Linear Canonical Transformations in relativistic quantum physics

Abstract

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.

Invariant quadratic operators associated with linear canonical transformations and their eigenstates

“…The eigenvalue equation (9) can be itself deduced from the relation (12) and the commutation relation in (11). We have also the well-known relations ð13Þ which justify the name ladder operators for and .…”

“…Some operators related to LCTs and their representations were already considered by various authors [5][6][7]. However, the quadratic operators that are identified in this work are new ones even if some of them can be considered as generalization of operators introduced in the references [10][11][12].…”

“…The operators (11) because its generalization plays an important role in the multidimensional generalization and in the construction of the phase space representation that are considered in the next section. This operator is a particular monodimensional case of the operator…”

“…Linear Canonical Transformations (LCTs) can be considered as generalization of some useful integral transformations like Fourier and Fractional Fourier Transforms. They are studied and used in many areas [1][2][3][4][5][6][7][8][9][10][11][12]. In quantum theory, they can be identified as the linear transformations which keep invariant the Canonical Commutation Relations (CCRs) defining the coordinates and momenta operators.…”

“…In quantum theory, they can be identified as the linear transformations which keep invariant the Canonical Commutation Relations (CCRs) defining the coordinates and momenta operators. In the references [11,12], it was shown that a covariance principle related with LCTs may play an important role in physics and in the establishment of a unified theory of fundamental interactions. In the present work, our main objective is to identify some quadratic operators that are invariant under the action of LCTs and to highlight their potential importance in physics.…”