《几何代数和统一场论》

辜英求

https://www.morebooks.shop/store/gb/book/clifford-algebra-and-unified-field-theory/isbn/978-620-2-81504-8

科学理论包括两个方面。首先，一个科学理论一定是一套整理好的演绎体系，从很少的逻辑相容的基本假设出发，推导出大量在各种特设条件下必然成立的逻辑结论。第二，基本假设涉及真理和信仰，是不能理性解释的，这只能通过实验检验其结论来确认。但是，基本假设的普遍性和有效性程度是有高低之分的，这反映创立者的洞察力，悟性，学识等思维能力和学术品味。怎样培养这种能力是我们的现行教育所欠缺的，因此很难培养出真正的思想大师，最后大家把一些异想天开的科普作家当作科学大师。

如何能用上帝的视角一睹物理规律的全貌，这是从古至今伟大思想家们的共同梦想。他们都认为自然是由很少几个极其简单的数学规则控制，纯粹思想可以把握现实。爱因斯坦一生的大部分时间都在追求这个梦想。本书通过分析现有统一场论的得失，提出了四条普遍适用的基本原理。从这四个基本原理出发，以Clifford代数或几何代数为主要数学工具，导出了所有基本物理方程，重建了方程之间的逻辑关系，并求解了一些 典型方程的解。 同时，对时空结构和量子理论进行了合理的解释，并得出了一些新的重要结论。

尽管本书已对原理做了大量的解释，对结论做了充分的论证，还有一些朋友协助检查，但也不能保证内容全部正确。真理总是在学术争论中明确和升华的，因此对本书的任何讨论和批评都是受欢迎的。

Clifford Algebra and Unified Field Theory

Ying-Qiu Gu (辜英求)

Abstract: The goal of this book is to study the unified field theory; that is, to logically derive all physical laws from a few simple principles. This goal is also the common dream of great thinkers who all agree that nature is governed by several extremely simple mathematical rules, and pure thought can grasp reality. Einstein spent most of his life in pursuit of the unified field theory.

Clifford algebra is a unification of real and complex numbers, quaternion and vector algebra, which accurately reflects the intrinsic properties of space-time. Clifford algebra provides a unified, standard, elegant and open language and tool for numerous complex mathematical and physical theories. Clifford algebra is a unified language for science and engineering, which can be expected to complete a new big synthesis of entire scientific knowledge.

Based on four basic principles and Clifford algebra, all basic physical equations were derived, the logical relations between equations were reconstructed, and the solutions of some typical equations were solved. Meanwhile, the concepts of space-time and quantum theory were reasonably explained, and some important new conclusions were obtained.

Publishing house: LAP LAMBERT Academic Publishing

Website: https://www.lap-publishing.com

By (author): Ying-Qiu Gu

Number of pages: 316

Published on: 2020-09-16

Stock: Available

Category: Theoretical physics

Book language: English

SBN-13: 978-620-2-81504-8

ISBN-10: 6202815043

EAN: 9786202815048

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1 The Holy Grail of Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1 Einstein's Ultimate Dream . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Weyl's Scale Invariant Unified Field Theory . . . . . . . . . . . . . . . . . . . . . 10

1.3 Five-Dimensional Space-Time of Kaluza . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Gauge Unified Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.5 String Theory and Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.6 Hamiltonian Formalism and Quantum Field Theory . . . . . . . . . . . . . . . . 21

1.7 New Concepts and Theoretical Framework . . . . . . . . . . . . . . . . . . . . . . 24

2 Clifford Algebra and Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.1 Brief Historical Review of Clifford Algebra . . . . . . . . . . . . . . . . . . . . . . 33

2.2 Clifford Algebra and Differential Geometry . . . . . . . . . . . . . . . . . . . . . 34

2.3 Representation of Clifford Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.4 Transformation of Clifford Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.5 Application in Classical Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3 Space-Time Geometry and Relativity . . . . . . . . . . . . . . . . . . . . . . . . . .59

