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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.


Introduction
In the attempt to shed light on the nature of gravity at the quantum scale, and to understand how continuum spacetime and its general-relativistic dynamics emerge in the classical regime, both string theory [1] and non-perturbative approaches to quantum gravity [2] have incorporated a pletora of concepts and tools from statistical mechanics, condensed matter physics and information theory, with an exceptional interdisciplinary effort. Nevertheless, a general covariant framework for describing the statistical fluctuations of the gravitational field, hence of spacetime geometry, is still missing, the very definition of a spacetime covariant formulation for statistical mechanics being a thorny open issue [3]. At the heart of the problem lies the conceptual clash of Einstein's general covariant scheme, characterised by systems with vanishing canonical Hamiltonian, with statistical mechanics, whose classical formulation grounds on the notion of Hamiltonian time flow and energy [4]. Important insights towards a general covariant formulation of statistical mechanics came from the thermal time hypothesis framework [5], inspired by the Tomita-Takasaki theorem in algebraic quantum field theory [6]. In this framework, timeless evolution is realised by the modular one-parameter flow of automorphisms of the covariant space algebra of the system, which can This leaves open the problem of characterizing the states that are in physical equilibrium, as any state is stationary with respect to its own flow. In [7], a physical characterisation of such general covariant Gibbs states has been proposed to be associated with states whose thermal time is a flow in spacetime.
Most work in this direction [7,8,9,10,11,12,13,14,15,16,17,18,19] grounded on these ideas, seeking for new physical characterizations of equilibrium, by means of a covariant extension of the fundamental postulates of ordinary statistical physics. These results have been so far limited to simple low dimensional parametrized mechanical systems, and a complete field-theoretic formulation of the problem is missing [7,17].
In this work, we consider the problem of the physical characterisation of a general covariant Gibbs' equilibrium within a multi-symplectic framework for relativistic field theories, by introducing an extended off-shell, epistemic notion of equilibrium [20], built in terms of the conserved covariant momentum map induced by the Hamiltonian flow of the diffeomorphism symmetry group on the extended phase space of the theory.
In Einstein's theory of gravity, as in any parametrized classical field theory, dynamics is fully characterized by the initial value constraints and inherently related to the gauge freedom of the theory [21,22,23,24,25,26,27]. Such an intimate relation between gauge symmetry and dynamics, between covariant and canonical formulation, is encoded in the notion of covariant momentum map in a multi-symplectic formalism [21,22,23,28,29]. For (first order) parametrized field theories, canonical initial value constraints coincide with the vanishing of the instantaneous reduction of the covariant momentum map [21,22,23], associated to the action of the gauge group of the theory on its extended phase-space. Such induced energy-momentum map appears to encode all the dynamical information carried by the theory [21,22,23]. In the canonical formalism, first class constraints generating the diffeomorphism symmetry are encoded in the normal and tangential components of the instantaneous reduction of the covariant momentum map so that the constraint surface is identified with its zero level set [22,23,26,27].
In the ADM formulation of gravity, for instance, the super-hamiltonian and super-momenta are the components of such energy-momentum map, reflecting the gauge symmetry (diffeomorphism covariance) of the full theory in the instantaneous setting [30,31].
In general, group-theoretic terms, the notion of momentum map is associated to a natural generalization of the Hamiltonian function, which comprises all conserved charges associated to the symplectic action of some dynamical group on a given phase space [32]. In this terms, the momentum map can be naturally used to generalize the Maxwell-Boltzmann-Gibbs approach to thermodynamics [33] to the case where the energy function is vector-valued and not just with m log(m) = 0 if m = 0, and it measures the amount of uncertainty (or broadness) represented by the probability density ρ. For a given constant mean value of the momentum map J, thermodynamic equilibria µ eq are states which maximize the entropy [20,33], and such that S(ρ eq ) is stationary with respect to all infinitesimal smooth variations of the probability density.
The quantity of the variable ξ ∈ g, taking value in R and in g * , respectively. In this sense, the (co)momentum map provides a natural vector-valued generalisation of the Hamiltonian function, retaining the operational information of all conserved charges associated to the symplectic action of a dynamical group on a given system's phase space. In particular, this leads to a remarkable covariant generalization of Gibbs's equilibrium as soon as we consider the action of the symmetry group of the system (e.g. Galileo, Poincaré) on its manifold of motions [34].
Souriau's geometric generalisation of Gibbs equilibrium essentially relies on the symplectic character of the phase space and on the Hamiltonian nature of the group action. Such a phase space coincides with the fully constrained phase space of the system, thereby providing an on-shell definition of covariant equilibrium state, analogous to the one later developed by the thermal time hypothesis.
In this work, we explore a radical conceptual extension of Souriau's Lie group thermodynamics to the framework of general covariant, or reparametrization invariant, field theories, where the covariant symmetry group of the system is gauge and the symplectic phase space consists of the full extended phase space of fields. We focus on parametrized field theories considered as paradigmatic models for diffeomorphism-covariant theories sharing important features with general relativity. In particular, to emphasise on the key geometric ingredients for our analysis, we work in the multi-symplectic framework for first order parametrized field theories, where a notion of covariant (multi)momentum map associated to the action of spacetime diffeomorphisms can be constructed. In absence of boundaries in the underlying spacetime base manifold, the pre-symplectic nature of both the constraint surface and the space of solutions of the field equations prevents us from representing the diffeomorphism group via a Poisson algebra of functions on the on-shell field configurations as the corresponding co-momentum Hamiltonian would be trivial [35]. This issue requires to move the entire analysis at the off-shell level. Here, building on the notion of covariant (multi)momentum map and on the geometric characterization of the parametrization procedure introduced in [36,37], where diffeomorphisms are promoted to dynamical fields themselves (covariance fields), we are able to define a spacetime covariant notion of equilibrium state for fully constrained field theories. The representation of the algebra of diffeomorphisms on the parametrized phase space of fields is achieved by taking into account the lifted action of spacetime diffeomorphisms on the embeddings naturally induced by that on the covariance fields. As a side result, this also allows us to recover the procedure introduced by Isham and Kuchař in [38] for the case of a scalar field.  [21,22,23] and references within to which we refer for a more detailed exposition.
Other works on this topic which include also different starting points are [39,40,41,42].

