The Hamiltonian description of classical gauge theories is a well studied subject. The two best known approaches, namely the covariant and canonical Hamiltonian formalisms, have received a lot of attention in the literature. The relation between them has also been extensively analyzed. However, a complete understanding of this relation is still not available, specially for gauge theories that are defined over regions with boundaries. Here we consider this issue, for spacetimes with isolated horizons as inner boundaries (representing black holes in equilibrium), and assess whether their corresponding descriptions can be seen as equivalent. First, we review and reanalyze both formalisms from anew and show that, if we compare them at face value, there are differences between them, for instance in the derivation of horizon energy, a physical observable of the theory, as well as in the number of boundary degrees of freedom. Here we construct and compare possible extensions of the corresponding phase spaces, in a way that has not been explored before. We analyze different possible interpretations in both approaches and show how one can achieve correspondence between results within covariant and canonical formalisms. Along the way, we shed light on the role of boundary terms in the symplectic structure in relation to boundary degrees of freedom.

The Generalized First Law (GFL) and the Generalized Second Law (GSL) of thermodynamics are searched for effective and alternative cosmic models using the entropy as a function of the apparent area, transforming the effective cosmological model into the standard form in cosmology. The general conditions of the GSL validity are analyzed for the entropy as a general function and for logarithmic corrections to the usual Black Hole entropy. In particular, we study the GFL and the regions where the GSL is valid for an effective Loop Quantum Cosmology model with spatially flat curvature taking every possible value of the logarithmic correction factor. In addition, we explore the possibility of having negative temperature, where the validity conditions for an alternative generalized second law (AGSL) are studied.

The Hamiltonian description of classical gauge theories is a very well studied subject. The two best known approaches, namely the covariant and canonical Hamiltonian formalisms, have received a lot of attention in the literature. However, a full understanding of the relation between them is not available, especially when the gauge theories are defined over regions with boundaries. Here, we consider this issue, by first making it precise what we mean by equivalence between the two formalisms. Then, we explore several first-order gauge theories and assess whether their corresponding descriptions satisfy the notion of equivalence. We shall show that, even when in several cases the two formalisms are indeed equivalent, there are counterexamples that signal that this is not always the case. Thus, non-equivalence is a generic feature of gauge field theories. These results call for a deeper understanding of the subject.

The notion of Isolated Horizons has played an important role in gravitational physics, being useful from the characterization of the endpoint of black hole mergers to (quantum) black hole entropy. In particular, the definition of weakly isolated horizons (WIHs) as quasilocal generalizations of event horizons is purely geometrical, and is independent of the variables used in describing the gravitational field. Here we consider a canonical decomposition of general relativity in terms of connection and vierbein variables starting from a first order action. Within this approach, the information about the existence of a (weakly) isolated horizon is obtained through a set of boundary conditions on an internal boundary of the spacetime region under consideration. We employ, for the self-dual action, a generalization of the Dirac algorithm for regions with boundary. While the formalism for treating gauge theories with boundaries is unambiguous, the choice of dynamical variables on the boundary is not. We explore this freedom and consider different canonical formulations for non-rotating black holes as defined by WIHs. We show that both the notion of horizon degrees of freedom and energy associated to the horizon is not unique, even when the descriptions might be self-consistent. This represents a generalization of previous work on isolated horizons both in the exploration of this freedom and in the type of horizons considered. We comment on previous results found in the literature.

The authors present concepts and mathematical developments which give rise to the Hamiltonian formulation of Einstein's general relativity, first introduced by Arnowitt, Deser, and Misner. All the geometrical quantities needed for the construction are explicitly obtained, and examples of some of the application of the formalism are given.

We investigate three issues that have been discussed in the context of inflation: Fading of the importance of quantum noncommutativity; the phenomenon of quantum squeezing; and the ability to approximate the quantum state by a distribution function on the classical phase space. In the standard treatments, these features arise from properties of mode functions of quantum fields in (near) de Sitter space-time. Therefore, the three notions are often assumed to be essentially equivalent, representing different facets of the same phenomenon. We analyze them in general Friedmann-Lemaitre-Robertson-Walker space-times, through the lens of geometric structures on the classical phase space. The analysis shows that: (i) inflation does not play an essential role; classical behavior can emerge much more generally; (ii) the three notions are conceptually distinct; classicality can emerge in one sense but not in another; and, (iii) the third notion is realized in a surprisingly strong sense; there is exact equality between completely general n-point functions in the classical theory and those in the quantum theory, provided the quantum operators are Weyl ordered. These features arise already for linear cosmological perturbations by themselves: considerations such as mode-mode coupling, decoherence, and measurement theory-although important in their own right-are not needed for emergence of classical behavior in any of the three senses discussed. Generality of the results stems from the fact that they can be traced back to geometrical structures on the classical phase space, available in a wide class of systems. Therefore, this approach may also be useful in other contexts.

