Clifford (Geometric) Algebras

1 Introduction.- 2 Clifford Algebras and Spinor Operators.- 2.1 A History of Clifford Algebras.- 2.2 Teaching Clifford algebras.- 2.2.1 The Clifford Product of Vectors.- 2.2.2 The Exterior Product.- 2.2.3 Components of a Vector in Given Directions.- 2.2.4 Perpendicular Projections and Reflections.- 2.2.5 Matrix Representation of Cl2.- 2.2.6 Exercises.- 2.2.7 Answers.- 2.3 Operator Approach to Spinors.- 2.3.1 Relation to the Convention of Bjorken-Drell.- 2.3.2 Bilinear Covariants.- 2.3.3 Fierz Identities (Discovered by Pauli and Kofink).- 2.3.4 Recovering the Spinor from its Bilinear Covariants.- 2.3.5 Boomerangs and the Reconstruction of Spinors.- 2.3.6 The Mother of All Real Spinors ? ? Cl1,3 ½(1 + ?0).- 2.3.7 Ideal Spinors ø ? Cl1,3 ½(1 - ?03).- 2.3.8 Spinor Operators ? ? Cl1,3+.- 2.3.9 Decomposition and Factorization of Boomerangs.- 1.4 Flags, Poles and Dipoles.- 1.4.1 A Geometric Classification of Spinors by their Bilinear Co-variants.- 1.4.2 Projection Operators in End(Cl1,3).- 1.4.3 Projection Operators for Majorana and Weyl Spinors.- 3 Introduction to Geometric Algebras.- 3.1 The Unipodal Number System.- 3.2 Clifford Algebra Matrix Algebra Connection.- 3.3 Geometric algebra.- 4 Linear Transformations.- 4.1 Structure of a Linear Operator.- 4.2 Isometries.- 4.3 Minimal Polynomials.- 4.4 Lie Algebras of Bivectors.- 5 Directed Integration.- 5.1 Simplices and Chains.- 5.2 Integral Definition of ?xF(x) on S.- 5.3 Classical Integration Theorems.- 5.4 Residue Theorem.- 5.5 Riemannian Geometry.- 6 Linear Algebra.- 6.1 Geometric Algebra.- 6.2 Spacetime Algebra.- 6.3 Geometric V’s Tensor Algebra.- 6.4 Index-Free Linear Algebra.- 6.5 Multivector Calculus.- 6.6 Adjoints and Inverses.- 6.7 Eigenvectors and Eigenbivectors.- 6.8 Invariants.- 6.9 Linear Functions in Spacetime.- 6.10 Functional Differentiation.- 7 Dynamics.- 7.1 Rigid Body Dynamics.- 7.1.1 The Rotor Equation.- 7.1.2 Kinetic Energy and the Inertia Tensor.- 7.1.3 The Equations of Motion of a Rigid Body.- 7.1.4 Free Precession of a Symmetric Top.- 7.2 Dynamics of Elastic Media.- 7.2.1 Energy Flow.- 7.2.2 Pre-Stressed Media.- 7.2.3 Linearised elasticity.- 7.2.4 The Elastic Filament.- 8 Electromagnetism.- 8.1 Electromagnetic Waves.- 8.1.1 Stokes Parameters.- 8.1.2 Reflection by a Conducting Plane.- 8.1.3 Waves in layered media.- 8.2 Diffraction Theory.- 8.2.1 The Boundary-Value Problem in Electrodynamics.- 8.3 The Electromagnetic Field of a Point Charge.- 8.4 Applications.- 8.4.1 Uniformly Moving Charge.- 8.4.2 Accelerated Charge.- 8.4.3 Circular Orbits.- 9 Electron Physics I.- 9.1 Pauli Spinors.- 9.1.1 Pauli Observables.- 9.1.2 Spinors and Rotations.- 9.2 Dirac Spinors.- 9.2.1 Alternative Representations.- 9.3 The Dirac Equation and Observables.- 9.3.1 Plane-Wave States.- 9.4 Hamiltonian Form.- 9.5 The Non-Relativistic Reduction.- 9.6 Angular Eigenstates and Monogenic Functions.- 9.6.1 The Spherical Monogenics.