Setup
Let X be an oriented (n + 1)-dimensional manifold, which in many examples is spacetime, and let Y π X Y − −− → X be a finite-dimensional fiber bundle over X whose fibers Y x over x ∈ X have dimension N . This is called the configuration bundle and is the field theoretic analogue of the configuration space in classical mechanics. Physical fields correspond to sections of this bundle. A set of local coordinates (x µ , y A ) on Y is provided by the n+1 local coordinates x µ , µ = 0, . . . , n, on X and the N fiber coordinates y A , A = 1, . . . , N , which represent the field components at a given point x ∈ X . As discussed in [43,44], this notion of extended configuration space for field theories has a nice operational motivation based on the observation that coordinates of Y (i.e., field values and spacetime positions) are the partial observables of the theory [45]. Indeed, one needs N measuring devices to measure the components of the field at a given point x ∈ X , and n + 1 devices to determine x 2 thus resulting in a (n+N +1)-dimensional configuration space. A point in Y represents a correlation between these observables, that is, a possible outcome of a simultaneous measurement of the partial observables.
The Lagrangian density for a first order classical field theory is given by where J 1 (Y) is the first jet bundle 3 of Y and Λ n+1 (X ) is the space of (n + 1)-forms on X . The first jet bundle J 1 (Y) of Y here plays the role of the field-theoretic analogue of the tangent bundle of classical mechanics 4 . Local coordinates (x µ , y A ) on Y induce coordinates v A µ on the 2 To determine an event in a n+1-dimensional spacetime we need one clock and n devices giving us the distance from n reference objects. 3 For higher order theories, a k th -order Lagrangian density will be defined on the k th jet bundle J k (Y) of Y [46]. 4 In this case Y = R × Q is the extended configuration space regarded as an R-bundle over Q, and J 1 (Q × R) is isomorphic to the bundle T Q × T R.
fibers of J 1 (Y) so that the first jet prolongation where y A ,µ = ∂ µ y A and ∂ µ = ∂/∂x µ . The Lagrangian then reads where d n+1 x = dx 0 ∧ · · · ∧ dx n is the volume form on X .
By introducing the multimomenta p µ A and the covariant Hamiltonian p defined via Legendre transformation as the field-theoretic analogue of the phase space of classical mechanics is provided by the socalled multiphase space Z defined as the sub-bundle of 2-horizontal (n + 1)-forms on Y whose elements can be uniquely written in terms of the fiber coordinates (p, p A µ ) as so that the contraction i V i W z with any two vertical vector fields V = V A ∂/∂y A , W = W A ∂/∂y A on Y vanishes. As proved in [21], the space Z is canonically isomorphic to the dual jet bundle J 1 (Y) * , the latter playing the role of the field-theoretic analogue of the cotangent bundle.
In complete analogy to standard symplectic mechanics [47,48], the canonical Poincaré-Cartan (n + 1)-form Θ on Z is given by and the canonical (n + 2)-form Ω on Z is then defined by A remarkable feature of this formalism is that for each field component y A there are multiple momenta p µ A , spatial in addition to temporal. The pair (Z, Ω) is an example of multisymplectic manifold 5 and the usual definitions of classical mechanics on the extended phase space are 5 According to [21,49,50], a multisymplectic manifold (M, Ω) is a manifold endowed with a closed nondegenerate k-form Ω (k = n + 2 in our case), i.e., such that dΩ = 0 and iV Ω = 0 for any nonzero tangent vector recovered when X is one-dimensional (i.e., n = 0) as reported into the following table which summarizes the analogies between classical symplectic mechanics and the multisymplectic formulation of classical field theories: Classical Mechanics (n = 0, X ≡ R) Field Theory (n > 0, dim X = n + 1) extended configuration space configuration bundle over spacetime

Parametrization and Covariance Fields
As it is well known from the pioneering work of Dirac [51], further developed by Kuchař and Isham [38,52,53], field theories with a fixed background metric can be made generally covariant, i.e., with the spacetime diffeomorphism group as symmetry group, by means of the so-called parametrization procedure. Roughly speaking, this amounts to introduce the diffeomorphisms themselves as new dynamical fields so that the covariance group of the theory can be enlarged while leaving the solution space unchanged. A precise geometric reformulation of the parametrization procedure within the context of multi-symplectic field theories, which provides an intrinsic extension and refinement of the idea of treating coordinate changes as fields ("parametrization"), was developed by Castrillón López, Gotay and Marsden in [36,37]. The main steps of the construction are the following: 1) One introduces the so-called covariance fields which are just (oriented) diffeomorphisms of X reinterpreted as sections η : X →X of the bundleX × Xπ − → X where (X , g) is a copy the (spacetime) base manifold.
2) Regarding η as new dynamical fields, the configuration bundle Y is then replaced by the fibered productỸ = Y × X (X × X ) whose sections are thought of as pairs (φ, η) according Y × X (X × X ) π y y r r r r r r r r r r r r where, denoting coordinates onX by u a (a = 0, . . . , n), π :Ỹ → Y and π r :Ỹ →X ×X are the projections on the first and second factor ofỸ respectively acting as π : (x µ , y A , u a ) → (x µ , y A ) and π r : (x µ , y A , u a ) → (x µ , u α ), and π X Y (y) =π(u).
3) The Lagrangian density L of the starting theory is modified by introducing the new Lagrangian densityL defined asL (j 1 φ, j 1 η) := L (j 1 φ, η * g) . (2.10) which, denoting coordinates on J 1 (Ỹ) by (x µ , y A , v A µ , u a , u a µ ) with u a µ the jet coordinates associated to u a , readsL Let then α X ∈ Diff(X ), we denote by α Y ∈ Aut(Y) its lift to Y. This can be extended to an action by bundle automorphisms onỸ by requiring that Diff(X ) acts trivially onX , i.e.
αX :X × X −→X × X by (u, x) −→ (u, α X (x)) . (2.13) The induced action on the spaceỸ ≡ Γ(X ,Ỹ) of sections ofỸ is then given by where generalizes the usual push-forward action on tensor fields, and To sum up, the key point of the construction is that now the fixed background metric g is no longer thought of as living on X , but rather just as a geometric object on the copyX in the fiber of the extended configuration bundleỸ. On the other hand, the metric variable G = η * g on X inherits a dynamical character via the covariance field η. The true dynamical fields of the parametrized theory are thus provided by φ and η. As discussed in [36,37], the Euler-Lagrange equations for the fields φ remains unchanged while those for the covariance fields η give the con-

