理论物理学数学透视
2003-10
Imperial College Press
Prakash, Nirmala
835
This book presents the basics of mathematics that are needed for learning the physics of today. It describes briefly the theories of groups and operators, finite- and infinite-dimensional algebras, concepts of symmetry and supersymmetry, and then delineates their relations to theories of relativity and black holes, classical and quantum physics, electroweak fields and Yang-ills. It concludes with a chapter on (the complex theory of) strings and superstrings and their link to black holes ?an idea that fascinates both the physicist and the mathematician.
PrefaceForewordAcknowledgementsChapter 0. Preliminaries 1. Basic Definitions 2. Topology Exercise (0.2) Hints to Exercise (0.2) 3. Differentiable Manifolds 3.1 Differentiable Manifolds 3.2 Tangent Space 3.3 Vector Fields, Tensors and Tensor Fields 3.4 Riemannian Metric and Covariant Derivation 3.5 Geodesics, Jacobi Fields, Curvature and Torsion 4. Measure, Exp H, Dirac t~-function 4.1 Measurable Spaces and Measurable Functions 4.2 Haar Measure 4.3 The SpaceExpH 4.4 Dirac d-function 5. Examples Based on Differential Geometry 5.1 Critical Points 6. Basic Definitions in Algebraic Topology 6.1 de Rham Complex and de Rham Cohomology 6.2 Category and Functors 6.3 Mayer-Vietoris Sequence 6.4 Homotopy References Chapter 1. Complex Functions, Riemann Surfaces and Two-Dimensional Conformal Field Theory (an Introduction) 1. Complex Functions 1.1 Complex Plane 1.2 Analytic Function 1.3 Harmonic Functions 1.4 Laurent Series 1.5 Simply Connected and Multiply Connected Domain 1.6 Residues and Poles 1.7 Elliptic Curves 2. Complex Structure on a Manifold, Kahler Metric 2.1 Complex Manifold M 2.2 Complex Structure on M 2.3 The Tangent and Cotangent Spaces to M 2.4 Holomorphic Vector Fields and Holomorphic Forms on M 2.5 Some Calculus on M 2.6 K~ihler Manifold 2.7 Harmonic Forms on a Kahler Manifold Exercise (1.2) Hints to Exercise (1.2) 3. Riemann Surfaces 3.1 Riemann Surface M 3.2 Holomorphic Mappings on M 3.3 Differential Forms on M, their Algebra and Calculus 3.4 The Star (*) Operator on M 3.5 Harmonic and Holomorphic Forms on M 3.6 Square-integrable 1-forms on M 3.7 Abelian Differentials on M 3.8 A Few Results Based on Transformation Groups of M Exercise (1.3) Hints to Exercise (1.3) 4. The Two-Dimensional Conformal Field Theory 4.1 Conformal Group 60 4.2 Light-cone Formalism and the Lorentz Group 4.3 Euclidean Space Formalism 4.4 Two-dimensional Conformal Group 4.5 M6bius Transformation 4.6 Conformal Tensor Calculus 4.7 Conserved Currents Exercise (1.4) Hints to Exercise (1.4) References Chapter 2. Elements of Group Theory and Group Representations 1. Introduction 1.1 Definition of a Group, Examples, and Conjugate Classes 1.2 Invariant Subgroups, Factor Groups, Simple and Semi-simple Groups 1.3 Products of Groups and Homomorphism 2. Lie Groups and Topological Groups 2.1 Topological Groups 2.2 Algebraic Groups ……Chapter 3 A Primer on OperatorsChapter 4 Basics of Algebras and Related ComceptsChapter 5 Infinite-Dimensional AlgebrasChapter 6 The Role of Symmertry in Physics and MathematicsChapter 7 All That's Super-An IntroductionChapter 8 Gravitation, Relativity and Black HolesChapter 9 Basics of Quantum THeoryChapter 10 Theory of Yang-Mills and The Yang-Mills-Higgs MechanismChapter 11 Strings and Superstrings (Elementary Aspects)SymbolsIndex
