We propose a novel BF-type formulation of real four-dimensional gravity, which generalizes previous models. In particular, it allows for an arbitrary Immirzi parameter. We also construct the analogue of the Urbantke metric for this model. PACS: 04.60.DsReal general relativity can be formulated as a constrained first-order BF-type theory of the form [1] (for an earlier alternative approach, seewhere. . = 0, 1, 2, 3 are raised and lowered with the Minkowski metric η IJ ). G(B, φ, µ) denotes a constraint quadratic in the 2-forms B IJ . Its role is to implement that, for some tetrad e I , the 2-forms take the form B IJ = * (e I ∧ e J ), with * the duality operator on Lorentz indices ( * 2 = −1). When the constraint is solved, substitution back in the action of this specific form would then recover general relativity in its first-order tetrad formulation. For Euclidean gravity, we have η IJ → δ IJ , the connection is valued in SO(4), and * 2 = +1. The constraint is of the formwith φ IJKL a Lagrange multiplier with obvious symmetries φ IJKL = −φ JIKL = −φ IJLK = φ KLIJ . It has 21 independent components. Since this is one too many, as the B IJ have 1

We describe a simple dynamical model characterized by the presence of two noncommuting Hamiltonian constraints. This feature mimics the constraint structure of general relativity, where there is one Hamiltonian constraint associated with each space point. We solve the classical and quantum dynamics of the model, which turns out to be governed by an SL(2, R) gauge symmetry, local in time. In classical theory, we solve the equations of motion, find a SO(2, 2) algebra of Dirac observables, find the gauge transformations for the Lagrangian and canonical variables and for the Lagrange multipliers. In quantum theory, we find the physical states, the quantum observables, and the physical inner product, which is determined by the reality conditions. In addition, we construct the classical and quantum evolving constants of the system. The model illustrates how to describe physical gauge-invariant relative evolution when coordinate time evolution is a gauge.

The covariant canonical formalism for four-dimensional BF theory is performed. The aim of the paper is to understand in the context of the covariant canonical formalism both the reducibility that some first class constraints have in Dirac's canonical analysis and also the role that topological terms play. The analysis includes also the cases when both a cosmological constant and the second Chern character are added to the pure BF action. In the case of the BF theory supplemented with the second Chern character, the presymplectic 3-form is different to the one of the BF theory in spite of the fact both theories have the same equations of motion while on the space of solutions they both agree to each other. Moreover, the analysis of the degenerate directions shows some differences between diffeomorphisms and internal gauge symmetries.Comment: Latex file, 22 pages (due to the macro). Revised version to match published versio

Abstract.We report a new internal gauge symmetry of the n-dimensional Palatini action with cosmological term (n > 3) that is the generalization of three-dimensional local translations. This symmetry is obtained through the direct application of the converse of Noether's second theorem on the theory under consideration. We show that diffeomorphisms can be expressed as linear combinations of it and local Lorentz transformations with field-dependent parameters up to terms involving the variational derivatives of the action. As a result, the new internal symmetry together with local Lorentz transformations can be adopted as the fundamental gauge symmetries of general relativity. Although their gauge algebra is open in general, it allows us to recover, without resorting to the equations of motion, the very well-known Lie algebra satisfied by translations and Lorentz transformations in three dimensions. We also report the analog of the new gauge symmetry for the Holst action with cosmological term, finding that it explicitly depends on the Immirzi parameter. The same result concerning its relation to diffeomorphisms and the open character of the gauge algebra also hold in this case. Finally, we consider the non-minimal coupling of a scalar field to gravity in n dimensions and establish that the new gauge symmetry is affected by this matter field. Our results indicate that general relativity in dimension greater than three can be thought of as a gauge theory.

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BF gravity comprises all the formulations of gravity that are based on deformations of BF theory. Such deformations consist of either constraints or potential terms added to the topological BF action that turn some of the gauge degrees of freedom into physical ones, particularly giving rise to general relativity. The BF formulations have provided new and deep insights into many classical and quantum aspects of the gravitational field, setting the foundations for the approach to quantum gravity known as spinfoam models. In this review, we present a self-contained and unified treatment of the BF formulations of D-dimensional general relativity and other related models, focusing on the classical aspects of them and including some new results.

The publishers would like to apologize for an error occurring in issue 5 (7 March 2001) of Classical and Quantum Gravity. The article by Capovilla et al was published as a Letter to the Editor, when it should have appeared as a Paper.

We present a manifestly Lorentz-covariant description of the phase space of general relativity with the Immirzi parameter. This formulation emerges after solving the second-class constraints arising in the canonical analysis of the Holst action. We show that the new canonical variables give rise to other Lorentz-covariant parametrizations of the phase space via canonical transformations. The resulting form of the first-class constraints in terms of new variables is given. In the time gauge, these variables and the constraints become those found by Barbero.