3.1 Geometry of Minkowski Space-Time . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2 Uniqueness of Realistic Simultaneity . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3 Kinematics: Coordinate and Measurement . . . . . . . . . . . . . . . . . . . . . . 65

3.3.1 Kinematics of a Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3.2 Kinematics of a Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.4 Paradoxes in Special Relativity and Resolution . . . . . . . . . . . . . . . . . . . 70

3.4.1 Twins Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.4.2 Ehrenfest Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.4.3 Ladder Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.5 Natural Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.6 Light-Cone Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.6.1 Construction of Light-Cone Coordinates . . . . . . . . . . . . . . . . . . . 79

3.6.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4 Dynamics of Elementary Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.1 Elementary Field and Transformation Law . . . . . . . . . . . . . . . . . . . . . . 91

4.1.1 Elementary Field of Spin s = 1/2  . . . . . . . . . . . . . . . . . . . . . . . . 91

4.1.2 Elementary Field of Spin s = 1 . . . . . . . . . . . . . . . . . . . . . . . . 93

4.1.3 Elementary Field of Spin s > 1 . . . . . . . . . . . . . . . . . . . . . . . . 96

4.2 Nonlinear Spinor Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.2.1 Covariant Quadratic Forms of Elementary Fields . . . . . . . . . . . . . . 96

4.2.2 Nonlinear Dark Spinor Field . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.2.3 Spinor Field with Electromagnetic Interaction . . . . . . . . . . . . . . . . 102

4.2.4 Spinor Field with Strong Interaction . . . . . . . . . . . . . . . . . . . . . 104

4.3 Coupling System and Nonrelativistic Approximation . . . . . . . . . . . . . . . . 106

4.4 Classical Mechanics of Spinor with Interaction . . . . . . . . . . . . . . . . . . . 111

4.4.1 Classical Parameter and Lorentz Transformation . . . . . . . . . . . . . . 112

4.4.2 Classical Approximation of Dirac Equation . . . . . . . . . . . . . . . . . 115

4.4.3 Test of Mass-Energy Relation E = mc² . . . . . . . . . . . . . . . . . . . 119

5 Theory of Spinor in Curved Space-Time . . . . . . . . . . . . . . . . . . . . . . . 125

5.1 Simplification of the Spinor Connection . . . . . . . . . . . . . . . . . . . . . . . 125

5.2 Relations between Tetrad and Metric . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.3 Classical Mechanics of Spinor in Gravity . . . . . . . . . . . . . . . . . . . . . . . 133

5.4 Energy-Momentum Tensor of Spinors . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.5 Origin of Celestial Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.6 Lagrangian of Tetrad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6 Solution and Property of Spinor . . . . . . . . . . . . . . . . . . . . . . . 151

6.1 Integrable Conditions of Eigen Equation . . . . . . . . . . . . . . . . . . . . . . . 151

6.1.1 Eigen Equation in Curvilinear Coordinate System . . . . . . . . . . . . . 151

6.1.2 Integrable Conditions for Dirac Equation . . . . . . . . . . . . . . . . . . 153

6.1.3 Integrable Conditions for Pauli Equation . . . . . . . . . . . . . . . . . . 157

6.2 Numerical Method for Eigenfunction . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.2.1 Simplification of Energy Functional . . . . . . . . . . . . . . . . . . . . . . 160

