Fauser - Treatise on Quantum Clifford Algebras.pdf - Algebra - Mathematics2014

A Treatise on

Quantum Clifford Algebras

Habilitationsschrift

Dr. Bertfried Fauser

Universit at Konstanz

Fachbereich Physik

Fach M 678

78457 Konstanz

January 25, 2002

To Dorothea Ida

and Rudolf Eugen Fauser

B ERTFRIED F AUSER — U NIVERSITY OF K ONSTANZ

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 Graßmann-Cayley algebras including

co-meet and co-join for Graßmann-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

Fauser - Treatise on Quantum Clifford Algebras.pdf