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Doctoral Thesis
DOI
https://doi.org/10.11606/T.45.2001.tde-20210729-124323
Document
Author
Full name
Sandro Vieira Romero
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2001
Supervisor
Title in Portuguese
Colchete de Poisson covariante na teoria geométrica dos campos
Keywords in Portuguese
Geometria Diferencial
Abstract in Portuguese
O principal resultado deste trabalho é a derivação do colchete de Peierls como colchete de Poisson associado à forma simplética no espaço de fase covariante, introduzida por Crnkovic, Witten e Zuckerman. A ligação entre estas duas estruturas, presentes na literatura há algum tempo mas estudadas de forma separada, permite o esclarecimento de ambas, contribuindo para o enriquecimento da teoria geométrica dos campos. Também é mostrado que a forma simplética de Witten pode ser obtida, de um modo bastante simples e direto, a partir da forma de Cartan, estabelecendo uma conexão com o formalismo multisimplético. Em todas essas abordagens, um papel importante é desempenhado por espaços e fibrados afins, ao contrário dos vetoriais, o que motivou um tratamento mais sistemático do que o normalmente encontrado na literatura. Esta maquinaria algébrica, discutida no primeiro capítulo, é particularmente útil para formular a transformação inversa de Legendre, introduzida no terceiro capítulo, que é essencial para caracterizar a equivalência do formalismo lagrangiano e hamiltoniano no caso não-degenerado mas é desenvolvido de maneira incompleta na literatura
Title in English
not available
Abstract in English
The main result of this thesis is the derivation of the Peierls bracked of covariant field theory as the Poisson bracket associated with the symplectic form on covariant phase space, as introduced by Crnkovic, Witten and Zuckerman. The link between these two structures, which have been present in the literature for some time but have up to now remained separate, sheds new light on both of them, enrichening the covariant formulation of geometric field theory. It also shown that Witten's symplectic form can be obtained in a very simple and direct manner from the Cartan form, thus establishing a connection with the multisymplectic formalism. In all these approaches, an important role is played by affine spaces and bundles, as opposed to vector spaces and vector bundles, which motivates a more systematic treatment of these concepts than that normally found in the literature. This algebraic machinery, discussed in the first chapter, is particularly useful for formulating the concept of the inverse Legendre transform introduced in third chapter, which is important to show the equivalence between the lagrangian and hamiltonian formulation in the non-degenerate case but is only incompletely developed in the literature
 
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Publishing Date
2021-07-29
 
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