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.
Published in | International Journal of Applied Mathematics and Theoretical Physics (Volume 5, Issue 3) |
DOI | 10.11648/j.ijamtp.20190503.12 |
Page(s) | 58-65 |
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This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
Copyright |
Copyright © The Author(s), 2019. Published by Science Publishing Group |
Linear Canonical Transformation, Special Pseudo-Orthogonal Transformation, Clifford Algebra, Spin Group, Spinorial Representation, Quantum Theory
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APA Style
Raoelina Andriambololona, Ravo Tokiniaina Ranaivoson, Hanitriarivo Rakotoson. (2019). Study on a Spinorial Representation of Linear Canonical Transformation. International Journal of Applied Mathematics and Theoretical Physics, 5(3), 58-65. https://doi.org/10.11648/j.ijamtp.20190503.12
ACS Style
Raoelina Andriambololona; Ravo Tokiniaina Ranaivoson; Hanitriarivo Rakotoson. Study on a Spinorial Representation of Linear Canonical Transformation. Int. J. Appl. Math. Theor. Phys. 2019, 5(3), 58-65. doi: 10.11648/j.ijamtp.20190503.12
AMA Style
Raoelina Andriambololona, Ravo Tokiniaina Ranaivoson, Hanitriarivo Rakotoson. Study on a Spinorial Representation of Linear Canonical Transformation. Int J Appl Math Theor Phys. 2019;5(3):58-65. doi: 10.11648/j.ijamtp.20190503.12
@article{10.11648/j.ijamtp.20190503.12, author = {Raoelina Andriambololona and Ravo Tokiniaina Ranaivoson and Hanitriarivo Rakotoson}, title = {Study on a Spinorial Representation of Linear Canonical Transformation}, journal = {International Journal of Applied Mathematics and Theoretical Physics}, volume = {5}, number = {3}, pages = {58-65}, doi = {10.11648/j.ijamtp.20190503.12}, url = {https://doi.org/10.11648/j.ijamtp.20190503.12}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20190503.12}, 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.}, year = {2019} }
TY - JOUR T1 - Study on a Spinorial Representation of Linear Canonical Transformation AU - Raoelina Andriambololona AU - Ravo Tokiniaina Ranaivoson AU - Hanitriarivo Rakotoson Y1 - 2019/08/26 PY - 2019 N1 - https://doi.org/10.11648/j.ijamtp.20190503.12 DO - 10.11648/j.ijamtp.20190503.12 T2 - International Journal of Applied Mathematics and Theoretical Physics JF - International Journal of Applied Mathematics and Theoretical Physics JO - International Journal of Applied Mathematics and Theoretical Physics SP - 58 EP - 65 PB - Science Publishing Group SN - 2575-5927 UR - https://doi.org/10.11648/j.ijamtp.20190503.12 AB - 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. VL - 5 IS - 3 ER -