Topological Decompositions of the Pauli Group and their Influence on Dynamical Systems

Abstract

In the present paper we show that it is possible to obtain the well known Pauli group P = 〈X,Y,Z | X² = Y² = Z² =?1,(Y Z)⁴ = (ZX)⁴ = (XY )⁴ =?1〉 of order 16 as an appropriate quotient group of two distinct spaces of orbits of the three dimensional sphere S³. The first of these spaces of orbits is realized via an action of the quaternion group Q₈ on S³; the second one via an action of the cyclic group of order four \(\mathbb {Z}(4)\) on S³. 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.