Covariant Momentum Map
Along the same steps discussed in Sec. 2.1, we can now develop a covariant Hamiltonian formalism for (first order) parametrized field theories by considering the covariant or parametrized multiphase spaceZ ∼ = J 1 (Ỹ) * equipped with a canonical Poincaré-Cartan (n + 1)-form where, taking into account that both φ and η are dynamical variables for the extended theory, the covariant Hamiltonianp and the multimomenta p µ A , µ a (respectively conjugate to the multivelocities v A µ and u a µ ) are defined w.r.t. the Lagrangian (2.11), i.e.
with T µν = 2 ∂L ∂Gµν the so-called Piola-Kirchhoff stress-energy-momentum tensor density [36]. The multisymplectic (n + 2)-formΩ = −dΘ onZ then reads Let now G be a Lie group (perhaps infinite-dimensional) realizing the gauge group of the theory and denote by g its Lie algebra. In the case of generally covariant field theories, G is a subgroup of Aut(Ỹ) covering diffeomorphisms on X . Given an element ξ ∈ g, we denote by ξ X , ξ Y , ξỸ , and ξZ the infinitesimal generators of the corresponding transformations on X , Y,Ỹ, andZ, i.e., the infinitesimal generators on X , Y,Ỹ, andZ of the one-parameter group generated by ξ. The group G is said to act onZ by covariant canonical transformation if the G-action corresponds to an infinitesimal multi-symplectomorphism, i.e.
where L ξZ denotes the Lie derivative along ξZ , while it is said to act by special covariant This is the Hamiltonian counterpart of the G-equivariance property (2.17) of the Lagrangian which in turn amounts to G-invariance of the Lagrangian form ΘL on J 1 (Ỹ) defined as the pull-back ofΘ along the covariant Legendre transform.
In analogy to the definition of momentum maps in symplectic geometry [47,48,55], a covariant momentum map (or a multimomentum map) associated to the action of G oñ Z by covariant canonical transformations is a map given by whereJ (ξ) is the n-form onZ whose value atz ∈Z is J (z), ξ with ·, · being the pairing between the Lie algebra g and its dual g * .
Let then α ∈ G be the transformation associated to ξ ∈ g, the covariant momentum mapJ is said to be Ad * -equivariant ifJ When G acts by special covariant canonical transformations, the (special) covariant momentum map admits an explicit expression given bỹ In particular -and this is the case of interest for parametrized field theories -if the G-action onZ is the lift of an action of G onỸ, then for is a smooth functional of ξ ν which for tensor field theories reads with coefficients C depending on x µ , y B and u b , and reduces to ξ A (x, y, u, in the case of a scalar field. Note that in Eq. (2.27) there are no components in the "u-directions" since, as discussed in the previous section (cfr. Eq. (2.13)), for any α X ∈ Diff(X ), there is a lift α Y ∈ Aut(Y) which is trivially extended toỸ by αỸ : (y, u, x) → (α Y (y), u, α X (x)). In coordinates, Eq. (2.26) then reads

Canonical Phase Space and Energy-Momentum Map
In the covariant formulation fields are defined as sections of bundles over spacetime. The canonical formalism instead relies on fields defined as "time-evolving" cross sections of bundles over a Cauchy surface. Therefore, in order to construct the canonical formulation of a field theory we need to introduce a foliation of spacetime and consequently of the bundles over it. The base manifold X can be thus decomposed into a smooth disjoint union of space-like hypersurfaces 7 . 7 In what follows, we assume the spacetime manifold to be globally hyperbolic, i.e., X ∼ = Σ × R, so that the foliation introduced to construct the canonical formalism covers the whole manifold thus avoiding technicalities concerning the possibility of defining the canonical formalism only locally.
Let then Σ be a compact, oriented, connected, boundaryless 3-manifold and let Emb G (Σ, X ) be the set of all space-like embeddings of Σ in X . A foliation s X of X then corresponds to a where x is a shorthand notation for the spatial coordinates x i , i = 1, . . . , n, on the space-like hypersurface Σ τ = τ (Σ). The generator of s X is a complete vector field ζ X on X everywhere transverse to the slices defined bẏ A foliation of X induces a compatible slicing of bundles over it whose generating vector fields project to ζ X . The flow of such a generating vector field defines a one-parameter group of bundle automorphisms. For parametrized field theories we are interested in a so-called Gslicing in which case the one-parameter group of automorphisms of the extended configuration bundle is induced by a one-parameter subgroup of the gauge group G, i.e., ζỸ = ξỸ for some ξ ∈ g. The corresponding slicing sZ ofZ is then generated by the canonical lift ζZ = ξZ of ξỸ toZ whose flow defines a one-parameter group of bundle automorphisms by special canonical transformations onZ (i.e., L ξZΘ = 0) 8 .
Spatial fields will then be identified with smooth sections of the pull-back bundle Y τ → Σ τ over a Cauchy surface given by ϕ := φ τ = τ * φ. Note that, as the subscript τ is meant to recall, the spatial fields ϕ( x) = φ τ ( x) are functionals of the embedding τ and at the same time functions of the point x on the spatial slice. Moreover, according to the parametrization procedure discussed in Sec. 2.2, the space-like embedding τ ∈ Emb G (Σ, X ) acquires a dynamical character through the covariance fields η. Indeed, we have τ = η −1 •τ for a given space-like embeddingτ ∈ Emb g (Σ,X ) of Σ intoX associated to the slicing ofX w.r.t. the fixed metric g.
The canonical parametrized configuration space then consists of the pairs (ϕ, τ ) of spatial fields defined over a Cauchy slice and the space-like embeddings identifying a G-slicing of spacetime w.r.t. one-parameter subgroups of diffeomorphisms. Let then (x 0 , x 1 , . . . , x n ) be a chart on X adapted to τ , i.e. such that Σ τ is locally a level set of x 0 . Denoting by (ϕ, Π, τ, P ) a point in the , the canonical symplectic structureω τ on T * Ỹ τ reads as [56] ω τ (ϕ, Π, τ, P ) = Following the construction of [22] (cfr. Ch. 5), the multisymplectic structure onZ induces a presymplectic structure on the spaceZ τ of sections of the bundleZ τ → Σ τ given bỹ relating in adopted coordinates the momenta Π A and P a respectively to the temporal components of the multimomenta p µ A and µ a as In particular [22], ker T σ R τ = kerΩ τ (σ) and the canonical parametrized phase space T * Ỹ τ is thus isomorphic to the quotientZ τ / kerΩ τ .
The covariant multimomentum map (2.24) associated to the G-action onZ will then induce a so-called (parametrized) energy-momentum map onZ τ defined bỹ The energy-momentum map is intimately related to the initial value constraints which generate the covariant gauge freedom thus providing on the one hand a fundamental link between dynamics and the gauge group, and on the other hand encoding in a single geometrical object all the physically relevant information about a given classical field theory [22].
To see this, let us denote byP τ the primary constraint submanifold in T * Ỹ τ defined as FL being the Legendre transform. For lifted actions -and this is the case for a G-slicing discussed before -the projectionJ H : P τ → g * onP τ of the parametrized energy-momentum map (2.37) encodes the first class secondary constraints respectively as its components in the transversal and tangential directions to the spatial slice. Indeed at the level of densities, using adapted coordinates and recalling the expressions (2.26), (2.29), we have for any σ ∈Z τ and ζZ the canonical lift of the generator of the G-slicing ζ X = ξ X toZ. Now, from which, by using the fact that it follows that the parametrized energy-momentum map (2.37) induces a functional onP τ given by where P µ = η a ,µ P a is the pull-back of P a to Σ along η. The functional (2.45) onP τ is nothing but the total Hamiltonian whose components in the tangential and transversal direction to the spatial slice yields the super-momenta and Hamiltonian constraints.
As it will be discussed in the next subsection, this functional is a equivariant moment map w.r.t. the G-action on the canonical parametrized phase space and it provides us with a representation of the Lie algebra of Diff(X ) on the parametrized phase space. Differently from the standard case of instantaneous canonical formalism where the equivariance of the corresponding functional is spoiled by the algebra of constraints, the key ingredient which give rise to a Lie algebra (anti)homomorphism between the algebra diff(X ) and the Poisson bracket algebra of observable functionals on the parametrized phase space relies on the introduction of the covariance fields which in turn allows to induce a corresponding G-action on the space of embeddings of Σ into X . The action of the diffeomorphism group on the embeddings τ induced via the action on η was actually what ensured the equivariance of the parametrized energy-momentum map (cfr. Eq. (2.39)).

