A general-covariant statistical framework capable of describing classical fluctuations of the gravitational field is a thorny open problem in theoretical physics, yet ultimately necessary to understand the nature of the gravitational interaction, and a key to quantum gravity. Inspired by Souriau’s symplectic generalization of the Maxwell–Boltzmann–Gibbs equilibrium in Lie group thermodynamics, we investigate a space–time-covariant formulation of statistical mechanics for parametrized first-order field theories, as a simplified model sharing essential general covariant features with canonical general relativity. Starting from a covariant multi-symplectic phase space formulation, we define a general-covariant notion of Gibbs state in terms of the covariant momentum map associated with the lifted action of the diffeomorphisms group on the extended phase space. We show how such a covariant notion of equilibrium encodes the whole information about symmetry, gauge and dynamics carried by the theory, associated with a canonical spacetime foliation, where the covariant choice of a reference frame reflects in a Lie algebra-valued notion of local temperature. We investigate how physical equilibrium, hence time evolution, emerges from such a state and the role of the gauge symmetry in the thermodynamic description.

中文翻译：

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参数化场论的协变动量图热力学

能够描述引力场经典涨落的一般协变统计框架是理论物理学中一个棘手的开放性问题，但最终对于理解引力相互作用的本质是必要的，也是量子引力的关键。受 Souriau 对李群热力学中 Maxwell-Boltzmann-Gibbs 平衡的辛推广的启发，我们研究了用于参数化一阶场论的统计力学的时空协变公式，作为与规范一般共享基本一般协变特征的简化模型相对论。从协变多辛相空间公式开始，我们根据与微分同胚群在扩展相空间上的提升作用相关的协变动量图来定义吉布斯状态的一般协变概念。我们展示了这种平衡的协变概念如何编码该理论所承载的有关对称性、规范和动力学的全部信息，与规范时空叶理相关，其中参考系的协变选择反映在局部温度的李代数值概念中. 我们研究了物理平衡是如何从这种状态中产生的，以及规范对称性在热力学描述中的作用。其中参考系的协变选择反映在局部温度的李代数值概念中。我们研究了物理平衡是如何从这种状态中产生的，以及规范对称性在热力学描述中的作用。其中参考系的协变选择反映在局部温度的李代数值概念中。我们研究了物理平衡是如何从这种状态中产生的，以及规范对称性在热力学描述中的作用。