375 research outputs found
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 . 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
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
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