A SPACETIME ALGEBRA GUIDE TO OCTONIONS AND THE SYMMETRY GROUPS OF PARTICLE PHYSICS	

Anthony Lasenby	University of Cambridge	

In the context of a meeting bringing together people interested in applications of Geometric Algebra (GA) across wide areas of engineering and science, it is of interest to see if fairly elementary GA tools can help us understand an area of modern physics normally considered quite difficult conceptually, and which uses a whole set of advanced mathematical tools. Here we wish to talk about the use of GA in an area at the interface between mathematics and particle physics. This concerns the use of octonions to motivate the symmetry groups arising in the standard model of particle physics, and also its extensions to `unified theories' based upon groups with a larger set of symmetries and working in higher dimensions. Remarkably, we will be able to show that the octonions themselves, and the actions of these higher symmetry groups, can all be expressed using just the Spacetime Algebra (STA), i.e. the Geometric Algebra of 4d spacetime. This makes computations within e.g. the SU(3) group of quantum chromodynamics, and the representation of quark and gluon states, all accessible to someone with a computer algebra program able to work in the STA (or alternatively the CGA of 2d Euclidean space). Proceeding beyond this, we show that the same approach can be applied to the `Sedenions', allowing larger groups such as SO(10), and potentially E8, which have long been possible candidates for unification groups in particle physics, to be understood wholly in STA terms.  Overall, although the individual topics mentioned may sound somewhat esoteric or complicated for a general audience, there is a underlying aim of making some advanced aspects of particle physics approachable in a new way, and accessible to anyone who is familiar with the basics of GA and the STA.