Abstract
This paper is a continuation of our works concerning Linear Canonical Transformations (LCT) and Phase Space Representation of Quantum Theory. The purpose is to study the spinorial representation of some particular LCT called Isodispersion LCT (ILCT) and to deduce a relation between them and some properties of the elementary fermions of the Standard Model of Particle Physics. After giving the definition of ILCT for the case of a general pseudo-Euclidean space and constructing their spinorial representation, we consider the particular case of a pentadimensional space with signature (1, 4). We then deduce a classification of quarks, leptons and their antiparticles according to the values of electric charge, weak hypercharge, weak isospin and colors after the introduction of appropriate operators defined from the generators of the Clifford Algebra corresponding to the ILCT spinorial representation. It is established that the electric charge is composed of four terms, the weak hypercharge of five terms and the weak isospin of two terms. Existence of sterile neutrinos and the possibility of describing a fermions generation with a single field are suggested.
Highlights
In our previous works [1,2,3,4,5,6,7,8], we have performed series of studies on a phase space representation of quantum theory and Linear Canonical Transformations (LCTs)
Our purpose is to highlight a relation which can be established between the LCTs and the elementary fermions of the Standard Model of Particle Physics [9,10,11,12,13,14] i.e. quarks and leptons
We consider a particular case of LCTs called Isodispersion Linear Canonical Transformations (ILCTs)
Summary
In our previous works [1,2,3,4,5,6,7,8], we have performed series of studies on a phase space representation of quantum theory and Linear Canonical Transformations (LCTs). Our purpose is to highlight a relation which can be established between the LCTs and the elementary fermions of the Standard Model of Particle Physics [9,10,11,12,13,14] i.e. quarks and leptons In this regard, we consider a particular case of LCTs called Isodispersion Linear Canonical Transformations (ILCTs). We apply this study to a pentadimensional pseudo- euclidian space, signature 1, 4 We find that it permits to describe the properties of the fermions belonging to a single generation of the Standard Model and their antiparticles. Linear Canonical Transformations Representation approach developed by Zenczykowski for nonrelativistic phase space [15,16] He obtains that the weak hypercharge is the sum of three terms. As in our previous works [1,2,3,4,5,6,7,8], we use for tensorial and matricial calculations the notations developed in the reference [17]
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