Clifford Algebra: A Computational Tool for Physicists

John Snygg0195098242, 9780195098242

Clifford algebras have become an indispensable tool for physicists at the cutting edge of theoretical investigations. Applications in physics range from special relativity and the rotating top at one end of the spectrum, to general relativity and Dirac’s equation for the electron at the other. Clifford algebras have also become a virtual necessity in some areas of physics, and their usefulness is expanding in other areas, such as algebraic manipulations involving Dirac matrices in quantum thermodynamics; Kaluza-Klein theories and dimensional renormalization theories; and the formation of superstring theories. This book, aimed at beginning graduate students in physics and math, introduces readers to the techniques of Clifford algebras.

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Table of contents :
Contents……Page 10
Introduction……Page 14
1.1 Reflections, Rotations, and Quaternions in E[sup(3)] via Clifford Algebra……Page 20
1.2 ‘The 4π Periodicity of the Rotation Operator……Page 28
1.3* The Spinning Top (One Point Fixed)—Without Euler Angles……Page 29
2.1 A Small Dose of Special Relativity……Page 42
2.2 Mass, Energy, and Momentum……Page 53
3.1 Clifford Numbers in n-Dimensional Euclidean or Pseudo-Euclidean Spaces……Page 58
3.2 Dirac Matrices in Real Euclidean or Pseudo-Euclidean Spaces……Page 61
3.3 The Metric Tensor and the Scalar Product for 1-Vectors……Page 64
3.4 The Exterior Product for p-Vectors and the Scalar Product for Clifford Numbers……Page 70
4.1 Gaussian Curvature and Parallel Transport on Two-Dimensional Surfaces in E[sup(3)]……Page 77
4.2 The Operator [omitted][sub(v)] on an m-Dimensional Surface Embedded in an n-Dimensional Flat Space……Page 85
4.3 Parallel Transport on an m-Dimensional Surface Embedded in an n-Dimensional Flat Space……Page 97
5.1 The Operator [omitted][sub(α)] and Dirac Matrices in Curved Spaces……Page 101
5.2 Connection Coefficients and Fock–Ivanenko 2-Vectors……Page 107
5.3 The Riemann Curvature Tensor and its Symmetries……Page 113
5.4 The Use of Fock–Ivanenko 2-Vectors to Compute Curvature 2-Forms……Page 118
5.5* The Interpretation of Curvature 2-Forms as Infinitesimal Rotation Operators……Page 121
6.1 The Use of Fock–Ivanenko 2-Vectors to Determine the Schwarzschild Metric……Page 128
6.2 The Precession of Perihelion for Mercury……Page 134
7.1 The Exterior Derivative d and the Codifferential Operator δ Related to the Operator [omitted] = γ[sup(J)][omitted][sub(J)]……Page 146
7.2 Maxwell’s Equations in Flat Space……Page 154
7.3* Is Gravity a Yang–Mills Field?……Page 161
7.4* The Migma Chamber of Bogdan Maglich……Page 171
7.5* The Generalized Stake’s Theorem……Page 178
8.1 Currents and Dipoles in Curved Space Resulting from Dirac’s Equation for the Electron……Page 187
8.2 Clifford Solutions for the Free Electron in Flat Space……Page 197
8.3 A Canonical Form for Solutions to Dirac’s Equation in Flat Space……Page 202
8.4 Spherical Harmonic Clifford Functions……Page 206
8.5 Clifford Solutions of Dirac’s Equation for Hydrogen-like Atoms……Page 221
9.1 The Kerr Metric……Page 234
10.1 Petrov’s Canonical Forms for the Weyl Tensor……Page 267
10.2 Principal Null Directions……Page 283
10.3 The Kerr Metric Revisited via its Petrov Matrix……Page 289
11.1 Matrix Representations of Clifford Algebras……Page 304
11.2 The Classification of all Real Finite Dimensional Clifford Algebras……Page 311
11.3 The Classification of all c-Unitary Groups……Page 318
A.1 The Product Decomposition of Restricted Lorentz Operators and Related Operators……Page 324
A.2 The Exponential Representation of Restricted Lorentz Operators……Page 331
A.3 The Bianchi Identity……Page 340
Bibliography……Page 342
C……Page 348
G……Page 349
N……Page 350
S……Page 351
Z……Page 352