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,(YZ)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.
Highlights
The Pauli group P is a finite group of order 16, introduced by W
We involve some methods of general nature, but develop a series of tools which are designed for P only. This choice is made for a specific motivation: we want to avoid a universal approach for the notions of amalgamated product and central product, even if these two notions may be formalized in category theory, or in classes of finite groups which are larger than the class of 2-groups
This is an interesting, and somehow unexpected, feature of the model: going from a larger Pμ to a smaller group Q8 does not affect at all the dynamical aspects of the system, since these are all contained in Q8
Summary
The Pauli group P is a finite group of order 16, introduced by W. The algebraic decomposition of P is not affected from the topological ones in Theorem 1.1 Another interesting result of the present paper is that P can be expressed in terms of pseudo-fermionic operators. It is relevant to note that some dynamical aspects of physical systems (involving pseudo-fermionic operators) are connected with a suitable decomposition of P. This will be shown later, and is stated by the following theorem. Let K be a field and A be a vector space over K with an additional internal operation : (x, y) ∈ A × A → x y ∈ A
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