Covariant Momentum Map Thermodynamics for Parametrized Field Theories

Abstract

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.

A review on geometric formulations for classical field theory: the Bonzom-Livine model for gravity

“…the Euler-Lagrange field equations ( 8) are completely equivalent to the De Donder-Weyl-Hamilton field equations (28). To see this, we start by emphasizing that for a non-singular Lagrangian system the covariant Legendre map ( 24) is a diffeomorphism [8].…”

“…To see this, we start by emphasizing that for a non-singular Lagrangian system the covariant Legendre map ( 24) is a diffeomorphism [8]. Therefore, given V ∈ X 1 (J 1 Y ) and W := T F DW L (V ) ∈ X 1 (P ) a pair of arbitrary vector fields, a straightforward calculation shows that, if a section j 1 φ ∈ J 1 Y X is a solution of the Euler-Lagrange field equations ( 8), then the section ̺ := F DW L•j 1 φ ∈ P X is a solution of the De Donder-Weyl Hamilton field equations (28), and vice versa.…”

“…, we can construct a conserved current for the solutions of the De Donder-Weyl-Hamilton field equations [6,8]. In other words, we have that, for all ξ η ∈ g and for any solution of the De Donder-Weyl-Hamilton field equations (28), ̺ ∈ P X , the quantity defined by…”

“…To see this, we start by emphasizing that, it is possible to write J (L) (ξ η ) = FL * J (M) (ξ η ). Furthermore, we know that, for a non-singular Lagrangian system the covariant Legendre map ( 24) is a diffeomorphism, and hence if a section ̺ ∈ P X is a solution of the De Donder-Weyl-Hamilton field equations (28), then the section…”

“…Such canonical form, known as the polysymplectic form, encodes the relevant physical data of a given classical field theory in order to construct a well-defined Poisson-Gerstenhaber bracket for a set of appropriately prescribed differential Hamiltonian forms, thus allowing us to analyze an arbitrary classical field theory in a covariant Poisson-Hamiltonian framework [20,21,22]. Some physically motivated examples for which the multisymplectic and polysymplectic formalisms have been applied may be encountered in references [5,8,9,20,23,24,25,26,27,28,29,30,31,32,33,34]. Despite their mathematical elegance, from our point of view, the analysis of the gauge content for a given classical field theory from the perspective of such geometric formulations has been rarely exploited, especially when considering certain highly non-trivial gauge models associated with General Relativity.…”