Dirac equation solution in the light front via linear algebra and its particularities
Autor(a) principal: | |
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Data de Publicação: | 2023 |
Outros Autores: | , , |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | The Journal of Engineering and Exact Sciences |
Texto Completo: | https://periodicos.ufv.br/jcec/article/view/16329 |
Resumo: | In undergraduate and postgraduate courses, it is customary to present the Dirac equation defined in a space of four dimensions: three spatial and one temporal. This article discusses aspects of the Dirac equation (QED) on the light front. This proposal of coordinate transformations comes from Dirac who originally introduced three distinct forms of relativistic dynamics possible depending on the choice we make of the different hypersurfaces constant in time. The first he called instantaneous, the most common form, the hypersurface of which is specified by the boundary conditions set at . The second, known as the point form, has as its characterizing surface, a hyperboloid, described by the initial conditions in , being one constant (chosen as the time of this system). The third relativistic form, known as the light front form, has its hypersurface tangent to the light cone; being defined by the initial conditions at , and is the time in the light front system. The method of this work is deductive. Therefore, one obtains the solution of the Dirac equation for the Free Electron and for the positron in the coordinates in the light front with the particularity of the energy associated with the system being given by , and for moments we have the electron and we have the positron. The result of this is that the positive energy states in the light front and negative are independently described in the equation, and with additional, the problem at the limit that does not converge. |
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Dirac equation solution in the light front via linear algebra and its particularities Solução da equação de Dirac na frente da luz via álgebra linear e suas particularidadesrelativityMinkowski spacecoordinate systemfermionsrelatividade, espaço de Minkowski, sistema de coordenadas, férmionsIn undergraduate and postgraduate courses, it is customary to present the Dirac equation defined in a space of four dimensions: three spatial and one temporal. This article discusses aspects of the Dirac equation (QED) on the light front. This proposal of coordinate transformations comes from Dirac who originally introduced three distinct forms of relativistic dynamics possible depending on the choice we make of the different hypersurfaces constant in time. The first he called instantaneous, the most common form, the hypersurface of which is specified by the boundary conditions set at . The second, known as the point form, has as its characterizing surface, a hyperboloid, described by the initial conditions in , being one constant (chosen as the time of this system). The third relativistic form, known as the light front form, has its hypersurface tangent to the light cone; being defined by the initial conditions at , and is the time in the light front system. The method of this work is deductive. Therefore, one obtains the solution of the Dirac equation for the Free Electron and for the positron in the coordinates in the light front with the particularity of the energy associated with the system being given by , and for moments we have the electron and we have the positron. The result of this is that the positive energy states in the light front and negative are independently described in the equation, and with additional, the problem at the limit that does not converge. Nos cursos de graduação e pós-graduação costuma-se apresentar a equação de Dirac definida num espaço de quatro dimensões: três espaciais e uma temporal. Este artigo, aborda aspectos da equação de Dirac (QED) na Frente de Luz. Essa proposta de transformações de coordenadas vem de Dirac que originalmente introduziu três formas distintas de dinâmica relativística possíveis, dependendo da escolha que fazemos das diferentes hipersuperfícies constantes no tempo. A primeira ele chamou de forma instantânea, a mais comum, cuja hipersuperfície é especificada pelas condições de contorno definidas em . A segunda, conhecida como forma pontual, tem como superfície caracterizadora, um hiperboloide, descrita pelas condições iniciais em , sendo “” uma constante (escolhida como o tempo desse sistema). A terceira forma relativística, conhecida como forma da frente de luz, tem sua hipersuperfície tangente ao cone de luz, sendo definido pelas condições iniciais em , e é o tempo para o sistema da frente de luz. O método deste trabalho é dedutivo. Portanto, obtém-se a solução da equação de Dirac para o elétron livre e para o pósitron nas coordenadas da frente de luz com a particularidade da energia associada ao sistema ser dada por , sendo que para momentos temos o elétron e temos o pósitron. O resultado disso é que os estados de energia positivo na frente de luz e negativo são descritos de forma independente na equação e, com adicional, o problema no limite que não converge. Universidade Federal de Viçosa - UFV2023-08-17info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfhttps://periodicos.ufv.br/jcec/article/view/1632910.18540/jcecvl9iss9pp16329-01eThe Journal of Engineering and Exact Sciences; Vol. 9 No. 9 (2023); 16329-01eThe Journal of Engineering and Exact Sciences; Vol. 9 Núm. 9 (2023); 16329-01eThe Journal of Engineering and Exact Sciences; v. 9 n. 9 (2023); 16329-01e2527-1075reponame:The Journal of Engineering and Exact Sciencesinstname:Universidade Federal de Viçosa (UFV)instacron:UFVenghttps://periodicos.ufv.br/jcec/article/view/16329/8137Copyright (c) 2023 The Journal of Engineering and Exact Scienceshttps://creativecommons.org/licenses/by/4.0info:eu-repo/semantics/openAccessSales, Jorge Henrique de OliveiraAragão, Gabriel de OliveiraNascimento, Diego Ramos doThibes, Ronaldo2023-11-26T15:36:25Zoai:ojs.periodicos.ufv.br:article/16329Revistahttp://www.seer.ufv.br/seer/rbeq2/index.php/req2/oai2527-10752527-1075opendoar:2023-11-26T15:36:25The Journal of Engineering and Exact Sciences - Universidade Federal de Viçosa (UFV)false |
