Clifford Algebras: An Introduction

Let E be a d-dimensional real vector space, and let q be a quadratic form on E with associated bilinear form b. Let A be a unital algebra, and suppose j maps E to A such that the image, j(E), avoids A’s unit and the square of j coincides with –q. If these conditions are met, j is called a Clifford mapping. If, additionally, the image j(E) generates A then the data (A,j) is the Clifford algebra for (E,q).

We learn all this on pp. 86–87 of the book under review, right after a historical prelude that addresses William Kingdom Clifford’s 1876 communication to the London Mathematical Society showing

how to modify the definition of the exterior algebra of an n-dimensional real inner product space [so as] to take the inner product into account … [H]e showed that the resulting algebra … has dimension 2ⁿ … and … can be decomposed into odd and even parts … Clifford’s definition extends easily to general quadratic spaces…

This is of course exceedingly exiting, given the huge importance played by quadratic forms in mathematics in general, as well as in algebra proper, of course. And this broad utility extends to Clifford algebras: as we read in the Introduction,

Clifford algebras find their use in many areas …[e.g,] in differential analysis, where operators of Dirac type are used in proofs of the Atiyah-Singer index theorem, in harmonic analysis, where the Riesz transforms provide a higher-dimensional generalization of the Hilbert transform, in geometry, where spin groups illuminate the structure of the classical groups, and in mathematical physics, where Clifford algebras provide a setting for electromagnetic theory, spin ½ particles, and the Dirac operator in relativistic quantum mechanics.

Fantastic! What a span of topics! Manifestly the subject of Clifford algebras needs no further justification.

The book under review, itself an Introduction, is another entry in the excellent series of Student Texts offered by the London Mathematical Society, and doesn’t disappoint. Garling starts slowly, with a review of groups, vector spaces, some representation theory, and multilinear algebra, with his discussion on p. 38 ff. of tensor products being particularly praiseworthy because of its emphasis on calculations and manipulations (all too often glossed over or suppressed). This meritorious focus on “getting your hands dirty,” to harken back to a popular phrase of my student days, remains in place in what follows, viz. the book’s second part, “Quadratic forms and Cliffordf algebras.” After doing proper justice to quadratic forms — and, by the way, Garling’s postscript comment (on p. 186) that “Lam is the standard work on quadratic forms” does my heart good: this is indeed a wonderful book! — Garling hits Clifford algebras proper and proceeds with the classification question and the matter of representing them; and after that it’s on to the subject of spin.

This all sets the stage for the third and final part of the book, “Some Applications,” which is itself split into three parts. The first concerns physics: Dirac is featured, of course, and we encounter Maxwell’s equations, too. The second concerns Clifford analyticity and touches on such topics as Cauchy’s integral formula extended to Clifford analytic functions, as well as the theme of “how Clifford algebras and augmented Dirac operators can be used to extend [e.g., the Riesz’ brothers’ theorem that states that “if the harmonic Dirichlet extension od a complex measure is analytic, then the measure is absolutely continuous”] to higher dimensions” (where one uses, instead of the Hilbert transform, a system of Riesz transforms). Finally, the third concerns Lie theory, i.e., “Representations of Spin_d and SO(d).”

Clifford Algebras: An Introduction is well-written and very accessible. Garling provides a decent number of exercises, and his closing section, “Some suggestions for further reading,” is very useful and points the reader in all sorts of interesting directions. This book certainly lives up the high standard set by LMS: it’s a clear winner.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.