- 9.7 Application — the Coulomb Problem.- 10 Electron Physics II.- 10.1 Propagation and Characteristic Surfaces.- 10.2 Spinor Potentials and Propagators.- 10.3 Scattering Theory.- 10.3.1 The Born Approximation and Coulomb Scattering.- 10.4 Plane Waves at Potential Steps.- 10.4.1 Matching Conditions for Travelling Waves.- 10.4.2 Matching onto Evanescent Waves.- 10.5 Spin Precession at a Barrier.- 10.6 Tunnelling of Plane Waves.- 10.7 The Klein Paradox.- 11 STA and the Interpretation of Quantum Mechanics.- 11.1 Tunnelling Times.- 11.2 Spin Measurements.- 11.2.1 A Relativistic Model of a Spin Measurement.- 11.2.2 Wavepacket Simulations.- 11.3 The Multiparticle STA.- 11.3.1 2-Particle Pauli States and the Quantum Correlator.- 11.3.2 Multiparticle Wave Equations.- 11.3.3 The Pauli Principle.- 11.3.4 8-Dimensional Streamlines and Pauli Exclusion.- 12 Gravity I — Introduction.- 12.1 Gauge Theories.- 12.2 Gauge Principles and Gravitation.- 12.3 The Gravitational Gauge Fields.- 12.3.1 The Rotation-Gauge Field.- 12.4 Observables and Covariant Derivatives.- 13 Gravity II — Field Equations.- 13.1 The Gravitational Field Equations.- 13.2 Covariant Forms of the Field Equations.- 13.3 Symmetries and Invariants of R(B).- 13.4 The Bianchi Identity.- 13.5 Symmetries and Conservation Laws.- 14 Gravity III — First Applications.- 14.1 Spherically-Symmetric Static Solutions.- 14.1.1 Point-Particle Trajectories.- 14.1.2 Particle Motion in a Spherically-Symmetric Background.- 14.2 Electromagnetism in a Gravitational Background.- 14.2.1 Application to a Black-Hole Background.- 15 Gravity IV — The ‘Intrinsic’ Method.- 15.1 Spherically-Symmetric Systems.- 15.2 Two Applications.- 15.3 Stationary, Axially-Symmetric Systems.- 15.4 The Kerr Solution.- 16 Gravity V — Further Applications.- 16.1 Collapsing Dust and Black Hole Formation.- 16.2 Cosmology.- 16.2.1 The Dirac Equation in a Cosmological Background.- 16.2.2 Point Charge in a k > 0 Cosmology.- 16.3 Cosmic Strings.- 17 The Paravector Model of Spacetime.- 17.1 A Brief Introduction to the Pauli Algebra.- 17.1.1 Generating Cl3.- 17.1.2 Bivectors as Operators.- 17.1.3 Complex Structure.- 17.1.4 Involutions of Cl3.- 17.2 Inverses and the metric.- 17.3 The Spacetime Manifold.- 17.4 Lorentz Transformations I.- 17.5 Vector Notation and Rotations.- 17.5.1 The Merry-Go-Round.- 17.5.2 Observers, Frames, and Vector Bases.- 17.6 Lorentz Transformations II.- 17.6.1 Proper Velocity.- 17.6.2 Covariant vs. Invariant.- 17.6.3 Spacetime Diagrams.- 18 Eigenspinors in Electrodynamics.- 18.1 Basic Electrodynamics.- 18.2 Eigenspinors.- 18.3 The Group SL(2,C): Diagrams.- 18.4 Time Evolution of Eigenspinor.- 18.4.1 Thomas Precession.- 18.5 Spinorial Lorentz-Force Equation.- 18.5.1 Solutions.- 18.6 Electromagnetic Waves in Vacuum.- 18.6.1 Maxwell’s Equation in a Vacuum.- 18.7 Projectors.- 18.8 Directed Plane Waves.- 18.8.1 Polarization.- 18.9 Motion of Charges in Plane Waves.