Topological decompositions of the Pauli group and their influence on dynamical systems

TL;DR

This paper explores topological decompositions of the Pauli group via orbit spaces of the 3-sphere, revealing new structural insights and potential physical interpretations related to pseudo-fermions.

Contribution

It introduces a novel topological decomposition of the Pauli group through orbit spaces of $S^3$, linking algebraic structure with geometric and physical concepts.

Findings

Decomposition of the Pauli group as a quotient of orbit spaces

Connection between topological decompositions and pseudo-fermions

Potential physical interpretation of the topological structure

Abstract

In the present paper we show that it is possible to obtain the well known Pauli group $P=\langle X,Y,Z \ | \ X^2=Y^2=Z^2=1, (YZ)^4=(ZX)^4=(XY)^4=1 \rangle $ of order $16$ as an appropriate quotient group of two distinct spaces of orbits of the three dimensional sphere $S^3$. The first of these spaces of orbits is realized via an action of the quaternion group $Q_8$ on $S^3$; the second one via an action of the cyclic group of order four $\mathbb{Z}(4)$ on $S^3$. We deduce a result of decomposition of $P$ of topological nature and then we find, in connection with the theory of pseudo-fermions, a possible physical interpretation of this decomposition.