Da teoria clássica à quântica de campos através da abordagem de Faddeev-Jackiw

• Ano de defesa: 2024
• Autor(a) principal: Caro Mendoza, Luis Gabriel [UNESP]
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: por
• Instituição de defesa: Universidade Estadual Paulista (Unesp)
• Palavras-chave em Português: Teoria quântica de campos; Geometria simplética; Integrais de trajetórias
• Link de acesso: https://hdl.handle.net/11449/258236

The advent of analytical mechanics entrenched the existence of two equivalent approaches to systematically study the behavior of a given system, namely, the Lagrangian and Hamiltonian formulations. Later, it was discovered that quantum mechanics could also be expressed in both languages, and therefore the respective quantization procedures commonly adopt one, and only one, of those alternatives. Notwithstanding, it is possible to show that in the Faddeev-Jackiw formulation, the underlying disjunction is not exclusive, since the former provides the necessary tools to develop both quantization alternatives in a unique formalism. This thesis presents a reformulation proposal for the Faddeev-Jackiw approach — in a field theory context — from a functional differential geometry perspective since all the involved objects behave as functionals over the space variables, as a consequence of the space-time foliation. Besides, the formalism is endowed with a $\mathbb{Z}_2-$grading to incorporate pseudoclassical fermionic fields (Grassmannian). Next, it is demonstrated that Darboux's theorem plays the role of a bridge to the path integral formulation, thus reaching the Lagrangian territory. It is worth mentioning that each step of such construction is accompanied by illustrative examples of different versions of electrodynamics, highlighting the model baptized as \textit{Generalized Maxwell-Stückelberg Electrodynamics} (GMSE), which is a second-order theory (à la Podolsky) in which the gauge field is massive, but $U(1)$ gauge freedom is preserved with the help of the Stückelberg mechanism. In addition, it is possible to get the other electromagnetic theories as different limiting cases of GMSE. One of the main features of this model is its finite behavior in the UV regime in the 1-loop approximation, as is shown within this thesis.