The research program started by Gull and Lasenby on the physical applications of geometric algebras has been strengthened by the addition of Doran from DAMTP and 3 research students. For the academic year 1994/5 we have been fortunate to have Prof. David Hestenes (Arizona State University) visiting our group. Although the origins of geometric algebras date back to the work of Hamilton, Grassmann and Clifford in the 19th Century, Hestenes has been the pioneer of the field for the last 30 years.
During the period of this report, there have been 10 publications, 4 of which form a series in the `Foundations of Physics' to celebrate Hestenes' 60th birthday. The central theme is that the geometric algebra of spacetime -- `SpaceTime Algebra' -- provides a universal and natural language for theoretical physics. These papers include a tutorial review (Gull, Lasenby & Doran, 1993a), applications to quantum mechanics (Doran, Lasenby & Gull 1993a) and Dirac theory (Gull, Lasenby & Doran 1993b), and a streamlined calculus for Lagrangian field theory (Lasenby, Doran & Gull, 1993a).
The ideas have resulted in a new gauge theory of gravity, which is novel in that it employs gauge fields in a flat background spacetime. These fields ensure that intrinsic physical relations are independent of the position and orientation of matter fields. Some preliminary astrophysical and cosmological results have been published, and a more extended version is now in press (Lasenby, Doran & Gull, in press). In the gauge theory approach, there is an insistence on finding a global solution for the gauge fields. This alters the physical picture of the horizon around a black hole, and enables one to discuss, for example, the properties of electric field lines inside the horizon due to a charge held outside it. Another difference from General Relativity is revealed by solving the Dirac equation in a cosmological background. The only models compatible with complete spatial homogeneity are those at critical density.
Elsewhere, Doran et al. . (1993) have shown that every Lie algebra can be represented as a bivector algebra, hence every Lie group can be represented as a spin group. The power of geometric algebra is therefore now available to simplify the analysis of Lie groups and Lie algebras.