Roberto Pereira

LQG vertex with finite Immirzi parameter

Abstract We extend the definition of the “flipped” loop-quantum-gravity vertex to the case of a finite Immirzi parameter γ . We cover both the Euclidean and Lorentzian cases. We show that the resulting dynamics is defined on a Hilbert space isomorphic to the one of loop quantum gravity, and that the area operator has the same discrete spectrum as in loop quantum gravity. This includes the correct dependence on γ , and, remarkably, holds in the Lorentzian case as well. The ad hoc flip of the symplectic structure that was required to derive the flipped vertex is not anymore required for finite γ . These results establish a bridge between canonical loop quantum gravity and the spinfoam formalism in four dimensions.

Loop-quantum-gravity vertex amplitude.

Spin foam models are hoped to provide the dynamics of loop-quantum gravity. However, the most popular of these, the Barrett-Crane model, does not have the good boundary state space and there are indications that it fails to yield good low-energy n-point functions. We present an alternative dynamics that can be derived as a quantization of a Regge discretization of Euclidean general relativity, where second class constraints are imposed weakly. Its state space matches the SO(3) loop gravity one and it yields an SO(4)-covariant vertex amplitude for Euclidean loop gravity.

Flipped spinfoam vertex and loop gravity

Abstract We introduce a vertex amplitude for 4d loop quantum gravity. We derive it from a conventional quantization of a Regge discretization of euclidean general relativity. This yields a spinfoam sum that corrects some difficulties of the Barrett–Crane theory. The second class simplicity constraints are imposed weakly, and not strongly as in Barrett–Crane theory. Thanks to a flip in the quantum algebra, the boundary states turn out to match those of SO ( 3 ) loop quantum gravity—the two can be identified as eigenstates of the same physical quantities—providing a solution to the problem of connecting the covariant SO ( 4 ) spinfoam formalism with the canonical SO ( 3 ) spin-network one. The vertex amplitude is SO ( 3 ) and SO ( 4 ) -covariant. It rectifies the triviality of the intertwiner dependence of the Barrett–Crane vertex, which is responsible for its failure to yield the correct propagator tensorial structure. The construction provides also an independent derivation of the kinematics of loop quantum gravity and of the result that geometry is quantized.

Lorentzian spin foam amplitudes: graphical calculus and asymptotics

The amplitude for the 4-simplex in a spin foam model for quantum gravity is defined using a graphical calculus for the unitary representations of the Lorentz group. The asymptotics of this amplitude are studied in the limit when the representation parameters are large, for various cases of boundary data. It is shown that for boundary data corresponding to a Lorentzian simplex, the asymptotic formula has two terms, with phase plus or minus the Lorentzian signature Regge action for the 4-simplex geometry, multiplied by an Immirzi parameter. Other cases of boundary data are also considered, including a surprising contribution from Euclidean signature metrics.

Lorentzian loop quantum gravity vertex amplitude

We generalize a model recently proposed for Euclidean quantum gravity to the case of Lorentzian signature. The main features of the Euclidean model are preserved in the Lorentzian one. In particular, the boundary Hilbert space matches the one of SU(2) loop quantum gravity. As in the Euclidean case, the model can be obtained from the Lorentzian Barrett-Crane model from a flipping of the Poisson structure, or alternatively, by adding a topological term to the action and taking the small Barbero-Immirzi parameter limit. Introduction Loop quantum gravity (LQG) [1] provides a well defined, background independent, construction of the kinematical Hilbert space of quantum general relativity. Spin foam techniques [2] have been developed as a possible framework to study the quantum dynamics. A spin foam is a two complex (union of edges, faces and vertices) colored by quantum numbers (faces are labelled by representations of a given group and edges by intertwiners). It can be interpreted as the history of a spin network (more precisely, the boundary of a spin foam is a spin network). A spin foam model is given by the assignment of amplitudes to faces, edges and vertices. The most studied model so far is the Barrett-Crane (BC) model for both Lorentzian [3] and Euclidean [4] signatures. It is obtained as a modification of a topological BF quantum field theory by imposing the discrete analogues of the constraints called simplicity constraints that, in the continuum limit, reduce BF theory to general relativity [5]. Much work has been carried out in recent years to extract the low energy behavior of this model [6] and it turns out that some components of the two-point functions are in disagreement with the expected behavior determined by standard perturbative quantum gravity [7]. As argued in [8, 9] the problem can be traced back to the way some of the constraints are imposed in the Barrett-Crane model. In fact, the simplicity constraints form a second class system [10] and in the BC model these are imposed as strong operator equations [11], killing then physical degrees of freedom. In [8, 9] a reformulation of these constraints has been proposed and this allows for a new sector of solutions. This can be obtained from the BC model from a flipping of the Poisson structure, or alternatively, by adding a topological term to the action and taking the small Barbero-Immirzi parameter limit. In [8, 9] only the Euclidean signature case was considered. Here we extend the construction to the Lorentzian case. The main features of the Euclidean model are preserved in the Lorentzian case. In particular, the boundary Hilbert space matches the one of SU(2) loop quantum gravity. ∗Unité mixte de recherche (UMR 6207) du CNRS et des Universités de Provence (Aix-Marseille I), de la Meditarranée (Aix-Marseille II) et du Sud (Toulon-Var); laboratoire affilié à la FRUMAM (FR 2291).We generalize a model recently proposed for Euclidean quantum gravity to the case of Lorentzian signature. The main features of the Euclidean model are preserved in the Lorentzian one. In particular, the boundary Hilbert space matches the one of SU(2) loop quantum gravity. As in the Euclidean case, the model can be obtained from the Lorentzian Barrett?Crane model from a flipping of the Poisson structure, or alternatively, by adding a topological term to the action and taking the small Barbero?Immirzi parameter limit.

