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 19^{th} 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].

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.

Last Modified 17 January 2001

Maintained by Chris Doran.