On the bundle of Clifford algebras over the space of quadratic forms
arXiv:2103.09767 · doi:10.1007/s00006-022-01251-x
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 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.
42 pages, Paper presented at the 12th International Conference on Clifford Algebras and their Applications in Mathematical Physics, Hefei, China, August 3-7, 2020. A bunch of misprints of the previous version corrected. Several formulations improved