Like the classification of Lie groups, the classification of Clifford algebras is a topic that is helpful, but not required, in understanding most of theoretical physics. However, Clifford algebras and related constructs such as spinors are central to many modern physical theories, and so are worth exploring in detail.

Recall that the Clifford algebra over a given \({n}\)-dimensional real vector space \({V}\) with a pseudo inner product is defined to be the tensor algebra modulo the identification \({vv\equiv\left\langle v,v\right\rangle }\). The isomorphism classes of such Clifford algebras are then determined by the signature of the associated inner product, which we denote \({C(r,s)}\).

Δ Notation for Clifford algebras varies widely; in particular, \({r}\) and \({s}\) in our above notation are sometimes reversed, and \({C(n)}\) sometimes refers to either \({C(0,n)}\) or \({C(n,0)}\). |