dc.title.none.fl_str_mv |
Dirac equation solution in the light front via linear algebra and its particularities Solução da equação de Dirac na frente da luz via álgebra linear e suas particularidades |
title |
Dirac equation solution in the light front via linear algebra and its particularities |
spellingShingle |
Dirac equation solution in the light front via linear algebra and its particularities Sales, Jorge Henrique de Oliveira relativity Minkowski space coordinate system fermions relatividade, espaço de Minkowski, sistema de coordenadas, férmions |
title_short |
Dirac equation solution in the light front via linear algebra and its particularities |
title_full |
Dirac equation solution in the light front via linear algebra and its particularities |
title_fullStr |
Dirac equation solution in the light front via linear algebra and its particularities |
title_full_unstemmed |
Dirac equation solution in the light front via linear algebra and its particularities |
title_sort |
Dirac equation solution in the light front via linear algebra and its particularities |
author |
Sales, Jorge Henrique de Oliveira |
author_facet |
Sales, Jorge Henrique de Oliveira Aragão, Gabriel de Oliveira Nascimento, Diego Ramos do Thibes, Ronaldo |
author_role |
author |
author2 |
Aragão, Gabriel de Oliveira Nascimento, Diego Ramos do Thibes, Ronaldo |
author2_role |
author author author |
dc.contributor.author.fl_str_mv |
Sales, Jorge Henrique de Oliveira Aragão, Gabriel de Oliveira Nascimento, Diego Ramos do Thibes, Ronaldo |
dc.subject.por.fl_str_mv |
relativity Minkowski space coordinate system fermions relatividade, espaço de Minkowski, sistema de coordenadas, férmions |
topic |
relativity Minkowski space coordinate system fermions relatividade, espaço de Minkowski, sistema de coordenadas, férmions |
description |
In undergraduate and postgraduate courses, it is customary to present the Dirac equation defined in a space of four dimensions: three spatial and one temporal. This article discusses aspects of the Dirac equation (QED) on the light front. This proposal of coordinate transformations comes from Dirac who originally introduced three distinct forms of relativistic dynamics possible depending on the choice we make of the different hypersurfaces constant in time. The first he called instantaneous, the most common form, the hypersurface of which is specified by the boundary conditions set at . The second, known as the point form, has as its characterizing surface, a hyperboloid, described by the initial conditions in , being one constant (chosen as the time of this system). The third relativistic form, known as the light front form, has its hypersurface tangent to the light cone; being defined by the initial conditions at , and is the time in the light front system. The method of this work is deductive. Therefore, one obtains the solution of the Dirac equation for the Free Electron and for the positron in the coordinates in the light front with the particularity of the energy associated with the system being given by , and for moments we have the electron and we have the positron. The result of this is that the positive energy states in the light front and negative are independently described in the equation, and with additional, the problem at the limit that does not converge. |
publishDate |
2023 |
dc.date.none.fl_str_mv |
2023-08-17 |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
https://periodicos.ufv.br/jcec/article/view/16329 10.18540/jcecvl9iss9pp16329-01e |
url |
https://periodicos.ufv.br/jcec/article/view/16329 |
identifier_str_mv |
10.18540/jcecvl9iss9pp16329-01e |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
https://periodicos.ufv.br/jcec/article/view/16329/8137 |
dc.rights.driver.fl_str_mv |
Copyright (c) 2023 The Journal of Engineering and Exact Sciences https://creativecommons.org/licenses/by/4.0 info:eu-repo/semantics/openAccess |
rights_invalid_str_mv |
Copyright (c) 2023 The Journal of Engineering and Exact Sciences https://creativecommons.org/licenses/by/4.0 |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.publisher.none.fl_str_mv |
Universidade Federal de Viçosa - UFV |
publisher.none.fl_str_mv |
Universidade Federal de Viçosa - UFV |
dc.source.none.fl_str_mv |
The Journal of Engineering and Exact Sciences; Vol. 9 No. 9 (2023); 16329-01e The Journal of Engineering and Exact Sciences; Vol. 9 Núm. 9 (2023); 16329-01e The Journal of Engineering and Exact Sciences; v. 9 n. 9 (2023); 16329-01e 2527-1075 reponame:The Journal of Engineering and Exact Sciences instname:Universidade Federal de Viçosa (UFV) instacron:UFV |
instname_str |
Universidade Federal de Viçosa (UFV) |
instacron_str |
UFV |
institution |
UFV |
reponame_str |
The Journal of Engineering and Exact Sciences |
collection |
The Journal of Engineering and Exact Sciences |
repository.name.fl_str_mv |
The Journal of Engineering and Exact Sciences - Universidade Federal de Viçosa (UFV) |
repository.mail.fl_str_mv |
|
_version_ |
1808845241351929856 |