Dispersion Operators Algebra and Linear Canonical Transformations

Abstract

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

Invariant quadratic operator associated with Linear Canonical Transformations

“…The LCT invariant operator identified through this work can also be considered as a generalization of a linear combination of the reduced dispersion-codispersion operators, denoted ℶ + , introduced in our previous work [9]. The results that are established here generalize and bring also more clarifications to the formulation developed and used in the references [9][10][11].…”

“…Linear Canonical Transformations (LCTs) are studied and used in several areas like signal processing, optics, and quantum physics [1][2][3][4][5][6][7][8][9]. In the fields of signal processing and optics, they are known to be the generalization of some useful integrals transforms such as Fourier and Fractional Fourier 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: their set can be considered as a symmetry group of these canonical commutation relations [10][11]. Some operators related to LCTs and their representations were already considered by various authors [5][6]9]. However no study has been done before regarding the identification of the invariant quadratic operator associated with LCTs, with a view to an application in relativistic quantum physics, as it is envisaged in the present work.…”

“…In order to simplify the presentation of the calculations and results associated with the identification of the LCT invariant operator in the next sections, we recall in this introduction section the definition and some of the main properties of the reduced dispersion operator ℶ + established in [9]. The notations that are used are not exactly the same as those utilized in [9] (especially for the mean values) but this change in notations is more suited to the current purpose. Some natural extensions which can be very easily deduced from the results obtained in [9] are also considered.…”

“…The notations that are used are not exactly the same as those utilized in [9] (especially for the mean values) but this change in notations is more suited to the current purpose. Some natural extensions which can be very easily deduced from the results obtained in [9] are also considered.…”