Representation of Spacetime Diffeomorphisms: Diff(X )-equivariant Momentum Map
Let G = Diff(X ) be the group of diffeomorphisms (i.e., smooth and inventible active point transformations 9 ) of the spacetime manifold X . The Lie algebra g = diff(X ) can be realized as the set of all (complete) vector fields on X . Indeed, for any element ξ ∈ diff(X ), we can associate to it a vector field ξ X ∈ X(X ) generating a one-parameter group α ξ λ of spacetime diffeomorphisms by d dλ In the standard setting of instantaneous canonical formalism, the generating vector field is decomposed into its "lapse" and "shift" components. i.e. it can be written as the sum of two vectors which are respectively normal and tangent vectors to the space-like hypersurface, say where n µ = G µν n ν is the future-pointing normal such that τ * n = 0 for any x ∈ Σ τ and G µν n ν n ν = −1, and N ∈ C ∞ (Σ, X ), N ∈ T Σ are respectively called the lapse function (N ) and the shift vector ( N ) of the foliation [26,31]. They respectively specify the magnitude of the normal and tangential deformation at every point on a spatial hypersurface. The location of a neighbouring slice (i.e. how the slices are embedded in a given spacetime) is determined by specifying lapse and shift which play the role of arbitrary Lagrange multipliers in the action implementing the first-class constraints of the theory.
The algebra diff(X ) of the spacetime diffeomorphism group is however "deformed" by the decomposition into perpendicular and tangential directions to the spatial hypersurfaces. Indeed, the projected constraint functions do not form a genuine Lie algebra but rather a Lie algebroid structure known as the Dirac hypersurface-deformation algebra [51,58,59] which, after smearing with lapse and shift, reads 10 and their connection we refer to [57] and references therein. 10 We refer to [38] Sec. 3.3 for the detailed calculation of the constraint algebra for the case of a canonical parametrized scalar field theory.
N k M ,k ), and K is such that given by d the dual of the space of lapses and shifts, i.e., Λ 0 d and Λ 1 d respectively denote the spaces of function densities and 1-form densities on Σ, is not a true momentum map. Indeed, as can be checked by direct computation, from the Dirac algebra (2.48) it follows that i.e., E τ is not (infinitesimally) equivariant. Only if we restrict to the spatial diffeomorphisms, i.e., the subgroup G τ = Diff(Σ τ ) of transformations which stabilize the image of τ , the energymomentum map (2.51) induces a momentum map w.r.t. the G τ -action, say The corresponding generating vector field, and similarly with L 2 -dual pairing given by where Q = τ * G is the induced metric on Σ.
This result is compatible with the procedure introduced by Isham and Kuchař in [38] for the case of a parametrized scalar field theory where the key step in representing spacetime diffeomorphisms was the observation that the embedding variables (which tells us how the model Cauchy surface Σ lies in the spacetime manifold) provide a link between the spatial and spatiotemporal pictures.
At the infinitesimal level, the equivariance property of the Hamiltonian (2.58) can be seen as follows. Using the fact that the Poisson bracket between the unprojected constraint functions vanishes strongly and the canonical Poisson brackets {τ µ ( x), P ν ( x )} = δ µ ν δ( x, x ) between the embeddings and their conjugate momenta as well as that the embedding commutes with H (ϕ) , for any two Lie algebra elements ξ, ζ ∈ diff(X ) whose corresponding vector fields ξ X (τ ( x)), ζ X (τ ( x)) generate one-parameter groups of spacetime diffeomorphisms, we have where the Lie bracket [ξ, ζ] is defined as the opposite of the commutator between the corresponding vector fields, say Finally, the total Hamiltonian (2.58) is constructed in such a way that the constraints are preserved along the flow generated by H(ξ), that iṡ vanishes on the constraint surface. Moreover, as any functional of the embedding commutes with H (ϕ) , we haveτ i.e., ξ X (τ ( x)) is the deformation vector of the foliation which can be decomposed into its transversal ξ (τ ( x)) and normal ξ ⊥ (τ ( x)) components which, unlike the Lagrange multipliers N and N entering the parametrized action, are now specific functionals of the embedding. On the other hand, since P (ξ) commutes with the field variables, the rates of change of the field ϕ and its conjugate momentum Π yield the Hamiltonian field equations with deformation vector ξ X (τ ( x)).
Therefore, the co-momentum map H(ξ) defined in Eq. Note that the explicit embedding-dependence of the induced vector field ξ X on X entering (2.58) implies that H(ξ) comes to be the Hamiltonian function along the flow lines generated by ξ X which correspond to a one-parameter family of emebeddings which in turn identifies a foliation with deformation vector ξ X (τ ( x)). Thus, for any ξ ∈ diff(X ), realized as a complete vector field on X , the induced vector field on Emb G (Σ, X ) is the tangent vector field to a curve of embeddings which identifies a foliation of spacetime. Different Lie algebra elements would identify different foliations whose corresponding deformation vectors are given by the induced vector fields on X restricted to the embedding. In this sense the momentum map H(ξ) (resp.J H (ξ)) provides a faithful representation of the Lie algebra of Diff(X ) on the parametrized phase space.

Covariant Gauge Group Thermodynamics
We can finally proceed to extend the generalized notion of thermodynamic equilibrium statesá la Souriau to parametrized field theories in which the Hamiltonian action we are interested in is that of the spacetime diffeomorphism group Diff(X ) or, more precisely, the lifted action to the parametrized phase space of the automorphisms of the extended configuration bundle covering diffeomorphisms on the base manifold X .

