自旋幾何筆記

本筆記旨在介紹自旋幾何的基本結構，包括 Clifford 代數、旋量表示、Spin主叢、自旋叢(spinor bundle)與 Dirac算子等核心觀念。

自旋幾何位於微分幾何、表示論與分析之交界，並在現代幾何拓撲(特別是 index theory 與四維流形理論)中扮演重要角色。 [Hydrogen Atom]

[Dirac equation]---[Spin Geometry]---Clifford代數---流形上的Dirac算子---指標定理

[Spin Geometry   Christian Bar]   [Clifford Algebra and Spinors]  by Pertti Lounesto]   [Dirac operators and Spectral Geometry   Joseph C. Varilly]

第一章 Differential operators on Manifold

  1. Differential operators on Manifolds
  2. Sobolev spaces
  3. Laplace-type and Dirac-type operators
  4. The Analysis of Dirac-type opeartors
  5. Hodge theory

第二章 Lorentz群與SL(2,C)

  1. Lorentz transformation
  2. Lorentz group SO(1,3)   [習作]  Lorentz spin group Spin(1,3) or SL(2,C)   [關於對稱性 :從SO(1,3) so(1,3) Spin(1,3)到Killing vector field]
  3. Lorentz群的表示       [表示隨處可見(席南華)] Represetations of the Lorentz Groups     [Topics in Representation Theory by (Giovanni Russo)]
    [SO(2)的表示]
  4. SO(4)    [SU(2)]   習作   SO(3)    [Spherical Harmonics]

第三章 Spinors and the classical Dirac operator

  1. Clifford Algebras Introduction       A child guide to spinors   Clifford product (Reflection and Rotation: (1) Infinitesimal Generators)   [cl3]    [cl4 and Spin(4) (rotation)]    習作   習作  習作(reflection)
  2. The spin Group ，Spin(3)，Spin(4)
  3. Spinors 立體投影將球面上的點和酉旋量聯繫起來  (2) 建構一個么正旋量  (3) [The Pauli matrices Pauli spinors]
    (4) Hermitian Vector Spaces    [Energy momentum equation]  [Klein-Gordon equation]   [自旋電子]   Spin Structure    Spinor Bundle   The Spin manifolds
  4. Spin strutures
  5. The classical Dirac operator on spinors
  6. Hypersurfaces
  7. Spacelike hypersurfaces of Lorentzian manifolds
  8. Maxwell，Dirac，Weyl，Majorana and Pauli spinors [習作 Dirac Spinors]
  9. Spin結構：[習作 Spin(4)]
  10. Spin 流形

第四章 [Dirac operator]      薛丁格方程

  1. The classical Dirac operator on spinors   [習作 Pauli spinors]  [Dirac方程]   [Dirac operator(1)]    [Dirac Operator(2)]   [The Dirac spectrum S^3的Dirac譜]
  2. 電磁場與Dirac current  Electromagnetic Field(電磁場與Dirac current J(x)] 非常困難 pdf檔亂掉了
  3. 聯絡與曲率
  4. Twisted Dirac Operator and Index Theory

第五章 量子力學中的旋量與Dirac方程式

  1. 氫原子的能階譜

第六章 The heat equation and index theory

  1. The heat kernel
  2. The formal heat kernel
  3. Growth of eigenvalues
  4. The index of Dirac-type operators
  5. 簡介Atiyah–Singe指標定理的幾何與拓撲意義
  6. 以Dirac 算子為例說明指標的計算
  7. 應用範例：手徵反常、零模計數、拓撲不變量

第七章 Holonomy groups

  1. 引入 Berger 分類的簡要說明
  2. 說明Spin(7)， 等特殊Holonomy流形與旋量的關係

第八章 Killing spinor fields

第九章 Open problems

 

  1. Spinors in 4-dimensional spaces   Gerardo F.Torres del Castillo(墨西哥)
    1.Spinor Algebra  2.Connection and Curveture  3.Applications to GR  4.Further Aplications
    Killing Spinors   [Exercise 1] [Exercise 2] [Exercise 3] [Exercise 4]
  2. The Dirac equation in Curved Spacetime   Peter Collas and David Klein
  3. The heat equation and index theory(熱核理論)   Atiyah-Singer-Index theorem  [Michael Atiyah and Isadore Singer]
  4. 正標量曲率(positive scalar curvature) : 一個給定的流形上是否存在一個具有正標量曲率的黎曼度量？
    Gromov-Lawson-Rosenberg conjecture [Open Problems]
  5. Kahler Manifolds and Complex Dirac Operators
  6. The Seiberg-Witten and Twistor Construtions(或Weitzenbock 公式應用)
  7. Applications to Gauge Theory and Topology
  8. Gemini建議的[自旋幾何講義]
  1. A Child Guide to Spinors      William O. Straub
  2. [Clifford Algebra and Spinors]  by Pertti Lounesto  習作
  3. [An Introduction Theory of Spinors]  by Moshe Carmeli   Shimon Malin   習作
  4. [Spinors in Physics]  by Jean Hladik  習作 (1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)
  5. [An Introduction to Clifford Algebras and Spinors]   by Jayme Vaz and Roldao da Rocha   習作
  6. [Spin Geometry] by  Christian Bar
  7. [Spinors in Four-Dimensional Spaces]  by    習作
  8. [Spinors and Space-time]  by  R.Penrose and W.Rindler
  9. [Clifford Algebra]  by Daniel Klawitter
  10. [Lecture Notes on Spin Geometry]  by Konstantin Wernli
  11. 張海潮先生的 (1)狹義相對論 (2)曲率公式 (3)伽利略與勞倫茲變換 (4)勞倫茲變換推導 (5)Minkowski metric  如何解愛因斯坦方程式
  12. George Uhlenbeck 1925       電子自旋
  13. 拓撲絕緣體(Topological insulator)  應用領域 : 量子計算 自旋電子學  低能耗電子元件