On the bundle of Clifford algebras over the space of quadratic forms
Arkadiusz Jadczyk
备注：
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
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.
The Bundles of Algebraic and Dirac-Hestenes Spinor Fields
The main objective of this paper is to clarify the ontology of Dirac-Hestenes spinor fields (DHSF) and its relationship with sum of even multivector fields, on a general Riemann-Cartan spacetime admitting a spin structure and to give a mathematically rigorous derivation of the so called Dirac-Hestenes equation (DHE) when spacetime is a Lorentzian. To this aim we introduce the Clifford bundle of multivector fields (Cl(M,g)) and the left (Cl_{ Spin_{1,3} ^{e}}^{l}(M)) and right (Cl_{ Spin_{1,3} ^{e}}^{r}(M)) spin-Clifford bundles on the spin manifold (M,g) The relation between left ideal algebra…
On the Transposition Anti-Involution in Real Clifford Algebras III: The Automorphism Group of the Transposition Scalar Product on Spinor Spaces
在[arXiv.1005.3554v1]中引入了实Clifford代数Cl_{p,q}对非退化二次型的一个依赖于（p，q）的转置反对合T.在[arXiv.1005.3558v1]中，我们证明了，依赖于（p-q）mod 8的值，映射T产生旋量表示中表示矩阵的转置、复厄米特或四元数厄米特共轭。由此产生的标量积一般不同于两个已知的标准标量积[Lounesto，Clifford algebras and Spinors 2001]。我们提供了一个完整的签名（p，q）相关的分类的不变性组Gpq_{p,q}的这个产品的p+q <= 9。映射T被认为是由（扭）群代数理论[Passmann，The Algebraic Structure of Group Rings 1985]所知的“星”映射，其中Clifford代数Cl_{p,q}被看作是扭群环k^t[（Z_2）^n]，n=p+q。讨论并列出了稳定群Gpq（f）_{p，q}的重要子群及其横截与旋量空间生成元的关系.
