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Unformatted text preview: arXiv:math.QA/0202059 v1 7 Feb 2002 A Treatise on Quantum Clifford Algebras Habilitationsschrift Dr. Bertfried Fauser Universitat Konstanz Fachbereich Physik Fach M 678 78457 Konstanz January 25, 2002 To Dorothea Ida and Rudolf Eugen Fauser BERTFRIED FAUSER UNIVERSITY OF KONSTANZ I ABSTRACT: Quantum Clifford Algebras (QCA), i.e. Clifford Hopf gebras based on bilinear forms of arbitrary symmetry, are treated in a broad sense. Five al ternative constructions of QCAs are exhibited. Grade free Hopf gebraic product formulas are derived for meet and join of GramannCayley algebras including comeet and cojoin for GramannCayley cogebras which are very efficient and may be used in Robotics, left and right contractions, left and right cocontractions, Clifford and coClifford products, etc. The Chevalley deformation, using a Clif ford map, arises as a special case. We discuss Hopf algebra versus Hopf gebra , the latter emerging naturally from a biconvolution. Antipode and crossing are consequences of the product and coproduct structure tensors and not subjectable to a choice. A frequently used Kuperberg lemma is revisited necessitating the def inition of nonlocal products and interacting Hopf gebras which are generically nonperturbative. A spinorial generalization of the antipode is given. The non existence of nontrivial integrals in lowdimensional Clifford cogebras is shown. Generalized cliffordization is discussed which is based on nonexponentially gen erated bilinear forms in general resulting in non unital, nonassociative products. Reasonable assumptions lead to bilinear forms based on 2cocycles. Cliffordiza tion is used to derive time and normalordered generating functionals for the SchwingerDyson hierarchies of nonlinear spinor field theory and spinor electro dynamics. The relation between the vacuum structure, the operator ordering, and the Hopf gebraic counit is discussed. QCAs are proposed as the natural language for (fermionic) quantum field theory. MSC2000: 16W30 Coalgebras, bialgebras, Hopf algebras; 1502 Research exposition (monographs, survey articles); 15A66 Clifford algebras, spinors; 15A75 Exterior algebra, Grassmann algebra; 81T15 Perturbative methods of renormalization II A Treatise on Quantum Clifford Algebras Contents Abstract I Table of Contents II Preface VII Acknowledgement XII 1 Peano Space and GramannCayley Algebra 1 1.1 Normed space normed algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Hilbert space, quadratic space classical Clifford algebra . . . . . . . . . . . . . 3 1.3 Weyl space symplectic Clifford algebras (Weyl algebras) . . . . . . . . . . . . 4 1.4 Peano space GramannCayley algebras . . . . . . . . . . . . . . . . . . . . . 5 1.4.1 The bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4.2 The wedge product join . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4.3 The veeproduct meet . . . . . . . . . . . . . . . . . . . . . . . . . .....
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