Study on a Spinorial Representation of Linear Canonical Transformations

TL;DR

This paper develops a spinorial representation for linear canonical transformations in quantum phase space, extending previous work by parameterizing these transformations through pseudo-orthogonal groups and their spinorial counterparts in multiple dimensions.

Contribution

It introduces a novel spinorial framework for linear canonical transformations, connecting pseudo-orthogonal and spin groups in quantum phase space representations.

Findings

Established parameterization of linear canonical transformations.

Derived spinorial representation using pseudo-orthogonal groups.

Applied theory to one-dimensional and multidimensional cases.

Abstract

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.