Geometric Algebra and Applications to Physics

Geometric Algebra and Applications to Physics

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Bringing geometric algebra to the mainstream of physics pedagogy, Geometric Algebra and Applications to Physics not only presents geometric algebra as a discipline within mathematical physics, but the book also shows how geometric algebra can be applied to numerous fundamental problems in physics, especially in experimental situations. This reference begins with several chapters that present the mathematical fundamentals of geometric algebra. It introduces the essential features of postulates and their underlying framework; bivectors, multivectors, and their operators; spinor and Lorentz rotations; and Clifford algebra. The book also extends some of these topics into three dimensions. Subsequent chapters apply these fundamentals to various common physical scenarios. The authors show how Maxwell's equations can be expressed and manipulated via space-time algebra and how geometric algebra reveals electromagnetic waves' states of polarization. In addition, they connect geometric algebra and quantum theory, discussing the Dirac equation, wave functions, and fiber bundles. The final chapter focuses on the application of geometric algebra to problems of the quantization of gravity. By covering the powerful methodology of applying geometric algebra to all branches of physics, this book provides a pioneering text for undergraduate and graduate students as well as a useful reference for researchers in the field.show more

Product details

  • Hardback | 184 pages
  • 157.5 x 236.2 x 17.8mm | 385.56g
  • Taylor & Francis Ltd
  • London, United Kingdom
  • English
  • 15 black & white illustrations
  • 1584887729
  • 9781584887720

Table of contents

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.show more