We study the canonical Hamiltonian analysis of gauge theories in the presence of boundaries. While the implementation of Dirac’s programme in the presence of boundaries, as put forward by Regge and Teitelboim, is not new, there are some instances in which this formalism is incomplete. Here we propose an extension to the Dirac formalism—together with the Regge–Teitelboim strategy—that includes generic cases of field theories. We see that there are two possible scenarios, one where there is no contribution from the boundary to the symplectic structure and the other case in which there is one, depending on the dynamical details of the starting action principle. As a concrete system that exemplifies both cases, we consider a theory that can be seen both as defined on a four dimensional spacetime region with boundaries—the bulk theory—or as a theory defined both on the bulk and the boundary of the region—the mixed theory—the bulk theory is given by the four-dimensional Maxwell + U(1) Pontryagin action while the mixed one is defined by the four-dimensional Maxwell + three-dimensional U(1) Chern–Simons action on the boundary. Finally, we show how these two descriptions of the same system are connected through a canonical transformation that provides a third description. The focus here is in defining a consistent formulation of all three descriptions, for which we rely on the geometric formulation of constrained systems, together with the extension of the Dirac–Regge–Teitelboim formalism put forward in the manuscript.

We perform the canonical Hamiltonian analysis of a topological gauge theory, that can be seen both as a theory defined on a four-dimensional spacetime region with boundaries — the bulk theory —, or as a theory defined on the boundary of the region — the boundary theory —. In our case, the bulk theory is given by the 4-dimensional [Formula: see text] Pontryagin action and the boundary one is defined by the [Formula: see text] Chern–Simons action. We analyze the conditions that need to be imposed on the bulk theory so that the total Hamiltonian, smeared constraints and generators of gauge transformations be well defined (differentiable) for generic boundary conditions. We pay special attention to the interplay between the constraints and boundary conditions in the bulk theory on the one side, and the constraints in the boundary theory, on the other side. We illustrate how both theories are equivalent, despite the different canonical variables and constraint structure, by explicitly showing that they both have the same symmetries, degrees of freedom and observables.

Alejandro Corichi 2, ∗ and Parampreet Singh † Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México, UNAM-Campus Morelia, A. Postal 61-3, Morelia, Michoacán 58090, Mexico Center for Fundamental Theory, Institute for Gravitation and the Cosmos, Pennsylvania State University, University Park PA 16802, USA Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA

The notion of Isolated Horizons has played an important role in gravitational physics, being useful from the characterization of the endpoint of black hole mergers to (quantum) black hole entropy. With an eye towards a canonical formulation we consider general relativity in terms of connection and vierbein variables and their corresponding first order actions. We focus on two main issues: (i) The role of the internal gauge freedom that exists, in the consistent formulations of the action principle, and (ii) the role that a 3 + 1 canonical decomposition has in the allowed internal gauge freedom. More concretely, we clarify in detail how the requirement of having well posed variational principles compatible with general weakly isolated horizons (WIHs) as internal boundaries does lead to a partial gauge fixing in the first order descriptions used previously in the literature. We consider the standard Hilbert-Palatini action together with the Holst extension (needed for a consistent 3 + 1 decomposition), with and without boundary terms at the horizon. We show in detail that, for the complete configuration space-with no gauge fixing-, while the Palatini action is differentiable without additional surface terms at the inner WIH boundary, the more general Holst action is not. The introduction of a surface term at the horizon-that renders the action for asymptotically flat configurations differentiable-does make the Holst action differentiable, but only if one restricts the configuration space and partially reduces the internal Lorentz gauge. For the second issue at hand, we show that upon performing a 3 + 1 decomposition and imposing the time gauge, there is a further gauge reduction of the Hamiltonian theory in terms of Ashtekar-Barbero variables to a U(1)-gauge theory on the horizon. We also extend our analysis to the more restricted boundary conditions of (strongly) isolated horizons as inner boundary. We show that even when the Holst action is indeed differentiable without the need of additional surface terms or any gauge fixing for Type I spherically symmetric (strongly) isolated horizons-and a preferred foliation-, this result does not go through for more general isolated or weakly isolated horizons. Our results represent the first comprehensive study of these issues and clarify some contradictory statements found in the literature.

