Geometric Algebra and Applications to Physics

Geometric Algebra and Applications to Physics

Published:
Content:
Author(s):
Free Standard Shipping

Purchasing Options

Hardback
$119.95
ISBN 9781584887720
Cat# C7729
Add to cart
eBook
ISBN 9781584887737
Cat# CE7729
 

Features

  • Introduces the mathematical fundamentals of geometric algebra, including bivectors, multivectors, and spinor theory
  • Provides examples of geometric algebra applications to the polarization of electromagnetic waves, neutron interferometry in gravitational fields, fiber bundles, and quantum theory
  • Includes helpful information on complex numbers in geometric algebra formulations of electrodynamics and on plane-wave solutions to Maxwell's equations
  • Explores the basic aspects of intrinsic spin and charge conjugation
  • Features authors well known for their research in general relativity, particularly in the quantization of gravity
  • Summary

    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.

    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 ?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.

    Editorial Reviews

    "The book is written for physicists with at least two years’ study of university physics, but Part I develops Clifford algebra from a standing start and is suitable for new undergraduates. I hope it finds a market among them as Clifford’s language spreads. Those who persist will find that Part II is a treat in store."

    – Dr. A.J.M. Garrett, Scitext, in Contemporary Physics, August 2007, Vol. 48, No. 4

    Textbooks
    Other CRC Press Sites
    Featured Authors
    STAY CONNECTED
    Facebook Page for CRC Press Twitter Page for CRC Press You Tube Channel for CRC Press LinkedIn Page for CRC Press Google Plus Page for CRC Press
    Sign Up for Email Alerts
    © 2014 Taylor & Francis Group, LLC. All Rights Reserved. Privacy Policy | Cookie Use | Shipping Policy | Contact Us