Clifford (Geometric) Algebras: With Applications to Physics, Mathematics, and Engineering

ISBN-13: 9781461286547 / Angielski / Miękka / 2011 / 517 str.

William E. Baylis

Opis

This volume is an outgrowth of the 1995 Summer School on Theoretical Physics of the Canadian Association of Physicists (CAP), held in Banff, Alberta, in the Canadian Rockies, from July 30 to August 12,1995. The chapters, based on lectures given at the School, are designed to be tutorial in nature, and many include exercises to assist the learning process. Most lecturers gave three or four fifty-minute lectures aimed at relative novices in the field. More emphasis is therefore placed on pedagogy and establishing comprehension than on erudition and superior scholarship. Of course, new and exciting results are presented in applications of Clifford algebras, but in a coherent and user-friendly way to the nonspecialist. The subject area of the volume is Clifford algebra and its applications. Through the geometric language of the Clifford-algebra approach, many concepts in physics are clarified, united, and extended in new and sometimes surprising directions. In particular, the approach eliminates the formal gaps that traditionally separate clas sical, quantum, and relativistic physics. It thereby makes the study of physics more efficient and the research more penetrating, and it suggests resolutions to a major physics problem of the twentieth century, namely how to unite quantum theory and gravity. The term "geometric algebra" was used by Clifford himself, and David Hestenes has suggested its use in order to emphasize its wide applicability, and b& cause the developments by Clifford were themselves based heavily on previous work by Grassmann, Hamilton, Rodrigues, Gauss, and others."

Recenzje

"Of interest due to the many provocative physical interpretations of quantum mechanics and gravitational theory suggested by the Clifford algebra approach to these theories."

-Mathematical Reviews

Spis treści

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.