Study on a Spinorial Representation of Linear Canonical Transformation

Raoelina Andriambololona; Ravo Tokiniaina Ranaivoson; Hanitriarivo Rakotoson

doi:doi:10.11648/j.ijamtp.20190503.12

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Study on a Spinorial Representation of Linear Canonical Transformation

Raoelina Andriambololona, Ravo Tokiniaina Ranaivoson, Hanitriarivo Rakotoson

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 5, Issue 3)

Received: 15 June 2019 Accepted: 30 July 2019 Published: 26 August 2019

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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.

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
Creative Commons	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), 2024. Published by Science Publishing Group

Keywords

Linear Canonical Transformation, Special Pseudo-Orthogonal Transformation, Clifford Algebra, Spin Group, Spinorial Representation, Quantum Theory

References

[1]	Raoelina Andriambololona, Ravo Tokiniaina Ranaivoson, Rakotoson Hanitriarivo, Wilfrid Chrysante Solofoarisina, “Study on Linear Canonical Transformation in a Framework of a Phase Space Representation of Quantum Mechanics”, arXiv: 1503.02449 [quant-ph], International Journal of Applied Mathematics and Theoretical Physics. Vol. 1, No. 1, 2015, pp. 1-8, 2015.
[2]	Raoelina Andriambololona, Ravo Tokiniaina Ranaivoson, Rakotoson Hanitriarivo, Damo Emile Randriamisy, “Dispersion Operator Algebra and Linear Canonical Transformation”, arXiv: 1608.02268v2 [quant-ph], International Journal of Theoretical Physics, Volume 56, Issue 4, pp 1258-1273, Springer, April 2017.
[3]	Ravo Tokiniaina Ranaivoson, Raoelina Andriambololona, Rakotoson Hanitriarivo, Roland Raboanary: “Study on a Phase Space Representation of Quantum Theory”, arXiv: 1304.1034 [quant-ph], International Journal of Latest Research in Science and Technology, ISSN(Online): 2278-5299, Volume 2,Issue 2: Page No.26-35, March-April, 2013.
[4]	Rakotoson Hanitriarivo, Raoelina Andriambololona, Ravo Tokiniaina Ranaivoson, Raboanary Roland: “Coordinate, momentum and dispersion operators in phase space representation”, arXiv: 1707.02223 [quant-ph], International Journal of Latest Research in Science and Technology ISSN (Online): 2278-5299 Volume 6, Issue 4: Page No. 8-13, July-August 2017.
[5]	Tatiana Alieva, Martin J. Bastiaans, “The Linear Canonical Transformations: Definition and Properties”, In: Healy J., Alper Kutay M., Ozaktas H., Sheridan J. (eds) Linear Canonical Transforms. Springer Series in Optical Sciences, vol 198. Springer, New York, 2016.
[6]	Tian-Zhou Xu, Bing-Zhao Li: Linear Canonical Transform and Its Applications, Science Press, Beijing, China, 2013.
[7]	M. Moshinsky and C. Quesne: Linear canonical transformations and their unitary representations, J. Math. Phys.12, 8, 1772-1783, 1971.
[8]	K. B. Wolf: A Top-Down Account of Linear Canonical Transforms, arXiv: 1206.1123 [Math.ph], SIGMA 8 (2012), 033, 13 pages.
[9]	J. J. Healy, M. A. Kutay, H. M. Ozaktas and J. T. Sheridan, "Linear Canonical Transforms: Theory and Applications", Springer, New York, 2016.
[10]	Robert Coquereaux, Espaces fibrés et connexions, Une introduction aux géométries classiques et quantiques de la physique théorique,Centre de Physique Théorique, Luminy- Marseille, 2002.
[11]	Robert Coquereaux, Clifford algebras, spinors and fundamental interactions: Twenty Years After, arXiv: math-ph/0509040, 2005.
[12]	I. Todorov, Clifford Algebras and Spinors, Bulg. J. Phys. 38 (2011) 3-28.
[13]	Raoelina Andriambololona: Algèbre linéaire et multilinéaire, Collection LIRA, INSTN- Madagascar, 1986.
[14]	Rakotoson Hanitriarivo, Raoelina Andriambololona, Ravo Tokiniaina Ranaivoson, Roland Raboanary, Linear Canonical Transformations in Relativistic Quantum Physics, arXiv: 1804.10053 [quant-ph], 2018.
[15]	Ravo Tokiniaina Ranaivoson, Raoelina Andriambololona, Rakotoson Hanitriarivo, Properties of Elementary Fermions of the Standard Model deduced from Linear Canonical Transformations, arXiv: 1806.07228 [physics.gen-ph], 2018.

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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

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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

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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

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@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}
}

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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
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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  -

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Author Information

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

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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

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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

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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

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@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}
}

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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  -

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