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

DOI10.1007/S00006-022-01251-XarXiv2103.09767MaRDI QIDQ6363095

A. Jadczyk

Publication date: 17 March 2021

Abstract: For each quadratic form $QinmboxQuad(V)$ over a given vector space over a field $mathbbR$ 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 $xotimesx-Q(x),,xinV.$ In the present paper we consider the whole family $Cl(V,Q),,QinmboxQuad(V)$ in a geometric way as a $Z_2$-graded vector bundle over the base manifold $mboxQuad(V).$ Bilinear forms $FinmboxBil(V)$ act on this bundle providing natural bijective linear mappings $$ between Clifford algebras for different $Cl(V,Q).$ Alternate (or antisymmetric) forms induce vertical automorphisms, which we propose to consider as 'gauge transformations'. We develop here the formalism of N. Bourbaki, which generalizes the well known Chevalley's isomorphism $$ In particular we realize the Clifford algebra twisting gauge trnsformations induced by antisymmetric bilinear forms as exponentials of contractions with elements of $$ 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 $Cl(Q^{\prime })$ if $Q^{\prime }=Q+Q_F,,FinmboxBil(V)$ and $Q_F(x)=F(x,x).$ Possible physical meaning of such transformations are also mentioned.

Mathematics Subject Classification ID

Clifford algebras, spinors (15A66) Quadratic and bilinear forms, inner products (15A63) Exterior algebra, Grassmann algebras (15A75) Deformations of fiber bundles (32G08)

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