Dispersion Operators Algebra and Linear Canonical Transformations

Abstract

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

Study on a Spinorial Representation of Linear Canonical Transformation

“…Following our paper [2], the 2 2 matrix $ Π Ξ Θ Λ belongs to the pseudo-symplectic group 2 , 2 . $ can be written in the form…”

“…In our previous papers [1][2][3][4], we have performed a series of study on a phase space representation of quantum theory and Linear Canonical Transformations (LCTs). LCTs have already been studied in various contexts [5][6][7][8][9] but our work is focused on their study in the framework of quantum theory.…”

“…LCTs have already been studied in various contexts [5][6][7][8][9] but our work is focused on their study in the framework of quantum theory. In the paper [2], we have established that there is an isomorphism between the dispersion operator algebra and the Lie algebra 2 , 2 of the Lie group 2 , 2 corresponding to the set of LCTs in a pseudo-euclidian vectorial space of signature , . This isomorphism permit to build an unitary representation of LCTs.…”

“…In the work [2], we have introduced operators defined from the momentum and coordinate operators of a particle. Some of these operators will be used throughout the present paper.…”

Properties of Phase Space Wavefunctions and Eigenvalue Equation of Momentum Dispersion Operator

“…The present work can be considered as part of a series of studies related to a phase space representation of quantum theory introduced and developed in [1], [2] and [3]. Because of the uncertainty relation [4], the problem of considering phase space, which mix momentum with coordinate, in quantum theory is an interesting challenge.…”