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
Abstract:A method for time-frequency analysis is given. The approach utilizes properties of Gaussian distribution, properties of Hermite polynomials and Fourier analysis. We begin by the definitions of a set of functions called Harmonic Gaussian Functions. Then these functions are used to define a set of transformations, noted , which associate to a function , of the time variable , a set of functions Ψ which depend on time, frequency and frequency (or time) standard deviation. Some properties of the transformations and the functions Ψ are given. It is proved in particular that the square of the modulus of each function Ψ can be interpreted as a representation of the energy distribution of the signal, represented by the function , in the time-frequency plane for a given value of the frequency (or time) standard deviation. It is also shown that the function can be recovered from the functions Ψ .
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