arXiv:1409.8163v3 [math-ph] 17 Apr 2020 Method of generalized Reynolds operators in Clifford algebras Dmitry Shirokov Abstract. We develop the method of averaging in Clifford algebras sug- gested by the author in the previous papers. Namely, we introduce gen- eralized Reynolds operators in real and complexified Clifford algebras of arbitrary dimension and signature and prove a number of new proper- ties of these operators. Using generalized Reynolds operators, we present the complete proof of generalization of Pauli’s theorem to the cases of Clifford algebras. Mathematics Subject Classification (2010). Primary 15A66; Secondary 11E88. Keywords. Clifford algebra, geometric algebra, method of averaging, Reynolds operator, Pauli’s theorem. 1. Introduction In the present paper, we develop the method of averaging in Clifford algebras suggested by the author in [18, 19, 25]. Namely, we introduce generalized Reynolds operators in Clifford algebras and prove a number of new properties of these operators in Sections 3 - 5. We use generalized Reynolds operators to prove generalized Pauli’s theorem (see Theorems 6.1 and 6.4), which has been formulated for the first time in a brief report [17] without a proof. The main idea of these theorems is to present an algorithm to compute the element T that connects two sets of Clifford algebra elements that satisfy the main anticommutative conditions. The proof of Theorems 6.1 and 6.4 is presented for the first time. Let e a be generators of the real Clifford algebra Cℓ p,q (or geometric algebra, see, for example, [9, 7, 4]) and e A = e a1a2...a k = e a1 e a2 ··· e a k are ba- sis elements enumerated by ordered multi-indices A = a 1 a 2 ...a k , a 1 <a 2 < ··· <a k with a length between 0 and n (the element e := e ∅ with empty multi- index is the identity element). The indices a, a 1 ,a 2 ,... take the values from 1 to n. The generators satisfy the main anticommutative conditions of Clifford