de Sabbata / Datta Geometric Algebra and Applications to Physics

Erscheinungsjahr 2006
ISBN: 978-1-58488-773-7
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 184 Seiten

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.

Zielgruppe

Students, mathematicians, and physicists.

Autoren/Hrsg.

de Sabbata, Venzo

Datta, Bidyut Kumar

Fachgebiete

• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Computeranwendungen in der Mathematik
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle

Inhaltsverzeichnis

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 ?µ
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-à-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.