Clifford algebra

A Clifford algebra is an associative algebra generated by a vector space equipped with a quadratic form, subject to the relation v² = Q(v)·1 for every vector v, where Q is the quadratic form. Introduced by William Kingdon Clifford in 1878, these algebras generalize complex numbers, quaternions, and the exterior algebra, and they provide the algebraic foundation for spin geometry and the Dirac operator. The Clifford algebra Cl(V, Q) of a real vector space V with a positive-definite quadratic form is isomorphic to a matrix algebra whose structure depends only on the dimension and signature of V modulo 8 — a periodicity known as Bott periodicity that connects Clifford algebras to K-theory and topological periodicity in a deep and still not fully understood way.

The importance of Clifford algebras extends far beyond their role as a technical tool for physicists. They encode the geometry of rotations and reflections, and their representation theory classifies the possible types of spinors that can exist on a manifold. A spinor is not merely a vector with extra components; it is an element of a representation of the spin group, the double cover of the special orthogonal group, and this double cover is visible only through the lens of Clifford algebra. Without Clifford algebras, the Dirac operator would be an unmotivated construction; with them, it is the natural first-order operator on a space that knows how to rotate.