Development of the Method of Averaging in Clifford Geometric Algebras

TL;DR

This paper introduces a generalized averaging method in Clifford algebras, extending Reynolds operators and proving a generalized Pauli's theorem applicable across various dimensions, with broad interdisciplinary applications.

Contribution

It develops a new averaging technique in Clifford algebras, generalizes Reynolds operators, and proves a comprehensive version of Pauli's theorem for arbitrary dimensions.

Findings

Generalized Reynolds operators in Clifford algebras

Complete proof of the generalized Pauli's theorem

Potential applications in multiple scientific fields

Abstract

We develop the method of averaging in Clifford (geometric) algebras suggested by the author in previous papers. We consider operators constructed using two different sets of anticommuting elements of real or complexified Clifford algebras. These operators generalize Reynolds operators from the representation theory of finite groups. We prove a number of new properties of these operators. Using the generalized Reynolds operators, we give a complete proof of the generalization of Pauli's theorem to the case of Clifford algebras of arbitrary dimension. The results can be used in geometry, physics, engineering, computer science, and other applications.