Projective Relativity: Present Status and Outlook

We give a critical analysis of projective relativity theory. Examining Kaluza's own intention and the following development by Klein, Jordan, Pauli, Thiry, Ludwig and others, we conclude that projective relativity was abused in its own terms and much more in the case of newer higher dimensional Kaluza–Klein theories with non-Abelian gauge groups. Reviewing the projective formulation of the Jordan isomorphy theorem yields some hints how one can proceed in a different direction. We can interpret the condition not as a field equation in a 5-dimensional Riemannian space, e.g. as vacuum Einstein-Hilbert equation, but can (or should) interpret it as a geometrical object, a null-quadric. Projective aspects of quantum (field) theory are discussed under this viewpoint.     

On the transposition anti-involution in real Clifford algebras I: the transposition map

A particular orthogonal map on a finite-dimensional real quadratic vector space (V,?Q) with a non-degenerate quadratic form Q of any signature (p,?q) is considered. It can be viewed as a correlation of the vector space that leads to a dual Clifford algebra C?(V*,?Q) of linear functionals (multiforms) acting on the universal Clifford algebra C?(V,?Q). The map results in a unique involutive automorphism and a unique involutive anti-automorphism of C?(V,?Q). The anti-involution reduces to reversion (resp. conjugation) for any Euclidean (resp. anti-Euclidean) signature. When applied to a general element of the algebra, it results in transposition of the element matrix in the left regular representation of C?(V,?Q). We also give an example for real spinor spaces. The general setting for spinor representations will be treated in part II of this work [R. Ab?amowicz and B. Fauser, On the transposition anti-involution in real Clifford algebras II: Stabilizer groups of primitive idempotents, Linear Multilinear Algebra, to appear].     

Idempotents of Clifford Algebras

A classification of idempotents of Clifford algebras C ^p,q is presented. It is shown that using isomorphisms between Clifford algebras C ^p,q and appropriate matrix rings, it is possible to classify idempotents in any Clifford algebra into continuous families. These families include primitive idempotents used to generate minimal one-sided ideals in Clifford algebras. Some low-dimensional examples are discussed.     

Focussing of laser beams by means of a z-pinch formed plasma guiding system

It is shown that during a capillary discharge, plasma channels for guiding light from high-power lasers are temporarily formed. The experiments are performed with alumina capillaries of 2-mm radius and 14-cm length filled with about 100 Pa of helium, methane or hydrogen. The transmission of probe light is enhanced due to light guiding by a factor up to 20 during a time window of 5 ns. The results indicate the formation of a first channel, when the shock wave reaches the axis, and of a second channel after the reflection of the shock wave on the axis. Compared to simple focussing by a lens, in the guiding device the intensity–length product increases by almost two orders of magnitude. Guiding is achieved over 180 times the Rayleigh length. Received: 20 June 2000 / Revised version: 4 August 2000 / Published online: 6 September 2000     

Quantum Clifford Hopf gebra for quantum field theory

We give arguments for the necessity to employ quantum Clifford Hopf gebras in quantum field theory. The role of the antipode is examined, Feynman diagrams are re-interpreted as tangles of graphical calculus. Regularization due to the design of convolution Hopf gebras is given as a program for future research.     

Products, Coproducts, and Singular Value Decomposition

Products and coproducts may be recognized as morphisms in a monoidal tensor category of vector spaces. To gain invariant data of these morphisms, we can use singular value decomposition which attaches singular values, i.e. generalized eigenvalues, to these maps. We show, for the case of Grassmannand Clifford products, that twist maps significantly alter these data reducing degeneracies. Since non group like coproducts give rise to non classical behavior of the algebra of functions, makeing them noncommutative, we hope to be able to learn more about such geometries. A remarkabe thechnicallity is that the coproduct for positive singular values of eigenvectors in A yields directly corresponding eigenvectors in A⊗ A.     

Mathematics of Clifford - a Maple package for Clifford and Graßmann algebras

CLIFFORD performs various computations in Gra?mann and Clifford algebras. It can compute with quaternions, octonions, and matrices with entries in C ℓ (B) - the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. Two user-selectable algorithms for the Clifford product are implemented: cmulNUM - based on Chevalley’s recursive formula, and cmulRS - based on a non-recursive Rota-Stein sausage. Gra?mann and Clifford bases can be used. Properties of reversion in undotted and dotted wedge bases are discussed.