Linear Canonical Transformations (LCTs) are known in signal processing and optics as the generalization of certain useful integral transforms. In quantum theory, they can be identified as the linear transformations which keep invariant the canonical commutation relations characterizing the coordinates and momenta operators. In this work, the possibility of considering LCTs to be the elements of a symmetry group for relativistic quantum physics is studied using the principle of covariance. It is established that Lorentz transformations and multidimensional Fourier transforms are particular cases of LCTs and some of the main symmetry groups currently considered in relativistic theories can be obtained from the contractions of LCTs groups. It is also shown that a link can be established between a spinorial representation of LCTs and some properties of elementary fermions. This link leads to a classification which suggests the existence of sterile neutrinos and the possibility of describing a generation of fermions with a single field. Some possible applications of the obtained results are discussed. These results may, in particular, help in the establishment of a unified theory of fundamental interactions. Intuitively, LCTs correspond to linear combinations of energy-momentum and spacetime compatible with the principle of covariance.

This work intends to present a study on relations between a Lie algebra called dispersion operators algebra, linear canonical transformation and a phase space representation of quantum mechanics that we have introduced and studied in previous works. The paper begins with a brief recall of our previous works followed by the description of the dispersion operators algebra which is performed in the framework of the phase space representation. Then, linear canonical transformations are introduced and linked with this algebra. A multidimensional generalization of the obtained results is givenComment: 13 page

The main purpose of this work is to identify invariant quadratic operators associated with Linear Canonical Transformations (LCTs) which could play important roles in physics. In quantum physics, LCTs are the linear transformations which keep invariant the Canonical Commutation Relations (CCRs). In this work, LCTs corresponding to a general pseudo-Euclidian space are considered and related to a phase space representation of quantum theory. Explicit calculations are firstly performed for the monodimensional case to identify the corresponding LCT-invariant quadratic operators then multidimensional generalizations of the obtained results are deduced. The eigenstates of these operators are also identified. A first kind of LCT-invariant operator is a second order polynomial of the coordinates and momenta operators. The coefficients of this polynomial depend on the mean values and the statistical variances-covariances of the coordinates and momenta operators themselves. It is shown that these statistical variances-covariances can be related with thermodynamic variables. In this context, new quantum corrections to the ideal gas state equation are deduced from correction to the Hamiltonian operator of non-relativistic free quantum particles that is suggested by LCT-covariance. Two other LCT-invariant quadratic operators, which can be considered as the number operators of some quasiparticles, are also identified: the first one is a number operator of bosonic type quasiparticles and the second one corresponds to fermionic type. This fermionic LCT-invariant quadratic operator is directly related to a spin representation of LCTs. It is shown explicitly, in the case of a relativistic pentadimensional theory, that the eigenstates of this operator can be considered as basic quantum states of elementary fermions. A classification of the fundamental fermions, compatible with the Standard Model of particle physics, is established from a classification of these states.

Abstract:This work intends to present a study on relations between a Lie algebra called dispersion

Sterile neutrinos are known to be hypothetical neutrinos which do not interact via the fundamental interactions described within the Standard Model of Particles Physics i.e. electroweak and strong interactions. They are expected to be important for the understanding of the physics beyond the current Standard Model. In the present work, it is shown that the existence of these particles can be suggested from covariance principle using a covariance group formed by Linear Canonical Transformations (LCTs) associated to a pentadimensional pseudo-Euclidian space. It is established that a spin representation of the LCT group gives a particle classification, applicable to the three families of leptons and quarks, which leads to the prediction of the existence of three sterile neutrinos and their antiparticles.

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This paper is a continuation of our previous works about coordinate, momentum, dispersion operators and phase space representation of quantum mechanics. It concerns a study on the properties of wavefunctions in the phase space representation and the momentum dispersion operator, its representations and eigenvalue equation. After the recall of some results from our previous papers, we give most of the main properties of the phase space wavefunctions and consider some examples of them. Then we establish the eigenvalue equation for the differential operator corresponding to the momentum dispersion operator in the phase space representation. It is shown in particular that any phase space wavefunction is solution of this equation.

This work is a continuation of our previous works concerning linear canonical transformations and phase space representation of quantum theory. It is mainly focused on the description of an approach which allows to establish spinorial representation of linear canonical transformations. This description is started with the presentation of a suitable parameterization of linear canonical transformations which permits to represent them with special pseudo-orthogonal transformations in an operator space. Then the establishment of the spinorial representation is deduced using the well-known relation existing between special pseudo-orthogonal and spin groups. The cases of one dimension and general multidimensional theory are both studied.