6.2.2 Finite Element Method for Eigenfunction . . . . . . . . . . . . . . . . . . 162

6.3 Spinor with Short Distance Strong Potential . . . . . . . . . . . . . . . . . .164

6.3.1 New Model for Strong Interaction . . . . . . . . . . . . . . . . . . . . . . 165

6.3.2 Mass Spectrum of the Eigenstates . . . . . . . . . . . . . . . . . . . . . . 168

6.4 Properties of Nonlinear Spinor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

6.4.1 Properties of Dark Nonlinear Spinor . . . . . . . . . . . . . . . . . . . . . 173

6.4.2 Nonlinear Spinor with Electromagnetic Potential . . . . . . . . . . . . . . 176

6.5 Quantum Entanglement and Interpretation . . . . . . . . . . . . . . . . . . . . . 182

6.5.1 EPR Correlation and Bell's Inequality . . . . . . . . . . . . . . . . . . . . 182

6.5.2 Correlation Function of Entangled Spin . . . . . . . . . . . . . . . . . . . 186

6.5.3 Polarization Correlation of Entangled Photons . . . . . . . . . . . . . . . 189

6.5.4 Reasonable Interpretation of Quantum Mechanics . . . . . . . . . . . . . 190

7 Solutions to Einstein Field Equation . . . . . . . . . . . . . . . . . . . . . . . 197

7.1 Solutions in Light-Cone Coordinate System . . . . . . . . . . . . . . . . . . . . . 197

7.1.1 Simplification of Einstein Tensor . . . . . . . . . . . . . . . . . . . . . . . 197

7.1.2 Exact Vacuum Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

7.2 Series Solution to Axisymmetrical Vacuum . . . . . . . . . . . . . . . . . . . . . 204

7.2.1 Solution in Weyl-Lewis-Papapetrou Metric . . . . . . . . . . . . . . . . . 205

7.2.2 Coefficients of the Series Solution . . . . . . . . . . . . . . . . . . . . . . . 209

7.3 Property of the Star of Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

7.3.1 Exact Solutions in Comoving Coordinate System . . . . . . . . . . . . . . 214

7.3.2 Equations of Stellar Structure . . . . . . . . . . . . . . . . . . . . . . . . . 218

7.3.3 Structural Functions of Static Stars . . . . . . . . . . . . . . . . . . . . . 222

7.4 Dynamics of a Star with Spherical Symmetry . . . . . . . . . . . . . . . . . . . . 224

8 Dynamics of Galaxy and Spiral Structure . . . . . . . . . . . . . . . . . . . . . . 231

8.1 Research Status and Working Hypotheses . . . . . . . . . . . . . . . . . . . . . . 231

8.2 Linearization of Einstein Field Equation . . . . . . . . . . . . . . . . . . . . . . . 233

8.3 Simplification of Galactic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 237

8.4 Structural Equations of Stationary Galaxies . . . . . . . . . . . . . . . . . . . . . 240

8.5 Solutions to Non-Warped Stationary Galaxy . . . . . . . . . . . . . . . . . . . . . 243

8.5.1 Barred Spiral Galaxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

8.5.2 Spiral Galaxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

9 Large Structure of the Universe . . . . . . . . . . . . . . . . . . . . . . . 247

9.1 Dynamics of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

9.2 Equation of State in Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

9.2.1 Equation of State of Ideal Gases . . . . . . . . . . . . . . . . . . . . . . . 251

9.2.2 Asymptotic Behavior of Scalar in the Early Era . . . . . . . . . . . . . . . 254

9.2.3 Equation of State of Spinor Gas . . . . . . . . . . . . . . . . . . . . . . . 255

9.2.4 Equation of State of Vector . . . . . . . . . . . . . . . . . . . . . . . . . . 256

9.3 Dynamical Constraints on K andΛ. . . . . . . . . . . . . . . . . . . . . . . . . 258

9.3.1 Current Situation of the Parameters . . . . . . . . . . . . . . . . . . . . . 258

9.3.2 Assumption and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 259

9.4 Towards a Realistic Cosmological Model . . . . . . . . . . . . . . . . . . . . . . . 265

10 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

10.1 Notes of Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

10.1.1 Set and Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

10.1.2 Group and Number Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

10.1.3 Space and Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

10.1.4 Structure of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

10.1.5 Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

10.2 Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

10.3 Fundamental Constants in Physics . . . . . . . . . . . . . . . . . . . . . . . . . . 289

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291