Formulação Feynman–Gell-Mann para o movimento fermiônico planar
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Data
2020-10-16
Autores
Mendrot, Vitor Bentes
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Editor
Universidade Estadual Paulista (Unesp)
Resumo
Realizamos um estudo minucioso de uma formulação alternativa de segunda ordem da equação
de Dirac, a formulação Feynman–Gell-Mann. Discutimos as condições que devem ser satisfeitas
para se obter uma corrente de probabilidade conservada, e a partir disso construímos a matriz
potencial de interação mais geral possível em 3+1 dimensões. Apresentamos a motivação,
seguida da discussão e derivação completa da formulação Feynman–Gell-Mann, generalizando-a
para incluir interações além das abordadas por Feynman e Gell-Mann, resultado que expande
o horizonte de possibilidades de emprego deste ferramental. Por mérito de completeza do
estudo, a partícula livre foi estudada. O limite não-relativístico e algumas considerações sobre a
exequibilidade do emprego da formulação em um mundo com 2+1 dimensões também foram
tratados. Neste trabalho focamos no estudo das interações vetoriais, da qual faz parte a interação
eletromagnética minimamente acoplada. Empregando o sistema de coordenadas cilíndricas no
estudo do movimento planar, obtivemos soluções analíticas para um férmion imerso em duas
configurações de campos circularmente simétricos, a saber, um campo magnético uniforme e um
campo elétrico radial inversamente quadrático. Verificamos que dada a natureza da formulação, é
possível mapear soluções de problemas da teoria não-relativística no nosso problema em ambos
os casos, estratégia que certamente pode ser usada para outros sistemas. Conjecturas sobre
futuras investigações foram feitas
A meticulous study about a second order alternative version of the Dirac equation, the Feynman– Gell-Mann formulation, was carried out. The necessary conditions to obtain a conserved probability current were discussed, and the most general interaction potential matrix in 3+1 dimensions was built based on it. It was presented a motivation, followed by a complete discussion and derivation of the Feynman–Gell-Mann formulation, generalizing it to include interactions beyond those covered by Feynman and Gell-Mann, a result which expands the horizon of possibilities of work with such a tool. For the sake of completeness of the discussion, the free particle was covered. The non-relativistic limit, and some considerations about the feasibility of the formulation in a (2+1)-dimensional world were also treated. In the present work, we focused on the study of vector interactions, which contains the minimally coupled electromagnetic interaction. Using the cylindrical coordinate system to describe the planar motion, we obtained analytical solutions for a fermion embedded in two circularly symmetric field configurations, namely, a uniform magnetic field and an inversely quadratic electric field. It was verified that due to the nature of the formulation, it is possible to map solutions of the non-relativistic theory to our problem in both cases, strategy which can certainly be used in other configurations. Conjectures about possible future investigations were made.
A meticulous study about a second order alternative version of the Dirac equation, the Feynman– Gell-Mann formulation, was carried out. The necessary conditions to obtain a conserved probability current were discussed, and the most general interaction potential matrix in 3+1 dimensions was built based on it. It was presented a motivation, followed by a complete discussion and derivation of the Feynman–Gell-Mann formulation, generalizing it to include interactions beyond those covered by Feynman and Gell-Mann, a result which expands the horizon of possibilities of work with such a tool. For the sake of completeness of the discussion, the free particle was covered. The non-relativistic limit, and some considerations about the feasibility of the formulation in a (2+1)-dimensional world were also treated. In the present work, we focused on the study of vector interactions, which contains the minimally coupled electromagnetic interaction. Using the cylindrical coordinate system to describe the planar motion, we obtained analytical solutions for a fermion embedded in two circularly symmetric field configurations, namely, a uniform magnetic field and an inversely quadratic electric field. It was verified that due to the nature of the formulation, it is possible to map solutions of the non-relativistic theory to our problem in both cases, strategy which can certainly be used in other configurations. Conjectures about possible future investigations were made.
Descrição
Palavras-chave
Equação de Dirac, Formulação Feynman–Gell-Mann, Movimento relativístico planar, Dirac equation, Feynman–Gell-Mann formulation, Relativistic planar motion, Dirac, equações de, Feynman, Diagramas de, Teoria quântica de campos