Spinor representations of Clifford algebras: a symbolic approach

Abstract

Clifford algebras Cℓ( B) of an arbitrary, not necessarily symmetric, bilinear form B provide an important computational tool for physicists and an interesting mathematical object to study. In this paper we explain step by step how to compute spinor representations of real Clifford algebras Cℓ( Q) of the quadratic form Q, Q(x) = B(x,x), with a new version of CLIFFORD, a Maple package for computations with Clifford algebras of an arbitrary bilinear form. New procedures in the package follow standard mathematical theory of such representations. When Cℓ( Q) is simple (resp. semisimple) its spinor representation is realized faithfully in a minimal left ideal S = Cℓ( Q) f (resp. S ⊕ S ̂ ) for some primitive idempotent |. The ideal S (resp. S ⊕ S ̂ ) is a right K -module (resp. K ⊕ Ǩ-module) where K is a subalgebra of Cℓ( Q) isomorphic with R, C or H depending on the dimension of V and the signature of Q. We show examples how gamma matrices with entries in K representing basis one-vectors of Cℓ( Q) and scalar products on spinor spaces S for various signatures of Q may be derived with CLIFFORD.