Coherent states in quantum gravity: a construction based on the flux representation of LQG

As part of a wider study of coherent states in (loop) quantum gravity, we introduce a modification to the standard construction, based on the recently introduced (non-commutative) flux representation. The resulting quantum states have some welcome features, in particular, concerning peakedness properties, when compared to other coherent states in the literature.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ?Coherent states: mathematical and physical aspects?.

Asymptotics of 4d spin foam models

We study the asymptotic properties of four-simplex amplitudes for various four-dimensional spin foam models. We investigate the semi-classical limit of the Ooguri, Euclidean and Lorentzian EPRL models using coherent states for the boundary data. For some classes of geometrical boundary data, the asymptotic formulae are given, in all three cases, by simple functions of the Regge action for the four-simplex geometry.

Coherent states, constraint classes, and area operators in the new spin-foam models

Recently, two new spin-foam models have appeared in the literature, both motivated by a desire to modify the Barrett–Crane model in such a way that the imposition of certain second class constraints, called cross-simplicity constraints, are weakened. We refer to these two models as the FKLS (Freidel–Krasnov–Livine–Speziale) model and the flipped model. Both of these models are based on a reformulation of the cross-simplicity constraints. This paper has two main parts. First, we clarify the structure of the reformulated cross-simplicity constraints and the nature of their quantum imposition in the new models. In particular, we show that in the FKLS model quantum cross-simplicity implies no restriction on states. The deeper reason for this seems to be that, with the symplectic structure relevant for FKLS, the reformulated simplicity constraints, among themselves, now form a first class system, and this seems to cause the coherent state method of imposing the constraints, key in the FKLS model, to fail to give any restriction on states. Nevertheless, the cross-simplicity can still be seen as implemented via suppression of intertwiner degrees of freedom in the dynamical propagation. In the second part of the paper, we investigate area spectra in the models. The results of these two investigations will highlight how, in the flipped model, the Hilbert space of states, as well as the spectra of area operators, exactly match those of loop quantum gravity, whereas in the FKLS (and Barrett–Crane) models the boundary Hilbert spaces and area spectra are different.

Coherent states for quantum gravity: towards collective variables

We investigate the construction of coherent states for quantum theories of connections based on graphs embedded in a spatial manifold, as in loop quantum gravity. We discuss the many subtleties of the construction, mainly relatedtothediffeomorphisminvarianceofthetheory.Aimingatapproximating a continuum geometry in terms of discrete, graph-based data, we focus on coherent states for collective observables characterizing both the intrinsic and extrinsic geometry of the hypersurface, and we argue that one needs to revise accordingly the more local definitions of coherent states considered in the literature so far. In order to clarify the concepts introduced, we work through a concrete example that we hope will be useful in applying coherent state techniques to cosmology.

Asymptotic analysis of Lorentzian spin foam models

We analyse the asymptotic properties of the amplitude for the 4-simplexes of the Lorentzian EPRL model using stationary phase methods. We compute the critical point equations and study the geometry of their solutions. For boundary data corresponding to a Lorentzian simplex, the asymptotic formula has two terms, with phase plus or minus the Lorentzian signature Regge action for the 4-simplex geometry, multiplied by an Immirzi parameter.