Linear Canonical Transformations in Relativistic Quantum Physics

TL;DR

This paper explores linear canonical transformations as symmetry elements in relativistic quantum physics, linking them to Lorentz transformations, Fourier transforms, and fermion properties, aiming toward a unified fundamental interactions theory.

Contribution

It demonstrates that LCTs encompass key relativistic symmetries and connects them to fermion properties, proposing a new framework for unification in quantum physics.

Findings

Lorentz transformations are special cases of LCTs

Main relativistic symmetry groups can be derived from LCT contractions

A link between LCT spinorial representations and fermion properties is established

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.