Volume 5, Issue 3, September 2019, Page: 58-65
Study on a Spinorial Representation of Linear Canonical Transformation
Raoelina Andriambololona, Information Technology and Theoretical Physics Department, National Institute of Nuclear Science and Techniques (INSTN-Madagascar), Antananarivo, Madagascar
Ravo Tokiniaina Ranaivoson, Information Technology and Theoretical Physics Department, National Institute of Nuclear Science and Techniques (INSTN-Madagascar), Antananarivo, Madagascar
Hanitriarivo Rakotoson, Information Technology and Theoretical Physics Department, National Institute of Nuclear Science and Techniques (INSTN-Madagascar), Antananarivo, Madagascar
Received: Jun. 15, 2019;       Accepted: Jul. 30, 2019;       Published: Aug. 26, 2019
DOI: 10.11648/j.ijamtp.20190503.12      View  98      Downloads  26
Abstract
This paper is mainly focused on the description of an approach for establishing a spinorial representation of linear canonical transformations. It can be considered as a continuation of our previous works concerning linear canonical transformations and phase space representation of quantum theory. The said method is based on the development of an adequate parameterization of linear canonical transformations which permits to represent them with special pseudo-orthogonal transformations. Obtaining this pseudo-orthogonal representation makes it possible to establish the spinorial representation exploiting the well-known relation existing between special pseudo-orthogonal and spin groups. The cases of one dimension and general multidimensional theories are both studied. The design of the pseudo-orthogonal transformation associated to a linear canonical transformation is achieved by introducing adequate operators which are linear combinations of reduced momentum and coordinate operators. It is shown that a linear canonical transformation is equivalent to a special pseudo-orthogonal transformation defined in the set formed by these adequate operators. The spinorial representation is then deduced by defining a composite operator which is linear combinations of the tensorial products of the generators of the Clifford algebra with the adequate operators defining the special pseudo-orthogonal representation. It is established that unlike the case of a spinorial representation associated with an ordinary commutative vector space, the main invariant corresponding to the transformation is not the square of the composite operator but a higher degree polynomial function of it.
Keywords
Linear Canonical Transformation, Special Pseudo-Orthogonal Transformation, Clifford Algebra, Spin Group, Spinorial Representation, Quantum Theory
To cite this article
Raoelina Andriambololona, Ravo Tokiniaina Ranaivoson, Hanitriarivo Rakotoson, Study on a Spinorial Representation of Linear Canonical Transformation, International Journal of Applied Mathematics and Theoretical Physics. Vol. 5, No. 3, 2019, pp. 58-65. doi: 10.11648/j.ijamtp.20190503.12
Copyright
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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