On the Bundle of Clifford Algebras Over the Space of Quadratic Forms

Abstract

For each quadratic form $$Q\in \text{ Quad }(V)$$ on a vector space over a field $$\mathbb {K},$$ we can define the Clifford algebra $${{\,\textrm{Cl}\,}}(V,Q)$$ as the quotient $${{\,\textrm{T}\,}}(V)/I(Q)$$ of the tensor algebra $${{\,\textrm{T}\,}}(V)$$ by the two-sided ideal generated by expressions of the form $$x\otimes x-Q(x),\, x\in V.$$ In the present paper we consider the whole family $$\{{{\,\textrm{Cl}\,}}(V,Q):\, Q\in \text{ Quad }(V)\}$$ in a geometric way as a $$\mathbb {Z}_2$$ -graded vector bundle over the base manifold $$\text{ Quad }(V).$$ Bilinear forms $$F\in \text{ Bil }(V)$$ act on this bundle providing natural bijective linear mappings $$\bar{\lambda }_F$$ between different Clifford algebras $${{\,\textrm{Cl}\,}}(V,Q).$$ Alternating (or antisymmetric) forms induce vertical automorphisms, which we propose to consider as ‘gauge transformations’. We develop here the formalism of Bourbaki, which generalizes the well known Chevalley’s isomorphism $${{\,\textrm{Cl}\,}}(V,Q)\rightarrow {{\,\textrm{End}\,}}(\bigwedge (V))\rightarrow \bigwedge (V).$$ In particular we realize the Clifford algebra twisting gauge transformations induced by antisymmetric bilinear forms as exponentials of contractions with elements of $$\bigwedge ^2(V^*)$$ representing these forms. Throughout 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 $${{\,\textrm{Cl}\,}}(Q)$$ can be naturally realized as acting via endomorphisms of any other Clifford algebra $${{\,\textrm{Cl}\,}}(Q')$$ if $$Q'=Q+Q_F,\, F\in \text{ Bil }(V)$$ and $$Q_F(x)=F(x,x).$$ Possible physical meaning of such transformations is also mentioned.