Covariant Momentum Map Thermodynamics for Parametrized Field Theories

Inspired by Souriau's symplectic generalization of the Maxwell-Boltzmann-Gibbs equilibrium in Lie group thermodynamics, we investigate a spacetime-covariant formalism for statistical mechanics and thermodynamics in the multi-symplectic framework for relativistic field theories. A general-covariant Gibbs state is derived, via a maximal entropy principle approach, in terms of the covariant momentum map associated with the lifted action of the diffeomorphisms group on the extended phase space of the fields. Such an equilibrium distribution induces a canonical spacetime foliation, with a Lie algebra-valued generalized notion of temperature associated to the covariant choice of a reference frame, and it describes a system of fields allowed to have non-vanishing probabilities of occupying states different from the diffeomorphism invariant configuration. We focus on the case of parametrized first-order field theories, as a concrete simplified model for fully constrained field theories sharing fundamental general covariant features with canonical general relativity. In this setting, we investigate how physical equilibrium, hence time evolution, emerge from such a state via a gauge-fixing of the diffeomorphism symmetry

Quantization of Midisuperspace Models

We give a comprehensive review of the quantization of midisuperspace models. Though the main focus of the paper is on quantum aspects, we also provide an introduction to several classical points related to the definition of these models. We cover some important issues, in particular, the use of the principle of symmetric criticality as a very useful tool to obtain the required Hamiltonian formulations. Two main types of reductions are discussed: those involving metrics with two Killing vector fields and spherically symmetric models. We also review the more general models obtained by coupling matter fields to these systems. Throughout the paper we give separate discussions for standard quantizations using geometrodynamical variables and those relying on loop quantum gravity inspired methods.Comment: To appear in Living Review in Relativit

Symmetry and Evolution in Quantum Gravity

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