The tensor Dirac equation in Riemannian space

We suggest a tensor equation on Riemannian manifolds which can be considered as a generalization of the Dirac equation for the electron. The tetrad formalism is not used. Also we suggest a new form of the tensor Dirac equation with a Spin(1,3) gauge symmetry in Minkowski space.Comment: Latex 19 page

A coordinateless form of the Dirac equation

We present a so called Dirac-type tensor equation (DTTE). This equation is written in coordinateless form with the aid of differential operators $d$ and $\delta$. A wave function of DTTE belongs to a minimal left ideal of the algebra of exterior forms with respect to the Clifford product. We show that a coordinate form of DTTE is identical to the Dirac equation in a fixed coordinate system

A model of composite structure of quarks and leptons

In the model every quark or lepton is identified with a quartet of four "more elementary" particles. One particle in a quartet is a massive spin-0 boson and other three particles are massless spin-1/2 fermions.Comment: 7 page

A tensor form of the Dirac equation

We prove the following theorem: the Dirac equation for an electron (invented by P.A.M.Dirac in 1928) can be written as a tensor equation. An equation is called a tensor equation if all values in it are tensors and all operations in it take tensors to tensors.Comment: LaTeX, 23 pages I correct mistyping in the third line of the formula on page 1

Dirac-type tensor equations with non-Abelian gauge symmetries on pseudo-Riemannian space

We suggest a so-called Dirac type tensor equation with nonabelian gauge symmetry on pseudo-Riemannian space. This equation reproduce some of the properties of spinor Dirac equation. A geometrical interpretation of results in terms of Riemannian geometry is given.Comment: 25 page

Dirac-type tensor equations on a parallelisable manyfolds

The goal of this work is to extend Dirac-type tensor equations to a curved space. We take four 1-forms (a tetrad) as a unique structure, which determines a geometry of space-time

Notions of determinant, spectrum, and Hermitian conjugation of Clifford algebra elements

We show how the matrix algebra notions of determinant, spectrum, and Hermitian conjugation transfer to the Clifford algebra and to differential forms on parallelisable manifolds

General solutions of one class of field equations

We find general solutions of some field equations (systems of equations) in pseudo-Euclidian spaces (so-called primitive field equations). These equations are used in the study of the Dirac equation and Yang-Mills equations. These equations are invariant under orthogonal O(p,q) coordinate transformations and invariant under gauge transformations, which depend on some Lie groups. In this paper we use some new geometric objects - Clifford field vector and an algebra of h-forms which is a generalization of the algebra of differential forms and the Atiyah-K\"{a}hler algebra.Comment: 22 page

Local generalization of Pauli's theorem

Generalized Pauli's theorem, proved by D. S. Shirokov for two sets of anticommuting elements of a real or complexified Clifford algebra of dimension $2^n$, is extended to the case, when both sets of elements depend smoothly on points of Euclidian space of dimension $r$. We prove that in the case of even $n$ there exists a smooth function such that two sets of Clifford algebra elements are connected by a similarity transformation. All cases of connection between two sets are considered in the case of odd $n$. Using the equation for the spin connection of general form, it is shown that the problem of the local Pauli's theorem is equivalent to the problem of existence of a solution of some special system of partial differential equations. The special cases $n=2$, $r\geq 1$ and $n\geq 2$, $r=1$ with more simpler solution of the problem are considered in detail.Comment: 17 page

A gauge model with spinor group for a description of local interaction of a fermion with electromagnetic and gravitational fields

We suggest model equations, which, from some point of view, describe local interaction of three physical fields: a field of matter, an electromagnetic field and a gravitational field. A base of the model is a field of matter described by the wave function of fermion satisfying the equation similar to Dirac equation for electron. Electromagnetic and gravitational fields appear as the gauge fields for this equation. We have found the connection between these fields and the curvature tensor of Riemannian manifold. We present a main Lagrangian from which the equations of the model are deduced. The covariance of the model equations under changes of coordinates is considered. We develop mathematical techniques needed for the model connected with an exterior algebra of Euclidean or Riemannian space. The exterior algebra is considered as a bialgebra with two operations of multiplications -- an exterior multiplication and Clifford multiplication. We define a structure of Euclidean or Riemannian space on the exterior algebra, which leads to the notions of Spin-isometric change of coordinates and Spin-isometric manifold used in the model.In the revised paper we correct an error with the formula $G_{ij}=-U^{-1}D_{ij}U/2$, (now U=1).Comment: 50 pages, LaTe