Geometric Algebra and Applications to Physics
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.
- Electronic book text | 184 pages
- 07 Dec 2006
- Taylor & Francis Inc
- CRC Press Inc
- Florida, United States
- 15 Illustrations, black and white
Table of contents
THE BASIS FOR GEOMETRIC ALGEBRAIntroduction Genesis of Geometric AlgebraMathematical Elements of Geometric Algebra Geometric Algebra as a Symbolic SystemGeometric Algebra as an Axiomatic System (Axiom A) Some Essential Formulas and DefinitionsMULTIVECTORS Geometric Product of Two Bivectors A and BOperation of Reversion Magnitude of a MultivectorDirections and Projections Angles and Exponential Functions (as Operators)Exponential Functions of MultivectorsEUCLIDEAN PLANE The Algebra of Euclidean PlaneGeometric Interpretation of a Bivector of Euclidean PlaneSpinor i-PlaneDistinction between Vector and Spinor PlanesThe Geometric Algebra of a PlaneTHE PSEUDOSCALAR AND IMAGINARY UNITThe Geometric Algebra of Euclidean 3-Space Complex ConjugationAppendix: Some Important ResultsREAL DIRAC ALGEBRAGeometric Significance of the Dirac Matrices ? Geometric Algebra of Space-TimeConjugationsLorentz RotationsSpinor Theory of Rotations in Three-Dimensional Euclidean SpaceSPINOR AND QUATERNION ALGEBRASpinor Algebra: Quaternion AlgebraVector AlgebraClifford Algebra: Grand Synthesis of Algebra of Grassmann and Hamilton and the Geometric Algebra of HestenesMAXWELL EQUATIONSMaxwell Equations in Minkowski Space-TimeMaxwell 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 WavesQuaternion Form of Maxwell Equations from the Spinor Form of STAMaxwell Equations in Vector Algebra from the Quaternion (Spinor) Formalism Majorana-Weyl Equations from the Quaternion (Spinor) Formalism of Maxwell EquationsAppendix A: Complex Numbers in ElectrodynamicsAppendix B: Plane-Wave Solutions to Maxwell Equations-Polarization of EM WavesGENERAL OBSERVATIONS AND GENERATORS OF ROTATIONS (NEUTRON INTERFEROMETER EXPERIMENT)Review of Space-Time Algebra (STA)The Dirac Equation without Complex NumbersObservables and the Wave FunctionGenerators of Rotations in Space-Time: Intrinsic SpinFiber Bundles and Quantum Theory vis-a-vis the Geometric Algebra ApproachFiber Bundle Picture of the Neutron Interferometer ExperimentCharge ConjugationAppendix QUANTUM GRAVITY IN REAL SPACE-TIME (COMMUTATORS AND ANTICOMMUTATORS)Quantum Gravity and Geometric AlgebraQuantum Gravity and TorsionQuantum Gravity in Real Space-TimeA Quadratic HamiltonianSpin FluctuationsSome Remarks and ConclusionsAppendix: Commutator and AnticommutatorINDEXReferences appear at the end of each chapter.