Merced Montesinos

BF gravity and the Immirzi parameter

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.

Covariant canonical formalism for four-dimensional BF theory

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.

SL(2,R) model with two Hamiltonian constraints

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.

Statistical mechanics of generally covariant quantum theories: a Boltzmann-like approach

We study the possibility of applying statistical mechanics to generally covariant quantum theories with a vanishing Hamiltonian. We show that (under certain appropriate conditions) this makes sense, in spite of the absence of a notion of energy and external time. We consider a composite system formed by a large number of identical components, and apply Boltzmann’s ideas and the fundamental postulates of ordinary statistical physics. The thermodynamical parameters are determined by the properties of the thermalizing interaction. We apply these ideas to a simple example, in which the component system has one physical degree of freedom and mimics the constraint algebra of general relativity.

Relational Evolution of the Degrees of Freedom of Generally Covariant Quantum Theories

We study the classical and quantum dynamics of generally covariant theories with vanishing Hamiltonian and with a finite number of degrees of freedom. In particular, the geometric meaning of the full solution of the relational evolution of the degrees of freedom is displayed, which means the determination of the total number of evolving constants of motion required. Also a method to find evolving constants is proposed. The generalized Heisenberg picture needs M time variables, as opposed to the Heisenberg picture of standard quantum mechanics where one time variable t is enough. As an application, we study the parametrized harmonic oscillator and the SL(2, R) model with one physical degree of freedom that mimics the constraint structure of general relativity where a Schrödinger equation emerges in its quantum dynamics.

Minimal Coupling and Feynman's Proof

The non quantum relativistic version of theproof of Feynman for the Maxwell equations is discussedin a framework with a minimum number of hypothesesrequired. From the present point of view it is clear that the classical equations of motioncorresponding to the gauge field interactions can bededuced from the minimal coupling rule, and we claimhere resides the essence of the proof ofFeynman.

Topological limit of gravity admitting an SU(2) connection formulation

We study the Hamiltonian formulation of the generally covariant theory defined by the Lagrangian 4-form L=e{sub I} and e{sub J} and F{sup IJ}({omega}), where e{sup I} is a tetrad field and F{sup IJ} is the curvature of a Lorentz connection {omega}{sup IJ}. This theory can be thought of as the limit of the Holst action for gravity for the Newton constant G{yields}{infinity} and Immirzi parameter {gamma}{yields}0, while keeping the product G{gamma} fixed. This theory has for a long time been conjectured to be topological. We prove this statement both in the covariant phase space formulation as well as in the standard Dirac formulation. In the time gauge, the unconstrained phase space of theory admits an SU(2) connection formulation which makes it isomorphic to the unconstrained phase space of gravity in terms of Ashtekar-Barbero variables. Among possible physical applications, we argue that the quantization of this topological theory might shed new light on the nature of the degrees of freedom that are responsible for black entropy in loop quantum gravity.

Alternative symplectic structures for SO(3,1) and SO(4) four-dimensional BF theories

The most general action, quadratic in the B fields as well as in the curvature F, having SO(3, 1) or SO(4) as the internal gauge group for a four-dimensional BF theory is presented and its symplectic geometry is displayed. It is shown that the space of solutions to the equations of motion for the BF theory can be endowed with symplectic structures alternative to the usual one. The analysis also includes topological terms and cosmological constant. The implications of this fact for gravity are briefly discussed.

Self-dual gravity with topological terms

The canonical analysis of the (anti-)self-dual action for gravity supplemented with the (anti-)self-dual Pontrjagin term is carried out. The effect of the topological term is to add a ‘magnetic’ term to the original momentum variable associated with the self-dual action leaving the Ashtekar connection unmodified. In the new variables, the Gauss constraint retains its form, while both vector and Hamiltonian constraints are modified. This shows that the contribution of the Euler and Pontrjagin terms is not the same as that coming from the term associated with the Barbero–Immirzi parameter, and thus the analogy between the θ -angle in Yang–Mills theory and the Barbero–Immirzi parameter of gravity is not appropriate. PACS number: 0460D

Geometric Thermodynamics: Black Holes and the Meaning of the Scalar Curvature

In this paper we show that the vanishing of the scalar curvature of Ruppeiner-like metrics does not characterize the ideal gas. Furthermore, we claim through an example that flatness is not a sufficient condition to establish the absence of interactions in the underlying microscopic model of a thermodynamic system, which poses a limitation on the usefulness of Ruppeiners metric and conjecture. Finally, we address the problem of the choice of coordinates in black hole thermodynamics. We propose an alternative energy representation for Kerr-Newman black holes that mimics fully Weinholds approach. The corresponding Ruppeiners metrics become degenerate only at absolute zero and have non-vanishing scalar curvatures.