- 19 Eigenspinors in Quantum Theory.- 19.1 Introduction.- 19.2 Spin.- 19.2.1 Magic of the Pauli Hamiltonian.- 19.2.2 Classical Spin Distribution.- 19.2.3 Quantum Form.- 19.2.4 Stern-Gerlach Filter.- 19.2.5 Linear Combinations of Spatial Rotations.- 19.3 Covaxiant Eigenspinors.- 19.3.1 Generalized Unimodularity.- 19.3.2 Eigenspinor of an Elementary Particle.- 19.4 Differential-Operator Form.- 19.4.1 The Electromagnetic Gauge Field.- 19.4.2 Linearity and Superposition.- 19.5 Basic Symmetry Transformations.- 19.6 Relation to Standard Form.- 19.6.1 Weyl Spinors.- 19.6.2 Momentum Eigenstates.- 19.6.3 Standing Waves.- 19.6.4 Zitterbewegung.- 19.7 Hamiltonians.- 19.7.1 Stationary States.- 19.7.2 Landau Levels.- 19.8 Fierz Identities of Bilinear Covariants.- 20 Eigenspinors in Curved Spacetime.- 20.1 Ideals, Spinors, and Symplectic Spaces.- 20.2 Bispinors.- 20.3 Flagpoles and Flags.- 20.4 Spinor Pairs.- 20.5 Time Evolution.- 20.6 Bispinor Basis of C4.- 20.7 Twistors.- 20.8 Relation to SO+ (1,3).- 20.9 Spinors in Curved Spacetime.- 20.10 Conclusions.- 21 Spinors: Lorentz Group.- 21.1 Introduction.- 21.2 Lorentz Group.- 21.2.1 Lorentz Lie Algebra.- 21.2.2 Lorentz Group Representations.- 21.3 Summary.- 22 Spinors: Clifford Algebra.- 22.1 Introduction.- 22.2 Clifford Algebra.- 22.2.1 Complex Clifford Algebra.- 22.2.2 Automorphism Group.- 22.2.3 Lorentz Group Redux.- 22.2.4 Poincaré Group.- 22.2.5 Conformai Group.- 22.3 Summary.- 23 Genersd Relativity: An Overview.- 23.1 Introduction.- 23.2 Tensor Analysis.- 23.2.1 Vectors and Tensors.- 23.2.2 Affine Connection and Covariant Differentiation.- 23.2.3 Torsion.- 23.2.4 Parallel Transport and Curvature.- 23.2.5 Bianchi Identities.- 23.2.6 Metric.- 23.2.7 Contracted Bianchi Identities.- 23.3 General Relativity.- 23.3.1 The Principle of Equivalence and the Einstein Equation.- 23.3.2 Gravitational Action.- 24 Spinors in General Relativity.- 24.1 Introduction.- 24.2 Local Lorentz Invariance.- 24.2.1 Vierbeins.- 24.2.2 Local Lorentz Invariance.- 24.2.3 Covariant Derivative and Spin Connection.- 24.2.4 Lorentz Field Strength Tensor.- 24.2.5 Action and Field Equations.- 24.3 Spinors in Genral Relativity.- 24.3.1 Dirac Equation & The Clifford Algebra.- 24.3.2 Lagrangian and Field Equations.- 24.4 Summary and Conclusions.- 25 Hypergravity I.- 25.1 Introduction.- 25.2 Automorphism Invariance.- 25.2.1 Automorphism Group.- 25.2.2 Covariant Derivative and Drehbeins.- 25.2.3 Curvatures and Field Strength Tensors.- 25.2.4 Bianchi Identities.- 25.3 Discussion.- 26 Hypergravity II.- 26.1 Introduction.- 26.2 Lagrangian.- 26.2.1 Gauge Field Terms.- 26.2.2 Spinor Field Terms.- 26.2.3 Drehbein Terms.- 26.3 Field Equations.- 26.3.1 Drehbein Field Equations.- 26.3.2 Spinor Field Equations.- 26.3.3 Gauge Field Equations.- 26.3.4 Gravitational Field Equations.- 26.4 Einstein Gravity Recovered.- 26.5 Discussion.- 27 Properties of Clifford Algebras for Fundamental Particles.- 27.1 Building Blocks of a Gauge Model.