The Multisymplectic Geometry of Classical Field Theories on Finite-Dimensional Covariant
[eng] This thesis presents new developments in the De Donder–Weyl formulation of classical field theories, which treats space and time on an equal footing. The formal Lagrangian geometric construction of field theories takes place on manifolds of jets, giving rise to the standard Euler–Lagrange equa...
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| Formato: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2025 |
| País: | España |
| Recursos: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/226530 |
| Acesso em linha: | https://hdl.handle.net/2445/226530 http://hdl.handle.net/10803/696523 |
| Access Level: | acceso abierto |
| Palavra-chave: | Geometria diferencial Física matemàtica Relativitat general (Física) Simetria (Física) Differential geometry Mathematical physics General relativity (Physics) Symmetry (Physics) |
| Resumo: | [eng] This thesis presents new developments in the De Donder–Weyl formulation of classical field theories, which treats space and time on an equal footing. The formal Lagrangian geometric construction of field theories takes place on manifolds of jets, giving rise to the standard Euler–Lagrange equations, while the equivalent Hamiltonian formulation of De Donder–Weyl takes place on affine dual jets, giving rise to the Hamilton–De Donder–Weyl equations. Manifolds of jets (and their duals) act as finite-dimensional phase spaces, and their multisymplectic geometry acts as a generalization of symplectic geometry in classical mechanics, underpinning variational calculus and providing a powerful set of tools for the investigation of Noether symmetries and bonds. In addition, the multisymplectic construction of relativistic field theories preserves covariance throughout the analysis of these theories and, by working on sections of jet manifolds (and their duals) over space-time, it is possible to derive the covariant phase space formalism from Zuckerman [152], Crnković and Witten [34], and Lee and Wald [108], best known in physical literature and, in general, of infinite dimension. The original contributions of this thesis include new properties of De Donder–Weyl bonds and natural symmetries, which we present as recently proven mathematical propositions and can be found in the following publications: [63, 75, 76]. This work also provides a multisymplectic construction of the Poisson parentheses introduced by Marsden et al. [116] to obtain the Hamilton–De Donder–Weyl equations from the action functional, in direct analogue with the analysis of classical mechanics. In addition, we show how our geometric interpretation of these Poisson parentheses also leads to field equations and discuss their relationship to the infinite-dimensional covariant formalism mentioned above. Finally, we offer a first step towards understanding the multisymplectic geometry associated with the BRST–BV analysis of gauge theories in the Lagrangian framework. After constructing the BRST symmetry, we find the corresponding multimoment application and show that it provides the standard BRST charge except for a total derivative. The novelties presented in this thesis offer a new selection of geometric techniques that allow us to move from the infinite-dimensional language of local functionals to the finite-dimensional language of vector fields and differential forms, treating space and time on an equal footing. |
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