In this review we consider first order gravity in four dimensions. In particular, we focus our attention in formulations where the fundamental variables are a tetrad $e_a^I$ and a SO(3,1) connection ${\omega_{aI}}^J$. We study the most general action principle compatible with diffeomorphism invariance. This implies, in particular, considering besides the standard Einstein-Hilbert-Palatini term, other terms that either do not change the equations of motion, or are topological in nature. Having a well defined action principle sometimes involves the need for additional boundary terms, whose detailed form may depend on the particular boundary conditions at hand. In this work, we consider spacetimes that include a boundary at infinity, satisfying asymptotically flat boundary conditions and/or an internal boundary satisfying isolated horizons boundary conditions. We focus on the covariant Hamiltonian formalism where the phase space $\Gamma$ is given by solutions to the equations of motion. For each of the possible terms contributing to the action we consider the well posedness of the action, its finiteness, the contribution to the symplectic structure, and the Hamiltonian and Noether charges. For the chosen boundary conditions, standard boundary terms warrant a well posed theory. Furthermore, the boundary and topological terms do not contribute to the symplectic structure, nor the Hamiltonian conserved charges. The Noether conserved charges, on the other hand, do depend on such additional terms. The aim of this manuscript is to present a comprehensive and self-contained treatment of the subject, so the style is somewhat pedagogical. Furthermore, along the way we point out and clarify some issues that have not been clearly understood in the literature.

The loop quantization of the Schwarzschild interior region, as described by a homogeneous anisotropic Kantowski-Sachs model, is re-examined. As several studies of different -inequivalent- loop quantizations have shown, to date there exists no fully satisfactory quantum theory for this model. This fact poses challenges to the validity of some scenarios to address the black hole information problem. Here we put forward a novel viewpoint to construct the quantum theory that builds from some of the models available in the literature. The final picture is a quantum theory that is both independent of any auxiliary structure and possesses a correct low curvature limit. It represents a subtle but non-trivial modification of the original prescription given by Ashtekar and Bojowald. It is shown that the quantum gravitational constraint is well defined past the singularity and that its effective dynamics possesses a bounce into an expanding regime. The classical singularity is avoided, and a semiclassical spacetime satisfying vacuum Einstein's equations is recovered on the "other side" of the bounce. We argue that such metric represents the interior region of a white-hole spacetime, but for which the corresponding "white-hole mass" differs from the original black hole mass. Furthermore, we find that the value of the white-hole mass is proportional to the third power of the starting black hole mass.

We study here a complete quantization of a Callan-Giddings-Harvey-Strominger vacuum model following loop quantum gravity techniques. Concretely, we adopt a formulation of the model in terms of a set of new variables that resemble the ones commonly employed in spherically symmetric loop quantum gravity. The classical theory consists of two pairs of canonical variables plus a scalar and diffeomorphism (first class) constraints. We consider a suitable redefinition of the Hamiltonian constraint such that the new constraint algebra (with structure constants) is well adapted to the Dirac quantization approach. For it, we adopt a polymeric representation for both the geometry and the dilaton field. On the one hand, we find a suitable invariant domain of the scalar constraint operator, and we construct explicitly its solution space. There, the eigenvalues of the dilaton and the metric operators cannot vanish locally, allowing us to conclude that singular geometries are ruled out in the quantum theory. On the other hand, the physical Hilbert space is constructed out of them, after group averaging the previous states with the diffeomorphism constraint. In turn, we identify the standard observable corresponding to the mass of the black hole at the boundary, in agreement with the classical theory. We also construct an additional observable on the bulk associated with the square of the dilaton field, with no direct classical analog.

We consider the loop quantization of the (diagonal) Bianchi type IX cosmological model. We explore different quantization prescriptions that extend the work of Wilson-Ewing and Singh. In particular, we study two different ways of implementing the so-called inverse triad corrections. We construct the corresponding Hamiltonian constraint operators and show that the singularity is formally resolved. We find the effective equations associated with the different quantization prescriptions, and study the relation with the isotropic $k$=1 model that, classically, is contained within the Bianchi IX model. We use geometrically defined scalar observables to explore the physical implications of each of these theories. This is the first part in a series of papers analyzing different aspects of the Bianchi IX model, with inverse corrections, within loop quantum cosmology.