Generally Covariant Gibbs State
To avoid weighing the notation down, in what follows we generically denote the canonical parametrized phase space of the theory by Υ, i.e., Υ ≡ T * Y τ × T * Emb G (Σ, X ).
A statistical state ρ : Υ → R([0, +∞[) on the parametrized phase space is a smooth probability density on Υ such that, for any Borel subset A of Υ, the integral defines a probability measure on Υ with the normalization condition where D[ϕ, Π, τ, P ] formally denotes the integration measure on Υ. To such a statistical state, we can associate a entropy functional The normalization condition (3.2) then implies where we limit b to the subset Ω ⊂ g such that the above integral converge. The generally covariant Gibbs statistical state is given by where ξ (b) denotes the vector field on X associated to b ∈ diff(X ) generating a one-parameter family of spacetime diffeomorphisms. Note that the statistical state is covariant in the sense that the momentum map is evaluated on any space-like hyper-surface without fixing the slicing a priori. The one-parameter group of automorphisms of the extended configuration space generated by b ∈ diff(X ) identifies a generalized concept of "time evolution" w.r.t. which the Gibbs state is of equilibrium.

Generalized Thermodynamic Potentials
Let us assume that the Hamiltonian action of the -G-slicing preserving -one-parameter subgroup of the diffeomorphisms, and its covariant moment mapJ H :P τ −→ g * are such that the ensemble of generalised (inverse) temperatures Ω ⊂ g is non-empty.
Whenever the generalized Gibbs states (3.13) can be defined, the infinite differentiability of the associated partition function Z : Ω → R in (3.12) allows us to define generalized macroscopic functions over Ω, in terms of differentials of Z(b) [32].
The thermodynamic free energy potential, F (b) ≡ − log Z(b), encodes complete thermodynamic information about the system. The equilibrium internal energy Q : Ω → g * is given by the first differential of F (b), taken as a smooth map from g in R, corresponding to the average momentum map in the generalised Gibbs ensemble. Exactly as for standard equilibrium thermodynamics, to a given Q corresponds at most one value of b, so that F (b) and the equilibrium probability density ρ b are uniquely determined.
The entropy function S : Ω → R, as defined in (3.3), has a strict maximum S(b) at equilibrium, given by From the second differential of F (b), for Y, Z ∈ g, b ∈ Ω, one gets a generalised geometric expression for the covariance matrix, In particular, ∀b ∈ Ω, ∀Y ∈ g, the contracted differential defines the concavity of the free energy. On the diagonal, ∀ Y ∈ g, Y = 0, such that J H , Y = const onP τ , the above formula defines a positive definite bilinear symmetric form, which corresponds, in thermodynamic terms [34], to a geometric notion of heat capacity. 12

Canonical vs Microcanonical Imposition of the Constraints
The state (3.13), together with the associated thermodynamic potentials define an equilibrium thermodynamic (or thermostatic) characterisation for the system of fields onP τ . At any fixed temperature, the system is governed by the principle of minimum free energy. Instead of occupying a single definite state, given by the Dirac measure δ(J H (ϕ, Π, τ, P )) which imposes the diffeomorphisms symmetry constraint on Υ, the system of fields is allowed to have different probabilities of occupying different states, and these probabilities will be chosen to minimize the generalized free energy In standard thermodynamics, the principle of minimum free energy reduces to the principle of minimum energy when b → ∞. In our generalised statistical framework, despiteJ H being not necessarily bounded below and the measure D not being finite, the free energy function withσ ∈J −1 H (0) ⊂P τ , corresponding to field configurations satisfying both primary and secondary first class constraints. From the second Noether theorem, we know that field configurationsσ such thatJ H (σ) = 0, represent solutions to the Euler-Lagrange equations [21]. In this sense, we can reformulate symplectic reduction problem for our first order parametrized field theory in thermodynamic terms, as a limiting case of a general principle of maximum entropy 14 [20], and in accordance with the geometrodynamics regained program of Kuchař [63]. To the idea that the geometry and symmetry together determine the field theory, here we add a further statistical characterisation, showing that from symmetry assumptions and the energy-momentum defines the Fisher information matrix, measuring the variance of the equilibrium distribution with respect to the given b. The positive definite form in (3.18) defines a pairing on g. In finite dimensions, for e.g. g R N , this gives a Riemannian metric on the N -dimensional parameter space, the Fisher metric of Information Geometry [60]. This defines an interesting connection between Lie group thermodynamics and Information Geometry first remarked in [62]. 13 Indeed, for (3.17), the derivative of the expected value of the momentum map with respect to the b parameter is always negative, so that the expected value decreases monotonically to zero as b → ±∞.
14 For b such that Q(b), b > 0, minimizing the free energy is the same as maximizing S.
map it is possible to recover the information on the solutions of the Euler-Lagrange equations in some thermodynamic limit, corresponding to the passage from a canonical description to a microcanonical description of the constrained theory space.
In a sense, it is natural to understand the generalized Gibbs state in (3.13) as an off-shell generalisation corresponding to a canonical statistical distribution in which a non-zero weight is assigned to configurations which do not solve the constraint equations, while at the same time weighting more such solutions compared to generic configurations in the thermodynamic limit.
From a statistical viewpoint, the passage from our canonical description to the microcanonical definition of the constrained theory space requires to coarse-grain the information encoded in the algebra element b. 15

