经典场论的几何表述：Bonzom-Livine引力模型
A review on geometric formulations for classical field theory: the Bonzom-Livine model for gravity
摘要：
出于与广义相对论相关的物理模型的研究，我们回顾了一些有限维，几何和协变的配方，使我们能够在一个简单的方式来表征的对称性，经典场论通过实施一个适当的纤维束结构，无论是在拉格朗日，多辛或多辛水平。特别是，我们能够制定诺特定理的协变动量映射的装置，并系统地介绍了协变泊松-哈密顿框架。此外，通过专注于一般的经典场论的空间加时间分解及其与这些几何公式的关系，我们能够成功地恢复规范内容和真正的局部自由度的理论。为了说明这些几何框架的相关性，我们集中我们的注意力分析的模型$3$维理论的广义相对论，涉及一个任意的Immirzi参数。在拉格朗日水平上，我们再现了系统的场方程，对于这个模型，它等价于消失挠率条件和爱因斯坦方程。我们还集中在系统的规范对称性的分析，以获得与理论相关的拉格朗日协变动量映射，因此，其相应的Noether电流。接下来，我们的目标是我们的注意力来描述如何的规范对称性的模型产生协变正则变换的协变多动量相空间，从而产生的存在的协变动量映射。此外，我们分析的物理系统中考虑的De Donder-Weyl正则理论在多辛水平上实现，从而建立从协变动量映射的理论的守恒电流的关系。
英文摘要：
Motivated by the study of physical models associated with General Relativity, we review some finite-dimensional, geometric and covariant formulations that allow us to characterize in a simple manner the symmetries for classical field theory by implementing an appropriate fibre-bundle structure, either at the Lagrangian, the multisymplectic or the polysymplectic levels. In particular, we are able to formulate Noether's theorems by means of the covariant momentum maps and to systematically introduce a covariant Poisson-Hamiltonian framework. Also, by focusing on the space plus time decomposition for a generic classical field theory and its relation to these geometric formulations, we are able to successfully recover the gauge content and the true local degrees of freedom for the theory. In order to illustrate the relevance of these geometric frameworks, we center our attention to the analysis of a model for $3$-dimensional theory of General Relativity that involves an arbitrary Immirzi-like parameter. At the Lagrangian level, we reproduce the field equations of the system which for this model turn out to be equivalent to the vanishing torsion condition and the Einstein equations. We also concentrate on the analysis of the gauge symmetries of the system in order to obtain the Lagrangian covariant momentum map associated with the theory and, consequently, its corresponding Noether currents. Next, we aim our attention to describing how the gauge symmetries of the model yield covariant canonical transformations on the covariant multimomenta phase-space, thus giving rise to the existence of a covariant momentum map. Besides, we analyze the physical system under consideration within the De Donder-Weyl canonical theory implemented at the polysymplectic level, thus establishing a relation from the covariant momentum map to the conserved currents of the theory.
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