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 (1 + ?0).- 2.3.7 Ideal Spinors ø ? Cl1,3 (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 - Fiel