Generalized Second Law
In the equilibrium, thermostatic setting described above, different b ∈ Ω identifies different slicings of the base manifold (spacetime), corresponding to different Cauchy problems. Each b corresponds to a choice of frame, which we can identify with a virtual observer, as well as a virtual reservoir for the system. This allows to recast relations among different equilibrium Cauchy slicings in terms of thermodynamic relations.
Transformations among equilibrium states induced by the Hamiltonian action of the group under concern (e.g. whenever two temperatures are related by the adjoint action of the one-d subgroup of the diffeomorphisms) are trivial due to the equivariance of the multi-momentum map. 16 Actual thermodynamical relations require infinitesimal departure from equilibrium.
In the following, we shall then consider how the equilibrium thermodynamic potentials vary due to infinitesimal perturbation our equilibrium state, by introducing an externally controlled parameter s, as shown in 3.1. We can think of s as a work parameter, as we perform work on the system by varying it. We can then denote an equilibrium state of the system ρ s,b by a temperature and a work parameter (see e.g. [64]).
Following a standard procedure, let us evolve the system from one equilibrium state to another, while generally driving it away from equilibrium in the interim. We start with a system at equilibrium ρ i,b , for fixed s = i, and we perturb it by varying the work parameter to a value s = f , while moving along the one-dimensional diffeos flow parameter. Finally, we allow 15 Indeed, we can see the uniform distribution partition function as a weighted superposition of canonical partition functions over various values of the intensive b parameter. 16 See [34] for an explicit derivation of the variations of the thermodynamic potentials under adjoint action of the group, in the generic case the system to somehow re-equilibrate with the equilibrium observer-reservoir and relax to ρ f,b .
We can measure the difference between initial and final states due to the work done while deviating from equilibrium via Kullback-Leibler divergence (KL) [60] The KL predicts that the external work performed on the system is no less than the free energy difference between the final state. Using our notation for the generalised thermodynamic functions, we have By interpreting W ≡J H f −J H i as the work associated to the jump in the energy-momentum where F s (b) is the free energy of the state ρ s,b as defined above. This is the Clausius inequality of classical thermodynamics, which express the essential statement of the second law of thermodynamics for an isothermal transformation.
When the parameter is varied slowly enough that the system remains in equilibrium along the flow, then the process is reversible and isothermal, and E i,b (W ) = ∆F . 17 The work performed during a reversible, isothermal process depends only on the initial and final states and not on the sequence of equilibrium states that mark the journey from Via KL divergence, we can describe the second law also for the case of an adiabathic transformation, where we imagine to keepJ fixed, while changing the temperature b i → b f . In this case, starting from the general expression in (3.23), we use the general definition of the entropy, 17 The case where the same two states at [i, b] and [f, b] are related via the action of a G-slicing preserving one-d diffeomorphism subgroup is apparent: moving along an adjoint orbit in g * , by definition, we have ∆F = 0. 18 The nonequilibrium work relation [64] further extends this statement to irreversible processes log e − W,b = −∆F.
By changing abruptly the temperature, hence the reference observer-reservoir, the entropy of the system increase, while the reservoir entropy decrease by an amount ∆Q, b f , with an overall change in entropy in accordance with the second law of thermodynamics.
A slow variation of the state realised via a series of equilibrium states defines a reversible adiabathic transformation, with ∆S = ∆Q, b f . Again, this quantity would be zero if measured along an adjoint orbit in g * .
Differently form the isothermal process, here we perform a change on the slicing of the manifold, which results in an increase of the entropy of the state of the fields.

Spacetime vs. Spatial Diff-Equilibrium State
As discussed in Sec. 2.5, the action of the diffeomorphism group on the covariance fields induces a natural action of Diff(X ) on the space of embeddings by left action (2.55). The natural action of the group Diff(Σ) of spatial diffeomorphisms on the space of embeddings is instead the right action, namely: and then this point is mapped by the embedding τ into the spacetime point τ ( x ) ∈ X .
The right action (4.1) of Diff(Σ) induces a left action of Diff(X ) on a given hypersurface τ (Σ) which preserves the hypersurface fixed, i.e., for a given τ ∈ Emb(Σ, X ), we have such that the following diagram commutes. In particular, considering a one-parameter subgroup of Diff(Σ) which induces a oneparameter subgroup of hypersurface preserving spacetime diffeomorphisms, the corresponding generating vector fields ξ Σ ∈ X(Σ) and ξ X ∈ X(X ) are related by Note that, unlike the fully spacetime covariant case discussed in the previous sections, the embedding τ is now fixed thus restricting ourselves to a given spatial hypersurface Σ τ = τ (Σ).
Correspondingly, the restriction of the covariant momentum mapJ H to the subalgebra g τ = diff Στ (X ) of spatial diffeomorphisms which preserves the image of τ identifies an equivariant momentum map w.r.t. the action of the subgroup G τ = Diff(Σ τ ). Indeed, using the relation =: J τ (ϕ, Π, τ, P ), ζ , ζ ∈ diff(Σ τ ) (4.5) so that, for ζ, ζ ∈ diff(Σ τ ), we have (4.6) as can be checked by direct computation using the fact that ζ j ( x), ζ k ( x ) do not depend on the embeddings and the Poisson bracket for the spatial diffeomorphism constraint. Thus, the map J τ =J H Στ provides a representation of the algebra of spatial diffeomorphisms on the extended phase space. Consistently, the map (4.5) gives the expected lift of the shift vector into the deformation vector tangential to the spatial slice, i.e. and we limit ζ to the subset Ω τ ⊂ g τ = diff(Σ τ ) such that the above integral converge. Due to the equivariance of the momentum map J τ under the action (4.2), the Gibbs statistical state (4.8) is then an equilibrium state w.r.t. the one-parameter family of spatial diffeomorphisms generated by the vector field (4.4) associated to ζ ∈ diff(Σ τ ).

