Method of generalized Reynolds operators in Clifford algebras

We develop the method of averaging in Clifford algebras suggested by the author in the previous papers. Namely, we introduce generalized Reynolds operators in real and complexified Clifford algebras of arbitrary dimension and signature and prove a number of new properties of these operators. Using generalized Reynolds operators, we present the complete proof of generalization of Pauli's theorem to the cases of Clifford algebras.Comment: 20 page

On computing the determinant, other characteristic polynomial coefficients, and inverse in Clifford algebras of arbitrary dimension

In this paper, we solve the problem of computing the inverse in Clifford algebras of arbitrary dimension. We present basis-free formulas of different types (explicit and recursive) for the determinant, other characteristic polynomial coefficients, adjugate, and inverse in real Clifford algebras (or geometric algebras) over vector spaces of arbitrary dimension $n$. The formulas involve only the operations of multiplication, summation, and operations of conjugation without explicit use of matrix representation. We use methods of Clifford algebras (including the method of quaternion typification proposed by the author in previous papers and the method of operations of conjugation of special type presented in this paper) and generalizations of numerical methods of matrix theory (the Faddeev-LeVerrier algorithm based on the Cayley-Hamilton theorem; the method of calculating the characteristic polynomial coefficients using Bell polynomials) to the case of Clifford algebras in this paper. We present the construction of operations of conjugation of special type and study relations between these operations and the projection operations onto fixed subspaces of Clifford algebras. We use this construction in the analytical proof of formulas for the determinant, other characteristic polynomial coefficients, adjugate, and inverse in Clifford algebras. The basis-free formulas for the inverse give us basis-free solutions to linear algebraic equations, which are widely used in computer science, image and signal processing, physics, engineering, control theory, etc. The results of this paper can be used in symbolic computation.Comment: 24 page

Development of the method of quaternion typification of Clifford algebra elements

In this paper we further develop the method of quaternion typification of Clifford algebra elements suggested by the author in the previous paper. On the basis of new classification of Clifford algebra elements it is possible to reveal and prove a number of new properties of Clifford algebra. We use k-fold commutators and anticommutators. In this paper we consider Clifford and exterior degrees and elementary functions of Clifford algebra elements.Comment: 15 page

On constant solutions of SU(2) Yang-Mills-Dirac equations

For the first time, a complete classification of all constant solutions of the Yang-Mills-Dirac equations with SU(2) gauge symmetry in Minkowski space ${\mathbb R}^{1,3}$ is given. The explicit form of all solutions is presented. We use the method of hyperbolic singular value decomposition of real and complex matrices and the two-sheeted covering of the group SO(3) by the group SU(2). In the degenerate case of zero potential, we use the pseudo-unitary symmetry of the Dirac equation. Nonconstant solutions can be considered in the form of series of perturbation theory using constant solutions as a zeroth approximation; the equations for the first approximation in the expansion are written.Comment: 19 page