Fauser - Treatise on Quantum Clifford Algebras

Fauser - Treatise on Quantum Clifford Algebras -...

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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 Gramann-Cayley algebras including co-meet and co-join for Gramann-Cayley co-gebras which are very efficient and may be used in Robotics, left and right contractions, left and right co-contractions, Clifford and co-Clifford 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 bi-convolution. Antipode and crossing are consequences of the product and co-product structure tensors and not subjectable to a choice. A frequently used Kuperberg lemma is revisited necessitating the def- inition of non-local products and interacting Hopf gebras which are generically non-perturbative. A spinorial generalization of the antipode is given. The non- existence of non-trivial integrals in low-dimensional Clifford co-gebras is shown. Generalized cliffordization is discussed which is based on non-exponentially gen- erated bilinear forms in general resulting in non unital, non-associative products. Reasonable assumptions lead to bilinear forms based on 2-cocycles. Cliffordiza- tion is used to derive time- and normal-ordered generating functionals for the Schwinger-Dyson hierarchies of non-linear 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; 15-02 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 Gramann-Cayley 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 Gramann-Cayley algebras . . . . . . . . . . . . . . . . . . . . . 5 1.4.1 The bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4.2 The wedge product join . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4.3 The vee-product meet . . . . . . . . . . . . . . . . . . . . . . . . . .....
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