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Doctoral Thesis
DOI
10.11606/T.45.2014.tde-07112014-152249
Document
Author
Full name
Luiz Henrique Pereira Pêgas
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2014
Supervisor
Committee
Forger, Frank Michael (President)
Bursztyn, Henrique
Hoefel, Eduardo Outeiral Correa
Mariano, Hugo Luiz
Struchiner, Ivan
Title in Portuguese
Grupoides de Lie e o teorema de Noether na formulação lagrangiana da teoria clássica de campos
Keywords in Portuguese
grupoides
simetrias
teoria clássica de campos
Abstract in Portuguese
O objetivo desta tese é oferecer um arcabouço que permita a modelagem de simetrias em fibrados suaves, que possuam um bom comportamento local. Para tanto, usa-se ferramentas de grupoides de Lie e correlatas, com a finalidade de reduzir, quando possível, simetrias dadas pela ação de um grupo diferenciável, possivelmente de dimensão infinita, sobre um fibrado suave, a problemas em dimensão finita. Uma definição de invariância de uma forma diferencial, definida no espaço total de um fibrado suave, sob a ação de um grupoide de Lie, é apresentada e desenvolvida. A seguir, discute-se estas ferramentas no contexto da formulação lagrangiana da teoria clássica de campos com o objetivo de descrever, simultaneamente, simetrias internas e no espaço-tempo, de maneira unificada. Obtém-se então, nesta linguagem, alguns objetos de estudo centrais da teoria, como os teoremas de Noether e, no caso das teorias de calibre, os teoremas de acoplamento mínimo e Utiyama. Por fim, discute-se brevemente o caso de simetrias a menos de elementos de contato e divergências totais.
Title in English
Lie groupoids and Noether's theorem in the Lagrangian formalism of classical field theory
Keywords in English
classical field theory
groupoids
symmetries
Abstract in English
The aim of this thesis is to provide a framework that allows the modelling of symmetries in smooth fibre bundles which have good local behaviour. For that, we use Lie groupoids and related tools in order to reduce, whenever possible, symmetries given by the action of a possibly infinite dimensional differentiable group on a smooth fibre bundle to finite dimensional problems. We give a definition of invariance of a differential form, defined on the total space of a fibre bundle, by the action of a Lie groupoid. Then, we discuss these tools in the case of a Lagrangian classical field theory to describe internal and space-time symmetries simultaneously, in a unified way. With this language, we get some central objects of the theory such as Noether's theorems and, in the case of gauge theories, the minimal coupling and Utiyama's theorems. Lastly, we briefly discuss the case of symmetries up to contact elements and a total divergence.
 
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Publishing Date
2015-01-05
 
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