Yes, We Have No Imaginary Numbers: A Gentle Introduction to Clifford Algebra

Matthew R. Francis

What would you say to the idea that you don't really know how to multiply vectors together? That imaginary numbers are not real? That adding together vectors and scalars is not only permissible, but a good idea?

This tossed salad of unusual ideas is the Clifford algebra, which is an extension of the vector algebra we know and love, but which also incorporates complex numbers, the Grassmann algebra (useful in fermionic systems), and some other more esoteric features into a single formalism. So far, it has found applications in such varied realms as quantum field theory, classical electromagnetics, general relativity, crystallography, statistical physics, and function theory. However, in this talk, I will focus on the basic concepts of the algebra itself: how it works, and how various mathematical objects we need in physics translate into its language.

So this is not a condensed matter talk! Nevertheless, I will try to keep the arguments as physical as possible, and emphasize applicability at all times. I hope to see you all there.

References:

Clifford Algebras in General

• S. Gull, A. Lasenby, and C. Doran, "Imaginary Numbers are Not Real---the Geometric Algebra of Spacetime".
  This is a piece of propaganda, so read it with a skeptical mind! However, it is an introductory paper intended for a general physics audience with an undergraduate's understanding of mathematics. You could get through this one in twenty minutes.
• P. Lounesto, Clifford Algebras and Spinors (Cambridge University Press, 1997).
  This book is a fairly gentle introduction to Clifford algebras by a mathematician. In spite of a lot of fairly technical language, the book is readable even by a non-mathematician such as myself. You could select worse books to begin your studies.
• D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (D. Reidel, 1984).
  NOT for beginners or the faint of heart! This is not a physics book, whatever its name implies. However, this hefty book covers a lot of ground, and may answer some of the more pressing questions raised over the course of my talk (i.e., analycity and tensor representations).

• D. Hestenes, Space-Time Calculus.
  David Hestenes' modern musings on relativistic field theory. This is a fairly long, detailed paper that could serve as a review article, if it were ever published in a place other than the Web.
• A. Lasenby, C. Doran, and S. Gull, Grassmann Calculus, Pseudoclassical Mechanics, and Geometric Algebra.
  The paper I recently discovered, which covers a formulation of the Grassmann algebra in terms of the Clifford algebra. As I said, I'm not completely satisfied with this paper, but one must begin somewhere.
• A. Lasenby, C. Doran, and S. Gull, A Multivector Derivative Approach to Lagrangian Field Theory.
  This paper covers generalized Lagrangians, which are not necessarily scalar entities, and an extension of Noether's theorem. I haven't had time to go through this one in detail, but it looks promising.
• M. R. Francis, Spinor Geometry Revisited, Part I: Spinet Algebra and Calculus.
  This is a work in progress, so judge it gently, dear friends, judge it gently! Spinors are a weak point in the usual development of Clifford algebras (in my opinion), so this is my attempt to incorporate spinor geometry as developed by Cartan, van der Waerden, and Penrose into the family.

In addition, see my Clifford algebra reference page for links to research groups.