Time Evolution Gibbs State via Gauge Fixing
As already discussed, the momentum map (2.58) provides on the one hand a representation of the algebra of spacetime diffeomorphisms on the parametrized phase space of the field theory under consideration. On the other hand it also generates the displacement of the spatial hypersurface embedded in spacetime setting at the same time the correctly evolved Cauchy data for fields and the constraints on the deformed hypersurface. The corresponding one-parameter subgroup of diffeomorphisms associated with the Lie algebra element b thus identifies a generalized notion of "dynamical evolution" w.r.t. which the covariant Gibbs state (3.13) is an equilibrium state.
By this we mean that the vector field ξ (b) associated with b ∈ diff(X ) comes to be the generating vector field of a foliation of spacetime and the corresponding flow defines a generalized concept of time evolution.
As expected for generally covariant systems, dynamics and gauge symmetry are deeply in- Following the strategy of [67,68,69], let us specialize the main steps of the construction to the general setting of parametrized field theories we are interested in the present paper.
Let then consider the parametrized phase space of our theory Υ = T * Y τ × T * Emb G (Σ, X ) with canonical variables (ϕ A ( x), Π A ( x), τ µ ( x), P µ ( x)), A = 1, . . . , N , µ = 0, . . . , n, first-class µ ≈ 0 generating gauge symmetries, and Hamiltonian H(ξ) given in (2.58). The canonical variables on Υ can be split into two disjoint sets (ϕ A , Π A ) and (τ µ , P µ ), the former being the physical field content of the theory (and their conjugate momenta) while the latter are the auxiliary field variables resulting from the parametrization procedure at the canonical level. These are the canonical variables we would like to eliminate by gauge-fixing.
To this aim, let us introduce the gauge-fixing conditions of the form , µ = 0, . . . , n (4.10) where the F µ are certain functions depending only on the configuration field variables ϕ A as well as on the time parameter 19 such that the conditions χ µ = 0 provide us with a complete set of gauge-fixing constraints. This means that the set of all constraints that we will collectively denote as ψ I = (H µ , χ ν ) is now a second-class set (i.e., {H µ , χ ν } ≈ 0), and the conditions χ µ = 0 eliminate all the gauge freedom, namely The new set of constraints ψ I = (H µ , χ ν ) identifies a reduced phase space Υ ⊂ Υ defined by ψ I = 0. Stability of time dependent constraints amounts to require that where now the weak equality ≈ denotes equality up to terms which vanish on Υ. Requiring with C IJ = (C) −1 IJ the inverse of the matrix C whose entries are given by the Poisson brackets of all constraints, say C IJ = {ψ I , ψ J } 20 . As by construction {f, ψ I } * = 0 holds strongly for any second class constraint ψ I and any function f on Υ, all constraints (now second class) can be imposed strongly inside the bracket to eliminate the corresponding number of variables and obtain a well-defined induced bracket on Υ.
Therefore, the crucial point is to look for the HamiltonianH which generates dynamical evolution equations written in terms of the Dirac bracket. The corresponding trajectories in Υ project onto trajectories in Υ lying in Υ for all time as implied by the stability of the constraints dψ I dλ ≈ 0. As discussed in [67,68,69], the HamiltonianH is given bȳ The generating function of such a transformation is given by and Hamiltonian given bȳ As can be checked by direct computation, from the constraint algebra it follows that the matrix C = {ψ I , ψ J } takes the block form so that for any function f on Υ the Dirac bracket yields which is the desired form (4.13) of the dynamical evolution.
Let us then consider the case in which the functions F µ do not depend on the fields ϕ A but only on λ, say χ µ = τ µ − F µ (λ) 21 . According to the expressions (4.17), in this case the canonical variables on Υ are just given by the physical fields and their conjugate momenta, i.e.: The constraints H µ = 0 can be then solved to express the momenta P µ =P µ in terms of the µ . The new Hamiltonian (4.15) then reads from which, restricting on Υ and taking into account that H = 0 imposing the constraints, it follows thatH Note thatḞ 0 and˙ F respectively play the role of lapse and shift consistently with ξ µ (τ ( x)) =τ µ being the deformation vector field of the foliation which after gauge-fixing yieldsτ µ =Ḟ µ . In particular, choosing the gauge the new Hamiltonian (4.25) is nothing but the usual field Hamiltonian 27) which as such generates the correct dynamics for the field variables (ϕ A , Π A ) after gauge-fixing.
Finally, we can define a Gibbs state on Υ given by This is an equilibrium state w.r.t. the dynamical evolution generated by the HamiltonianH.
In particular, choosing the temporal gauge F 0 = x 0 , F = 0, we get the standard relativistic equilibrium state with temperature given by the inverse of the lapse.

On the Thermodynamic Characterization of Covariant Equilibrium
In this section we comment on the relation between the generalized covariant Gibbs state for parametrized field theories presented in this work and the thermodynamic characterization of covariant statistical equilibrium arising in previous investigations based on the thermal time hypothesis. Inspired by the Tomita-Takasaki theorem in algebraic quantum field theory [6], thermal time hypothesis [3,5] states that timeless evolution is given by whereω is the symplectic structure on Υ given in (2.33) and, with a slight abuse of notation, we still denote by d the exterior differential on the extended parametrized phase space of fields although it should not be confused with that on spacetime. More explicitly, as already anticipated in Sec. 2.5, the equations of motion associated to the Hamiltonian H(ξ) are given bẏ so that the vector field generates the correct dynamical evolution for the fields as well as the constraints (see Eq. (2.61)).
In particular, regarding Υ as a bundle over Emb(Σ, X ) whose fiber above τ ∈ Emb(Σ, X ) is the field phase space T * Y τ coordinatized by the spatial matter fields and their conjugate momenta, we see from (4.33) that the above vector field restricted to Emb(Σ, X ) generates a one-parameter curve of embeddings c : R → Emb(Σ, X ) by c(λ) = τ (λ) which in turn identifies a slicing s X ( x, λ) = τ (λ)( x) in spacetime generated by the vector field ξ X (τ ( x)). The corresponding curve c τ (λ) = (ϕ(λ), Π(λ)) in T * Y τ identifies then a compatible G-slicing of the field space.
Therefore, as schematically reported in Fig. 1, the thermal flow associated to the Gibbs state (3.13) with thermal time parameter λ defines a one-parameter group of bundle automorphisms as well as a compatible slicing of spacetime so that the Lie algebra-valued temperature ξ identifies a direction in spacetime along which geometry and matter fields evolve.
Moreover, along the line of [7], we can define a local temperature function, i.e. a map T : Υ × Σ → R, given by with the vector field ξ µ (τ ( x)) playing the role of multi-fingered time. Consistently, with the gauge choice F 0 = x 0 discussed in the previous section, the above temperature function reduces on-shell to the inverse of the lapse function, say T = 1 x 0 , thus locally yielding the rate of change of the thermal time parameter λ and the coordinate time of di↵eomorphisms. Let then (x 0 , x 1 , . . . , x n ) be a chart on X adapted to ⌧ , i.e. such that ⌃ ⌧ is locally a level set of x 0 . Denoting by (', ⇧, ⌧, P ) a point in the canonical parametrized phase space T ⇤Ỹ ⌧ = T ⇤ Y ⌧ ⇥ T ⇤ Emb G (⌃, X ), the canonical symplectic structure! ⌧ on T ⇤Ỹ ⌧ reads as [?] Following the construction of [?] (cfr. Ch. 5), the multisymplectic structure onZ induces a presymplectic structure on the spaceZ ⌧ of sections of the bundlẽ Z ⌧ ! ⌃ ⌧ given bỹ ⌧ relating in adopted coordinates the momenta ⇧ A and P a respectively to the temporal components of the multimomenta p µ A and % µ a as (2.37) In particular [?], ker T R ⌧ = ker⌦ ⌧ ( ) and the canonical parametrized phase space T ⇤Ỹ ⌧ is thus isomorphic to the quotientZ ⌧ / ker⌦ ⌧ . Let now 2Z ⌘ (X ,Z) be a section of the bundleZ over X , and let ↵Z :Z !Z be a covariant canonical transformation covering a di↵eomorphism ↵ X : X ! X whose induced action on sections is given by ↵Z ( ) = ↵Z ↵ 1 X (cfr. Eq. (2.17)). The corresponding transformation onZ ⌧ ⌘ (⌃ ⌧ ,Z ) given by The covariant multimomentum map (2.26) associated to the G-action onZ will then induce a so-called (parametrized) energy-momentum map onZ ⌧ defined bỹ of di↵eomorphisms. Let then (x 0 , x 1 , . . . , x n ) be a chart on X adapted to ⌧ , i.e. such that ⌃ ⌧ is locally a level set of x 0 . Denoting by (', ⇧, ⌧, P ) a point in the canonical parametrized phase space T ⇤Ỹ ⌧ = T ⇤ Y ⌧ ⇥ T ⇤ Emb G (⌃, X ), the canonical symplectic structure! ⌧ on T ⇤Ỹ ⌧ reads as [?] Following the construction of [?] (cfr. Ch. 5), the multisymplectic structure onZ induces a presymplectic structure on the spaceZ ⌧ of sections of the bundlẽ Z ⌧ ! ⌃ ⌧ given bỹ ⌧ relating in adopted coordinates the momenta ⇧ A and P a respectively to the temporal components of the multimomenta p µ A and % µ a as (2.37) In particular [?], ker T R ⌧ = ker⌦ ⌧ ( ) and the canonical parametrized phase space T ⇤Ỹ ⌧ is thus isomorphic to the quotientZ ⌧ / ker⌦ ⌧ . Let now 2Z ⌘ (X ,Z) be a section of the bundleZ over X , and let ↵Z :Z !Z be a covariant canonical transformation covering a di↵eomorphism ↵ X : X ! X whose induced action on sections is given by ↵Z ( ) = ↵Z ↵ 1 X (cfr. Eq. (2.17)). The corresponding transformation onZ ⌧ ⌘ (⌃ ⌧ ,Z ) given by The covariant multimomentum map (2.26) associated to the G-action onZ will then induce a so-called (parametrized) energy-momentum map onZ ⌧ defined bỹ