We consider a class of two-dimensional dilatonic models, and revisit them from the perspective of a new set of 'polar type' variables. These are motivated by recently defined variables within the spherically symmetric sector of 4D general relativity. We show that for a large class of dilatonic models, including the case with matter, one can perform a series of canonical transformations in such a way that the Poisson algebra of the constraints becomes a Lie algebra. Furthermore, we construct Dirac observables and a reduced Hamiltonian that accounts for the time evolution of the system.

In this paper, we review the issue of defining energy for test particles on a background stationary spacetime. We revisit different notions of energy as defined by different observers. As is well-known, the existence of a timelike isometry allows for the notion of total conserved energy to be well defined. We use this well-known quantity to show that a gravitational potential energy can be consistently defined. As examples, we study the case of the exterior regions of an asymptotically flat black hole and of the [Formula: see text] Schwarzschild–de Sitter (SdS) case, where an asymptotic region is not available. We then consider the situation in which the test particle is absorbed by the black hole and analyze the energetics in detail. In particular, we show that the notion of horizon energy as defined by the isolated horizons formalism provides a satisfactory notion of energy compatible with the particle’s total conserved energy. With these choices, there is a global conservation of energy. Finally, we comment on a recent proposal to define energy of the black hole as seen by a nearby observer at rest, for which this feature is lost.

We consider Lambda = 0 three-dimensional gravity with asymptotically flat boundary conditions. This system was studied by Ashtekar and Varadarajan within the second-order formalism-with metric variables-who showed that the Regge-Teitelboim formalism yields a consistent Hamiltonian description where, surprisingly, the energy is bounded from below and from above. The energy of the spacetime is, however, determined up to an arbitrary constant. The natural choice was to fix that freedom such that Minkowski spacetime has zero energy. More recently, Marolf and Patino started from the Einstein-Hilbert action supplemented with the Gibbons-Hawking term and showed that, in the (2 + 1) decomposition of the theory, the energy is shifted from the Ashtekar-Varadarajan analysis in such a way that Minkowski spacetime possesses a negative energy. In this contribution we consider the first-order formalism, where the fundamental variables are a so(2, 1) connection w(a)(I) (J) and a triad e(a)(I). We consider two actions. A natural extension to 3 dimensions of the consistent action in 4D Palatini gravity is shown to be finite and differentiable. For this action, the (2 + 1) decomposition (that we perform using two methods) yields a Hamiltonian boundary term that corresponds to energy. It assigns zero energy to Minkowski spacetime. We then put forward a totally gauge invariant action and show that it is also well defined and differentiable. Interestingly, it turns out to be related, on shell, to the 3D Palatini action by an additive constant in such a way that its associated energy is given by the Marolf-Patino expression. Thus, we conclude that, from the perspective of the first-order formalism, Minkowski spacetime can consistently have either zero, or a negative energy equal to -1/4G, depending on the choice of consistent action employed as starting point.

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We consider the issue of attaining a consistent Hamiltonian formulation, after a 3+1 splitting, of a well-defined action principle for asymptotically flat gravity. More precisely, our starting point is the gravitational first order Holst action with surface terms and fall off conditions that make the variational principle and the covariant phase space formulation well-defined for asymptotically flat spacetimes. Keeping all surface terms and paying due attention to subtleties that arise from the different cut-offs at infinity, we give a derivation of the gravitational Hamiltonian starting from this action. The 3+1 decomposition and time gauge fixing results in a well-defined Hamiltonian action and a well-defined Hamiltonian formulation for the standard—and more general—asymptotic ADM conditions. Unlike the case of the Einstein–Hilbert action with Gibbons–Hawking–York or Hawking–Horowitz terms, here we do recover the ADM energy momentum from the covariant surface term also when more general variations respecting asymptotic flatness are allowed. Additionally, our strategy yields a derivation of the parity conditions for connection variables independent of the conditions given by Regge and Teitelboim for ADM variables. Finally, we exhibit the other Poincaré generators in terms of real Ashtekar–Barbero variables. We complement previous constructions in self-dual variables by pointing out several subtleties and refining the argument showing that—on shell—they coincide with the ADM charges. Our results represent the first consistent treatment of the Hamiltonian formulation for the connection-tetrad gravitational degrees of freedom, starting from a well posed action, in the case of asymptotically flat boundary conditions.