Geometric Algebra and Applications to Physics

ISBN: 9781584887720 出版年：2006 页码：186 de Sabbata, Venzo Datta, Bidyut Kumar CRC Press

THE BASIS FOR GEOMETRIC ALGEBRA Introduction Genesis of Geometric Algebra Mathematical Elements of Geometric Algebra Geometric Algebra as a Symbolic System Geometric Algebra as an Axiomatic System (Axiom A) Some Essential Formulas and Definitions MULTIVECTORS Geometric Product of Two Bivectors A and B Operation of Reversion Magnitude of a Multivector Directions and Projections Angles and Exponential Functions (as Operators) Exponential Functions of Multivectors EUCLIDEAN PLANE The Algebra of Euclidean Plane Geometric Interpretation of a Bivector of Euclidean Plane Spinor i-Plane Distinction between Vector and Spinor Planes The Geometric Algebra of a Plane THE PSEUDOSCALAR AND IMAGINARY UNIT The Geometric Algebra of Euclidean 3-Space Complex Conjugation Appendix: Some Important Results REAL DIRAC ALGEBRA Geometric Significance of the Dirac Matrices ? Geometric Algebra of Space-Time Conjugations Lorentz Rotations Spinor Theory of Rotations in Three-Dimensional Euclidean Space SPINOR AND QUATERNION ALGEBRA Spinor Algebra: Quaternion Algebra Vector Algebra Clifford Algebra: Grand Synthesis of Algebra of Grassmann and Hamilton and the Geometric Algebra of Hestenes MAXWELL EQUATIONS Maxwell Equations in Minkowski Space-Time Maxwell Equations in Riemann Sace-Time (V4 Manifold) Maxwell Equations in Riemann-Cartan Space-Time (U4 Manifold) Maxwell Equations in Terms of Space-Time Algebra (STA) ELECTROMAGNETIC FIELD IN SPACE AND TIME (POLARIZATION OF ELECTROMAGNETIC WAVES) Electromagnetic (EM) Waves and Geometric Algebra Polarization of Electromagnetic Waves Quaternion Form of Maxwell Equations from the Spinor Form of STA Maxwell Equations in Vector Algebra from the Quaternion (Spinor) Formalism Majorana-Weyl Equations from the Quaternion (Spinor) Formalism of Maxwell Equations Appendix A: Complex Numbers in Electrodynamics Appendix B: Plane-Wave Solutions to Maxwell Equations-Polarization of EM Waves GENERAL OBSERVATIONS AND GENERATORS OF ROTATIONS (NEUTRON INTERFEROMETER EXPERIMENT) Review of Space-Time Algebra (STA) The Dirac Equation without Complex Numbers Observables and the Wave Function Generators of Rotations in Space-Time: Intrinsic Spin Fiber Bundles and Quantum Theory vis-a-vis the Geometric Algebra Approach Fiber Bundle Picture of the Neutron Interferometer Experiment Charge Conjugation Appendix QUANTUM GRAVITY IN REAL SPACE-TIME (COMMUTATORS AND ANTICOMMUTATORS) Quantum Gravity and Geometric Algebra Quantum Gravity and Torsion Quantum Gravity in Real Space-Time A Quadratic Hamiltonian Spin Fluctuations Some Remarks and Conclusions Appendix: Commutator and Anticommutator INDEX References appear at the end of each chapter.