Covariant Momentum Map Thermodynamics for Parametrized Field Theories

TL;DR

This paper develops a covariant statistical mechanics framework for parametrized field theories, capturing gravitational fluctuations and symmetry, and introduces a covariant notion of equilibrium linked to gauge and spacetime foliation.

Contribution

It proposes a novel covariant Gibbs state formulation for parametrized field theories, extending thermodynamics to a general covariant setting inspired by symplectic geometry.

Findings

Defines a covariant Gibbs state using the momentum map and diffeomorphism group action.

Shows how equilibrium encodes symmetry, gauge, and dynamics in a covariant manner.

Explores emergence of time evolution and the role of gauge symmetry in thermodynamics.

Abstract

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 spacetime-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 to 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.