On the bundle of Clifford algebras over the space of quadratic forms

TL;DR

This paper explores the geometric structure of Clifford algebras over quadratic forms, introducing gauge transformations via antisymmetric forms and formalizing their algebraic relations without relying on the traditional 'quantum' terminology.

Contribution

It develops a geometric formalism for Clifford algebra families as vector bundles, introduces gauge transformations induced by antisymmetric forms, and generalizes Chevalley's isomorphism without using the standard 'quantum' Clifford algebra terminology.

Findings

Clifford algebras form a Z2-graded vector bundle over quadratic forms.

Antisymmetric bilinear forms induce gauge transformations as exponentials of contractions.

Clifford algebra actions can be realized through endomorphisms of other Clifford algebras.

Abstract

For each quadratic form Q in Quad(V) over a given vector space over a field R we have the Clifford algebra Cl(V,Q) defined as the quotient T(V)/I(Q) of the tensor algebra T(V) over the two-sided ideal generated by expressions of the form $x x-Q(x),x in V. In the present paper we consider the whole family Cl(V,Q) in a geometric way as a Z2-graded vector bundle over the base manifold Quad(V). Bilinear forms F from Bil(V) act on this bundle providing natural bijective linear mappings lambda_F between Clifford algebras for different Cl(V,Q). Alternate (or antisymmetric) forms induce vertical automorphisms, which we propose to consider as 'gauge transformations'. We develop here the formalism of N. Bourbaki, which generalizes the well known Chevalley's isomorphism Cl(V,Q)->End(Wedge(V)->Wedge(V). In particular we realize the Clifford algebra twisting gauge trnsformations induced by antisymmetric bilinear forms as exponentials of contractions with elements of $Wedge(V^* representing these forms. Throughtout all this paper we intentionally avoid using the so far accepted term "Clifford algebra of a bilinear form" (known otherwise as "Quantum Clifford algebra"), which we consider as possibly misleading, as it does not represent any well defined mathematical object. Instead we show explicitly how any given Clifford algebra Cl(Q) can be naturally realized as acting via endomorphisms of any other Clifford algebra Cl(Q') if Q'=Q+Q_F,and Q_F(x)=F(x,x). Possible physical meaning of such transformations are also mentioned.