Conclusions and Outlook
In this work, we consider the problem of defining statistical mechanics and thermodynamics A straightforward application of Souriau's Lie group thermodynamics to the case of a fully constrained system leads to define a Gibbs-like state with respect to the gauge group action generated by the first-class constraints. In this framework, the Hamiltonian Lie group action is characterized by the vanishing of the associated momentum map. The vanishing momentum map is equivalent to a first class constraint, reflecting the presence of a gauge symmetry for the system, and the constraint surface is identified with the zero-level set of the associated momentum map [70,23,26].
However, in generally covariant theories the super-Hamiltonian and super-momentum constraints do not close a genuine Lie algebra which reflects into the non-equivariance of the energy-momentum map preventing a straightforward representation of the algebra of the spacetime diffeomorphism group. The application of Jaynes' entropy maximization principle [20] with constant mean values of the first-class constraints necessarily leads to a Gibbs-like state which, due to the non-equivariance of the energy-momentum map, can not be an equilibrium state under the gauge flow generated by the constraints. In terms of the classification of different kinds of Gibbs states given in [71], this state would correspond to a thermodynamical rather than dynamical statistical state. In particular, it would not be of equilibrium with respect to a one-parameter group of spacetime diffeomorphisms, but at most only with respect to spatial diffeomorphisms for which there is a momentum map induced by the energy-momentum map restricted to the Lie algebra of spatial diffeomorphisms (cfr. Eqs. (2.54) and (4.5)).
We overcome this issue by extending our multi-symplectic phase space via the introduction of covariance fields. This allows us to recover the action of the spacetime diffeomorphism group by means of the observable Hamiltonian functional H(ξ) defined in (2.58) which identifies a equivariant momentum map homomorphically relating the diffeomorphism algebra to the Poisson bracket algebra of functionals on the parametrized phase space. The derivation of a Gibbs state associated to this equivariant momentum map via the prescriptions of Lie group thermodynamics then provides us with a dynamical (in the sense of [71]) equilibrium state on the parametrized phase space. Such a state is of equilibrium with respect to the one-parameter group of diffeomorphisms generated by the vector field ξ X associated to ξ ∈ g = diff(X ). In this sense, it defines a spacetime covariant notion of thermodynamical equilibrium.
The gauge character of spacetime diffeomorphisms implies a radical conceptual shift in the definition of equilibrium state with respect to Souriau's work. By replacing a dynamical symmetry with a gauge one, we move our analysis from the fully reduced symplectic space of motions (on-shell) to the unconstrained extended phase space of the system (off-shell). While being defined off-shell, the covariant Gibbs state is by construction an observable of the theory, and it encodes, via the covariant momentum map functional, all the dynamical information carried by the given parametrized field theory: its canonical Hamiltonian, its initial value constraints, its gauge freedom, and its stress energy-momentum tensor [21]. Therefore, we expect the off-shell equilibrium to play a role similar to a generating functional partition function in field theory, with a Gibbs state corresponding to a "soft" imposition of the constraints of the theory. This might allow for a probabilistic approach to symplectic reduction, a further interesting perspective which we leave open for future investigation.
The proposed result points to a deep connection between geometrical methods, information theory and field theories. We expect our approach to open the road for a spacetime covariant formulation of statistical mechanics, possibly capable of describing the fluctuations of the gravitational field in a general relativistic context (see e.g. [17,72]). From an information-geometric viewpoint, we further expect the derived covariant Gibbs state functional to provide a useful tool for exploring a statistical generalization of symplectic reduction in field theory, as well as a further support to the use of momentum map and Lie group formalism in the study of covariant generative models in machine learning [73].
As our original motivation is general relativity, the present analysis needs to be extended to gravity for which a covariant Hamiltonian formalism has been extensively studied in the last years, see e.g. [44,30,74,75,76,77] and references within. However, the intrinsic parametrized nature of gravity [78] makes the achievement of a phase space representation of spacetime diffeomorphisms a non-trivial task. Useful insights in this respect may be found in the work by Isham and Kuchař [38].
A further interesting and closely related aspect to explore consists in including boundaries in our covariant description (see e.g. [79,80,81] for the analysis of boundaries in the framework of multi-symplectic field theories). The inclusion of boundaries would be needed to identify a notion of subsystems which is crucial in any thermodynamical analysis. As remarked above, in this covariant setting the constraints are beautifully encoded in the momentum map associated with the gauge group action on the field space. The latter plays a key role in the construction of the covariant Gibbs state. As shown in [82], in presence of boundaries, the kinematical constraint algebra can be written as conservation laws for boundary charges. These charges would then enter the boundary momentum map out of which a statistical mechanic framework for the on-shell boundary modes can be constructed. This may open fruitful connections with recent work on boundary modes in quantum gravity [83]. Of particular interest would be the case in which the finite boundaries describe horizons, with potential application to black hole thermodynamics.
From a broader point of view, the application of our covariant statistical mechanics formalism to discrete gravity models may provide interesting insights to the study of coarse-graining approaches and the continuum limit from a thermodynamical perspective, as initiated for instance in [84,85].