field are generalized onto such types of noncommutative spaces. Applications to some phenomenological situations, like strongly quartic scalar coupling, lattice Dirac
operator and gauge fields, quark masses, are explored. Our new mathematics results include classification of vector fields on on discrete sets so that the expression of connections is more intuitive; accordingly there are derivative coefficients on finite groups; signature rule of involution on finite groups and the compatibility condition, deformation of manifold and presentation by a star-super Lie algebra; canonical first order differential on Abelian groups, stagger representation, antisymmetric realization of higher order differentials; classification of Hermitian structures, definition of Laplacian and spectral theorem; one- dimensional Connes' distant. Our new physics results include general scalar theory and definition of conserved currents, strong coupled scalar quartic theory; fermion over semi-direct product groups and currents, cubic-order improved Dirac-Wilson operator; mass matrix on semi-direct product group; non-unitary gauge theory over lattices, solutions to gauge theory over periodic lattices.
Hereafter we quote chapter one "introduction" and chapter five "finale" in the thesis as the guide for readers.
The subject of this thesis is mathematical basis of discrete noncommutative geometry and field theory construction on discrete noncommutative spaces. In this introduction, we first give a very brief nontechnical history review, to illustrate the interrelation between the growth of modern mathematical physics and theoretical physics, as well as the coarsest characteristic of noncommutative geometry; then we introduce the organization of this thesis, and meantime define precisely the motif and scope of this thesis. It is beneficial at least at the level of methodology to have a retrospect over the history of science. The development of modern
theoretical physics and that of differential geometry are interwound closely and expediated mutually. Here we concern the algebraic representation of geometry. The geometric objects: in common sense, the algebraic presentation of geometry and geometric objects, at least, can be traced to Descartes; the coordinatization description of geometric objects in Euclidean space facilitated the precise local analysis, hence supplying a manipulable formalism to Newtonian mechanics, geometric optics, and Frenet's theory of curves after two centuries. In fact even though Isaac Newton endeavored to formulate mechanics in terms of classical axiomatization, the complete axiomatization of the theory of infinitesimal has not been widely accepted hitherto. The concept of differential manifolds is abstracted from the logic extension of coordinate geometry and Lagrangian mechanics on configuration space and Hamiltonian mechanics on phase space are developed from the application of vector mechanics. Geometric structures: Although wave optics and classical electrodynamics were expressed perfectly adopting the language of classical field theory in three- dimensional Euclidean space, the underlying geometric connotation, as gauge connection over vector bundle was revealed after nearly one hundred years. On the contrary, Gauss disclosed the intrinsic metric characteristics of two-dimensional surface, which was ahead about one hundred years to the geometrization of gravity by Einstein. Note that such geometric notions as connection can be presented by differential operators over smooth functions and corresponding modules. In summarization, the above-mentioned experience displays that the kinematics of media of interactions can be usually expressed by algebraic formulation of some geometric structure on differential manifold. Actually noncommutative geometry method enlarges the concept of differential manifold, accordingly generalizing corresponding geometric structures. Simply speaking, noncommutative geometry method enters into manifold from considering the algebra of functions on manifold; this change of view point enables to geometrize a broad category of objects and to describe them by algebraic language; once geometric structure can be defined in this sense, such objects are regarded as generalized manifolds. In a narrow sense, the term "noncommutative geometry" refers to the axiomatized formalism from the mathematical school in Paris and Marseilles, leading by Allen Connes. After two-years' development, this formalism has become a mature branch of mathematics, which is signified by a series of monographs, from the early review "Noncommutative Differential Geometry" by Connes in 1985 to the masterpiece in this arena, "Noncommutative Geometry" by Connes in 1994 (referred as "the Bible of the subject"), as well as the latest textbook "Elements of Noncommutative Geometry" by J. M. Gracia-Bondia et al in 2001 (we also list a collection of other authoritative reviews in our bibliography. To the extent of our knowledge the earliest attempt to apply this formalism in physics is Edward Witten's open string field theory in 1986. Except the latest application in string/M-theory, it provides a new interpretation to Higgs mechanism in standard model and describes the integer quantum Hall effect. Speaking of physics, the physical realization in nature is as important, if not more, as the logic exactness of framework; one completely abstracted mathematical model
is not necessarily apt to describe observational phenomena. In this sense, the idea of noncommutative geometry possesses a broader extension to Connes' geometry formalism. Indeed, there exists a class of widely-applied geometric models, lattices, are not able to be brought into Connes' axiom system, which is what we set out. The following three chapters fulfil three missions of this thesis. Chapter two reviews briefly but comprehensively Connes' geometry system; whereas China lacks such level of references, we believe this review for physicists is completely necessary. Chapter three serves to construct a complete noncommutative geometry over discrete objects such as lattices. Chapter four discusses the corresponding field theory construction. It is worthwhile to note that notwithstanding the straightforwardness of generalization of
scalar, vector and tensor on discrete objects, there is no "canonical" guide to generalize fermionic field (spinor); what is presented here is merely a possibility and attempt. This indicates on the other hand the non-triviality of spin-statistics connection.
Physicists, whatever experimentalists or theoreticians, can be categorized into two classes, pragmatic vs metaphysical, according whether he concerns the eternal laws; topics like noncommutative geometry are apt to inspire people to explore the final theory. Ever since 1985 the mainstream of physicists have believed that
string theory is the unifying machine; however the logic breakthrough within string theory framework from 1994 obscures the physical picture of string theory, which is a conformal field theory over worldsheet, such that what is the fundamental degree(s) of freedom becomes an essential problem. From then on expertise resorts to noncommutative geometry. Henceforth, if the intuition from these experts is correct, then we can expect that insight into the physics around Planck energy, especially such profound problems as quantum gravity, can be possibly acquired through the deployment of this new geometry. From a more positive point of view, Higgs boson is the last piece of missing jigsaw puzzle within standard model, the latest "Philosophiae Naturalis Principia Mathematica", our human being's deepest knowledge of material structure and motion; models based on noncommutative geometry predicts generically that Higgs particle is necessary. Furthermore, this prediction enables it to be falsified whether noncommutative geometry is or not an appropriate tool to describe nature around electro-weak energy, such that we can answer whether noncommutative, which is true under mathematical logic, is true or not under physical experiment. The macroscopic low-velocity physical world is peacefully described by the non-commeasurable quantities discovered about two thousand years ago; to analyze the symmetry property of wave function under one nanometer or at law temperature, complex field has to be utilized; so far no human being knows where between one TeV and Planck energy numbers become noncommutative with each other...
