Clifford Geometrodynamics

We examine the structure of the Clifford algebra associated with a Hermitian bilinear form and apply the result to a dynamical model of the relativistic point particle. The dynamics of the particle is described by a Dirac spinor with components in a Clifford algebra. This spinor determines, through the Clifford algebra, both the space-time coordinates and their conjugate momenta and satisfies a first order equation of motion which leads to the usual space-time canonical equat...

We further explore the idea that physics takes place in Clifford space which should be considered as a generalization of spacetime. Following the old observation that spinors can be represented as members of left ideals of Clifford algebra, we point out that the transformations which mix bosons and fermions could be represented by means of operators acting on Clifford algebra-valued (polyvector) fields. A generic polyvector field can be expanded either in terms of bosonic, or...

We demonstrate the emergence of the conformal group SO(4,2) from the Clifford algebra of spacetime. The latter algebra is a manifold, called Clifford space, which is assumed to be the arena in which physics takes place. A Clifford space does not contain only points (events), but also lines, surfaces, volumes, etc..., and thus provides a framework for description of extended objects. A subspace of the Clifford space is the space whose metric is invariant with respect to the co...

A geometric approach to the standard model in terms of the Clifford algebra Cl_7 is advanced. A key feature of the model is its use of an algebraic spinor for one generation of leptons and quarks. Spinor transformations separate into left-sided ("exterior") and right-sided ("interior") types. By definition, Poincare transformations are exterior ones. We consider all rotations in the seven-dimensional space that (1) conserve the spacetime components of the particle and antipar...

Based on a fact that complex Clifford algebras of even dimension are isomorphic to the matrix ones, we consider bundles in Clifford algebras whose structure group is a general linear group acting on a Clifford algebra by left multiplications, but not a group of its automorphisms. It is essential that such a Clifford algebra bundle contains spinor subbundles, and that it can be associated to a tangent bundle over a smooth manifold. This is just the case of gravitation theory. ...

In this paper we present a multipartite formulation of gauge theory gravity based on the formalism of space-time algebra for gravitation developed by Lasenby and Doran (Lasenby, A. N., Doran, C. J. L, and Gull, S.F.: Gravity, gauge theories and geometric algebra. Phil. Trans. R. Soc. Lond. A, 582, 356:487 (1998)). We associate the gauge fields with description of fermionic and bosonic states using the generalized graded tensor product. Einstein's equations are deduced from th...

Expository notes on Clifford algebras and spinors with a detailed discussion of Majorana, Weyl, and Dirac spinors. The paper is meant as a review of background material, needed, in particular, in now fashionable theoretical speculations on neutrino masses. It has a more mathematical flavour than the over twenty-seven-year-old "Introduction to Majorana masses" by P.D. Mannheim and includes historical notes and biographical data on past participants in the story.

It is shown in the present paper that the transformation relating a parallel transported vector in a Weyl space to the original one is the product of a multiplicative gauge transformation and a proper orthochronous Lorentz transformation. Such a Lorentz transformation admits a spinor representation, which is obtained and used to deduce the transportation properties of a Weyl spinor, which are then expressed in terms of a composite gauge group defined as the product of a multi...

The Standard Model of elementary particles is a theory unifying three of the four basic forces of the Nature: electromagnetic, weak, and strong interactions. In this paper we consider the Standard Model in the presence of a classical (non-quantized) gravitation field and apply a bundle approach for describing it.

These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra construction (then called geometric algebra) with geometric applications in mind, as well as in an algebraically more general form which is well suited for combinatorics, and for defining and understanding the numerous products and operations of t...