Noncommutativity and Duality through the Symplectic Embedding Formalism
This work is devoted to review the gauge embedding of either commutative and noncommutative (NC) theories using the symplectic formalism framework. To sum up the main features of the method, during the process of embedding, the infinitesimal gauge generators of the gauge embedded theory are easily a...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2010 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2010
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/146359 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Noncommutativity and Duality through the Symplectic Embedding Formalism / Everton M.C. Abreu, Albert C.R. Mendes, W. Oliveira // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 58 назв. — англ. |
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Everton M.C. Abreu Albert C.R. Mendes Oliveira, W. 2019-02-09T09:39:13Z 2019-02-09T09:39:13Z 2010 Noncommutativity and Duality through the Symplectic Embedding Formalism / Everton M.C. Abreu, Albert C.R. Mendes, W. Oliveira // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 58 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 70S05; 70S10; 81Q65; 81T75 DOI:10.3842/SIGMA.2010.059 https://nasplib.isofts.kiev.ua/handle/123456789/146359 This work is devoted to review the gauge embedding of either commutative and noncommutative (NC) theories using the symplectic formalism framework. To sum up the main features of the method, during the process of embedding, the infinitesimal gauge generators of the gauge embedded theory are easily and directly chosen. Among other advantages, this enables a greater control over the final Lagrangian and brings some light on the so-called ''arbitrariness problem''. This alternative embedding formalism also presents a way to obtain a set of dynamically dual equivalent embedded Lagrangian densities which is obtained after a finite number of steps in the iterative symplectic process, oppositely to the result proposed using the BFFT formalism. On the other hand, we will see precisely that the symplectic embedding formalism can be seen as an alternative and an efficient procedure to the standard introduction of the Moyal product in order to produce in a natural way a NC theory. In order to construct a pedagogical explanation of the method to the nonspecialist we exemplify the formalism showing that the massive NC U(1) theory is embedded in a gauge theory using this alternative systematic path based on the symplectic framework. Further, as other applications of the method, we describe exactly how to obtain a Lagrangian description for the NC version of some systems reproducing well known theories. Naming some of them, we use the procedure in the Proca model, the irrotational fluid model and the noncommutative self-dual model in order to obtain dual equivalent actions for these theories. To illustrate the process of noncommutativity introduction we use the chiral oscillator and the nondegenerate mechanics. This paper is a contribution to the Special Issue “Noncommutative Spaces and Fields”. The full collection is available at http://www.emis.de/journals/SIGMA/noncommutative.html. EMCA would like to thank the hospitality and kindness of the Dept. of Physics of the Federal University of Juiz de Fora where part of this work was done. This work was supported in part by Funda¸c˜ao de Amparo a Pesquisa do Estado de Minas Gerais (FAPEMIG) and Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq), Brazilian Research Agencies. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Noncommutativity and Duality through the Symplectic Embedding Formalism Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Noncommutativity and Duality through the Symplectic Embedding Formalism |
| spellingShingle |
Noncommutativity and Duality through the Symplectic Embedding Formalism Everton M.C. Abreu Albert C.R. Mendes Oliveira, W. |
| title_short |
Noncommutativity and Duality through the Symplectic Embedding Formalism |
| title_full |
Noncommutativity and Duality through the Symplectic Embedding Formalism |
| title_fullStr |
Noncommutativity and Duality through the Symplectic Embedding Formalism |
| title_full_unstemmed |
Noncommutativity and Duality through the Symplectic Embedding Formalism |
| title_sort |
noncommutativity and duality through the symplectic embedding formalism |
| author |
Everton M.C. Abreu Albert C.R. Mendes Oliveira, W. |
| author_facet |
Everton M.C. Abreu Albert C.R. Mendes Oliveira, W. |
| publishDate |
2010 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
This work is devoted to review the gauge embedding of either commutative and noncommutative (NC) theories using the symplectic formalism framework. To sum up the main features of the method, during the process of embedding, the infinitesimal gauge generators of the gauge embedded theory are easily and directly chosen. Among other advantages, this enables a greater control over the final Lagrangian and brings some light on the so-called ''arbitrariness problem''. This alternative embedding formalism also presents a way to obtain a set of dynamically dual equivalent embedded Lagrangian densities which is obtained after a finite number of steps in the iterative symplectic process, oppositely to the result proposed using the BFFT formalism. On the other hand, we will see precisely that the symplectic embedding formalism can be seen as an alternative and an efficient procedure to the standard introduction of the Moyal product in order to produce in a natural way a NC theory. In order to construct a pedagogical explanation of the method to the nonspecialist we exemplify the formalism showing that the massive NC U(1) theory is embedded in a gauge theory using this alternative systematic path based on the symplectic framework. Further, as other applications of the method, we describe exactly how to obtain a Lagrangian description for the NC version of some systems reproducing well known theories. Naming some of them, we use the procedure in the Proca model, the irrotational fluid model and the noncommutative self-dual model in order to obtain dual equivalent actions for these theories. To illustrate the process of noncommutativity introduction we use the chiral oscillator and the nondegenerate mechanics.
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| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/146359 |
| citation_txt |
Noncommutativity and Duality through the Symplectic Embedding Formalism / Everton M.C. Abreu, Albert C.R. Mendes, W. Oliveira // Symmetry, Integrability and Geometry: Methods and Applications. — 2010. — Т. 6. — Бібліогр.: 58 назв. — англ. |
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2025-12-07T15:31:11Z |
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2025-12-07T15:31:11Z |
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