On the Decomposition of Clifford Algebras of Arbitrary Bilinear Form

Abstract

Clifford algebras are naturally associated with quadratic forms. These algebras are ℤ2 -graded by construction. However, only a ℤn -gradation induced by a choice of a basis, or even better, by a Chevalley vector space isomorphism Cl(V) ↔ Λ V and an ordering, guarantees a multi-vector decomposition into scalars, vectors, tensors, and so on, mandatory in physics. We show that the Chevalley isomorphism theorem cannot be generalized to algebras if the ℤn-grading or other structures are added, e.g., a linear form. We work with pairs consisting of a Clifford algebra and a linear form or a ℤn -grading which we now call Clifford algebras of multi-vectors or quantum Clifford algebras. These quantum Clifford algebras are in fact Clifford algebras of a bilinear form in a functorial way. It turns out that in this sense, all multi-vector Clifford algebras of the same quadratic but different bilinear forms are non-isomorphic. The usefulness of such algebras in quantum field theory and superconductivity was shown elsewhere. Allowing for arbitrary bilinear forms however spoils their diagonaliz-ability which has a considerable effect on the tensor decomposition of the Clifford algebras governed by the periodicity theorems, including the Atiyah-Bott-Shapiro mod 8 periodicity. We consider real algebras Clp,q which can be decomposed in the symmetric case into a tensor product Clp-1,q-1 ⊕ Cl1,1. The general case used in quantum field theory lacks this feature. Theories with non-symmetric bilinear forms are however needed in the analysis of multi-particle states in interacting theories. A connection to q -deformed structures through nontrivial vacuum states in quantum theories is outlined.