Physical Applications of Geometric Algebra

Synopsis

This course introduces a new approach to solving problems in physics based on the mathematical language of Geometric Algebra. Geometric Algebra was discovered in the 19th Century by the Cambridge mathematician W.K. Clifford and has been attracting much interest recently as a tool for unifying the study of a wide range of topics in physics. The first part of the course introduces Geometric Algebra and illustrates its use with some problems in non-relativistic physics. These include orbital motion, rigid-body dynamics and quantum spin. The second half of the course studies applications in relativistic physics. These are built on the geometric algebra of spacetime - the spacetime algebra or STA. This will be introduced to solve problems in relativistic electrodynamics and point particle mechanics, which provide a solid set of applications to build on. A look at relativistic quantum mechanics then paves the way for the introduction of gravitation as a gauge theory in the STA. This provides an alternative route to the Einstein equations to that of traditional General Relativity, and establishes a clear link between gravitation and gauge theories of the other fundamental forces. This will be of particular interest to students taking either the General Relativity or Gauge Theory courses, though knowledge of either will not be assumed.

Consistent application of a common language aids appreciation of the links between fields, and a strong emphasis is placed on encouraging inter-disciplinarity. This course is intended for the more theoretically minded, interested in ideas and applications that are not covered in other courses, but which are relevant to current research.

NON-RELATIVISTIC PHYSICS

Some history and the course aims; the geometric product, planes and bivectors; complex numbers and the Geometric Algebra of space; reflections, rotations, elliptic orbits and rigid body dynamics [4].

Spinors and quantum spin; magnetic resonancs [2].

RELATIVISTIC PHYSICS

An introduction to Spacetime Algebra; the equations of motion for a point particle, acceleration as a bivector and a rotor form of the equations of motion; applications in electrodynamics and particles with spin; the field due to a point charge and synchrotron radiation [5].

GAUGE THEORIES AND GRAVITATION

The Dirac equation, minimal coupling and gauge theories; gauge principles for gravitation and the field equations; curvature viewpoint versus the gauge viewpoint; the Newtonian gauge for spherical systems - the Schwarzschild solution and cosmology; the geodesic equation and its rotor form [5].

Reading List

A comprehensive set of handouts will be provided by the lecturers. Useful additional sources include:

Space-time Algebra by D Hestenes (Gordon and Breech 1966 )

New Foundations for Classical Mechanics D Hestenes (Reidel 1985 )

Clifford Algebra to Geometric Calculus by D Hestenes and G. Sobczyk (Reidel 1984 )

Clifford (Geometric) Algebras WE Baylis, Editor (Birkhauser 1996)

Multivectors and Clifford Algebra in Electrodynamics by B Jancewicz (World Scientific 1989)

A number of further papers can be obtained from the geometric algebra group publications page.

[ Cavendish | Geometric Algebra | Lecture Course ]


Last Modified 17 January 2001
Maintained by Chris Doran.