Table of Contents
Part I 1
1 The Basis for Geometric Algebra 3
1.1 Introduction 3
1.2 Genesis of Geometric Algebra 4
1.3 Mathematical Elements of Geometric Algebra 10
1.4 Geometric Algebra as a Symbolic System 13
1.5 Geometric Algebra as an Axiomatic System (Axiom A) 18
1.6 Some Essential Formulas and Definitions 23
References 26
2 Multivectors 27
2.1 Geometric Product of Two Bivectors A and B 27
2.2 Operation of Reversion 29
2.3 Magnitude of a Multivector 30
2.4 Directions and Projections 30
2.5 Angles and Exponential Functions (as Operators) 34
2.6 Exponential Functions of Multivectors 37
References 39
3 Euclidean Plane 41
3.1 The Algebra of Euclidean Plane 41
3.2 Geometric Interpretation of a Bivector of Euclidean Plane 44
3.3 Spinor i-Plane 45
3.3.1 Correspondence between the i-Plane of Vectors and the Spinor Plane 47
3.4 Distinction between Vector and Spinor Planes 47
3.4.1 Some Observations 49
3.5 The Geometric Algebra of a Plane 50
References 51
4 The Pseudoscalar and Imaginary Unit 53
4.1 The Geometric Algebra of Euclidean 3-Space 53
4.1.1 The Pseudoscalar of E3 56
4.2 Complex Conjugation 57
Appendix A Some Important Results 57
References 58
5 Real Dirac Algebra 59
5.1 Geometric Significance of the Dirac Matrices γμ 59
5.2 Geometric Algebra of Space-Time 60
5.3 Conjugations 64
5.3.1 Conjugate Multivectors (Reversion) 64
5.3.2 Space-Time Conjugation 65
5.3.3 Space Conjugation 65
5.3.4 Hermitian Conjugation 65
5.4 Lorentz Rotations 66
5.5 Spinor Theory of Rotations in Three-Dimensional Euclidean Space 69
References 72
6 Spinor and Quaternion Algebra 75
6.1 Spinor Algebra: Quaternion Algebra 75
6.2 Vector Algebra 77
6.3 Clifford Algebra: Grand Synthesis of Algebra of Grassmann and Hamilton and the Geometric Algebra of Hestenes 78
References 80
Part II 81
7 Maxwell Equations 83
7.1 Maxwell Equations in Minkowski Space-Time 83
7.2 Maxwell Equations in Riemann Space-Time (V4 Manifold) 85
7.3 Maxwell Equations in Riemann-Cartan Space-Time (U4 Manifold) 86
7.4 Maxwell Equations in Terms of Space-Time Algebra (STA) 88
References 91
8 Electromagnetic Field in Space and Time (Polarization of Electromagnetic Waves) 93
8.1 Electromagnetic (e.m.) Waves and Geometric Algebra 93
8.2 Polarization of Electromagnetic Waves 94
8.3 Quaternion Form of Maxwell Equations from the Spinor Form of STA 97
8.4 Maxwell Equations in Vector Algebra from the Quaternion (Spinor) Formalism 99
8.5 Majorana-Weyl Equations from the Quaternion (Spinor) Formalism of Maxwell Equations 100
Appendix A Complex Numbers in Electrodynamics 103
Appendix B Plane-Wave Solutions to Maxwell Equations-Polarization of e.m. Waves 105
References 107
9 General Observations and Generators of Rotations (Neutron Interferometer Experiment) 109
9.1 Review of Space-Time Algebra (STA) 109
9.1.1 Note 110
9.1.2 Multivectors 111
9.1.3 Reversion 111
9.1.4 Lorentz Rotation R 111
9.1.5 Two Special Classes of Lorentz Rotations: Boosts and Spatial Rotations 112
9.1.6 Magnitude 112
9.1.7 The Algebra of a Euclidean Plane 113
9.1.8 The Algebra of Euclidean 3-Space 114
9.1.9 The Algebra of Space-Time 116
9.2 The Dirac Equation without Complex Numbers 116
9.3 Observables and the Wave Function 118
9.4 Generators of Rotations in Space-Time: Intrinsic Spin 120
9.4.1 General Observations 121
9.5 Fiber Bundles and Quantum Theory vis-à-vis the Geometric Algebra Approach 122
9.6 Fiber Bundle Picture of the Neutron Interferometer Experiment 122
9.6.1 Multivector Algebra 125
9.6.2 Lorentz Rotations 127
9.6.3 Conclusion 129
9.7 Charge Conjugation 132
Appendix A 133
References 134
10 Quantum Gravity in Real Space-Time (Commutators and Anticommutators) 137
10.1 Quantum Gravity and Geometric Algebra 137
10.2 Quantum Gravity and Torsion 140
10.3 Quantum Gravity in Real Space-Time 142
10.4 A Quadratic Hamiltonian 146
10.5 Spin Fluctuations 149
10.6 Some Remarks and Conclusions 154
Appendix A Commutator and Anticommutator 156
References 158
Index 159