- 27.1.1 Introduction.- 27.1.2 The Elements of the Algebra.- 27.1.3 Operations within the Algebra.- 27.2 Spinors in the Clifford Algebra Clp,q.- 27.2.1 Minimal Left Ideals of Clp,q.- 27.2.2 Spinors in Clp,q.- 27.2.3 Bar-Conjugate Spinors in Clp,q.- 27.2.4 The Spinor Norm in Clp,q.- 27.2.5 Left- and Right-Handed Spinors in Clp,q.- 27.3 Selecting a Higher-Dimensional Algebra for a Gauge Model.- 27.3.1 The Principles of the Model.- 27.3.2 A Model Based onCl1,6.- 27.3.3 A Gauge Model inCl1,6.- 28 The Extended Grassmann Algebra of R3.- 28.1 Introduction.- 28.2 Multivectors and pseudo-multivectors.- 28.3 Forms and pseudoforms.- 28.4 Linear spaces.- 28.5 Various products.- 28.6 Physical quantities.- 28.7 Quadratic spaces.- 28.8 Conclusion.- 29 Geometric Algebra: Applications in Engineering.- 29.1 Applications in Computer Vision.- 29.1.1 Projective Space and Projective Transformations.- 29.1.2 Geometric Invariance in Computer Vision.- 29.1.3 Motion and Structure from Motion.- 29.2 Applications in Robotics/Mechanisms.- 29.2.1 Screw Transformations.- 29.2.2 A simple robot arm.- 29.2.3 Dual Quaternions.- 29.3 Further Applications.- 30 Projective Quadrics, Poles, Polars, and Legendre Transformations.- 30.1 Introduction.- 30.2 The Legendre Transformation (1).- 30.3 Projective Spaces and Geometric Algebra.- 30.4 Quadrics, poles, and polars.- 30.5 The Legendre Transformation (2).- 30.6 Comments.- 31 Spacetime Algebra and Line Geometry.- 31.1 Introduction.- 31.2 Projective geometry.- 31.3 The null polarity belonging to R2,3.- 31.4 Line geometry in spacetime algebra.- 32 Generalizations of Clifford Algebra.- 32.1 Generalizations of Clifford Algebra.- 32.1.1 Introduction.- 32.2 Dimensions of Zero Extent.- 32.3 Bosonic Vectors.- 32.4 Higher Cycling Dimensions.- 32.5 Conclusion.- 33 Clifford Algebra Computations with Maple.- 33.1 Introduction.- 33.2 Basic chores.- 33.2.1 ‘cliterms’.- 33.2.2 ‘clisort’.- 33.2.3 ‘clicollect’.- 33.2.4 ‘reorder’.- 33.2.5 ‘scalarpart’.- 33.2.6 ‘vectorpart’.- 33.3 Ring operations in Clifford algebra and computation of a symbolic inverse.- 33.3.1 ‘LC’: left contraction by a vector and ‘wedge’ multiplication.- 33.3.2 Clifford multiplication ‘cmul’ or ‘&c’.- 33.3.3 ‘cinv’: symbolic inverse of a multivector.- 33.4 Clifford algebra automorphisms: grade involution, reversion and conjugation.- 33.4.1 Grade involution and reversion.- 33.4.2 Clifford conjugation and complex conjugation.- 33.5 Matrix representations.- 33.5.1 Left regular representations.- 33.5.2 Spinor representations in left minimal ideals.- 33.6 Clifford and exterior exponentiations.- 33.6.1 ‘cexp’: Clifford exponentiation.- 33.6.2 ‘wexp’: exterior exponentiation.- 33.7 Quaternions and three dimensional rotations.- 33.7.1 Quaternion type, conjugation, norm and inverse.- 33.7.2 ‘rot3d’: rotations in three dimensions.- 33.8 Octonions: type, conjugation, norm and inverse.- 33.9 Working with homomorphisms of algebras.