  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.
