Spin-foams for all loop quantum gravity (original) (raw)

Canonical ``Loop'' Quantum Gravity and Spin Foam Models

1999

The canonical "loop" formulation of quantum gravity is a mathematically well defined, background independent, non perturbative standard quantization of Einstein's theory of General Relativity. Some among the most meaningful results of the theory are: 1) the complete calculation of the spectrum of geometric quantities like the area and the volume and the consequent physical predictions about the structure of the space-time at the Planck scale; 2) a microscopical derivation of the Bekenstein-Hawking black-hole entropy formula. Unfortunately, despite recent results, the dynamical aspect of the theory (imposition of the Wheller-De Witt constraint) remains elusive.

New Spin Foam Models of Quantum Gravity

Modern Physics Letters A, 2005

We give a brief and a critical review of the Barret-Crane spin foam models of quantum gravity. Then we describe two new spin foam models which are obtained by direct quantization of General Relativity and do not have some of the drawbacks of the Barret-Crane models. These are the model of spin foam invariants for the embedded spin networks in loop quantum gravity and the spin foam model based on the integration of the tetrads in the path integral for the Palatini action.

Path integral representation of spin foam models of 4D gravity

Classical and Quantum Gravity, 2008

We give a unified description of all recent spin foam models introduced by Engle, Livine, Pereira & Rovelli (ELPR) and by Freidel & Krasnov (FK). We show that the FK models are, for all values of the Immirzi parameter γ, equivalent to path integrals of a discrete theory and we provide an explicit formula for the associated actions. We discuss the relation between the FK and ELPR models and also study the corresponding boundary states. For general Immirzi parameter, these are given by Alexandrov's & Livine's SO(4) projected states. For 0 ≤ γ < 1, the states can be restricted to SU(2) spin networks.

Spin Foam Models of Quantum Gravity

Mathematical, Theoretical and Phenomenological Challenges Beyond the Standard Model - Perspectives of the Balkan Collaborations - BW2003 Workshop, 2005

We give a short review of the spin foam models of quantum gravity, with an emphasis on the Barret-Crane model. After explaining the shortcomings of the Barret-Crane model, we briefly discuss two new approaches, one based on the 3d spin foam state sum invariants for the embedded spin networks, and the other based on representing the string scattering amplitudes as 2d spin foam state sum invariants.

Finite spin-foam-based theory of three- and four-dimensional quantum gravity

Physical Review D, 2002

Starting from Ooguri's construction for BF theory in three (and four) dimensions, we show how to construct a well defined theory with an infinite number of degrees of freedom. The spin network states that are kept invariant by the evolution operators of the theory are exact solutions of the Hamiltonian constraint of quantum gravity proposed by Thiemann. The resulting theory is the first example of a well defined, finite, consistent, spin-foam based theory in a situation with an infinite number of degrees of freedom. Since it solves the quantum constraints of general relativity it is also a candidate for a theory of quantum gravity. It is likely, however, that the solutions constructed correspond to a spurious sector of solutions of the constraints. The richness of the resulting theory makes it an interesting example to be analyzed by forthcoming techniques that construct the semiclassical limit of spin network quantum gravity.

A field-theoretic approach to Spin Foam models in Quantum Gravity

Proceedings of Corfu Summer Institute on Elementary Particles and Physics - Workshop on Non Commutative Field Theory and Gravity — PoS(CNCFG2010)

We present an introduction to Group Field Theory models, motivating them on the basis of their relationship with discretized BF models of gravity. We derive the Feynmann rules and compute quantum corrections in the coherent states basis.

Spin Foam Models for Quantum Gravity

2003

In the second part we concentrate on the definition of the Barrett-Crane model. We present the main results obtained in this framework from a critical perspective. Finally we review the combinatorial formulation of spin foam models based on the dual group field theory technology. We present the Barrett-Crane model in this framework and review the finiteness results obtained for both its Riemannian as well as its Lorentzian variants.

Simplicity and closure constraints in spin foam models of gravity

Physical Review D, 2008

We revise imposition of various constraints in spin foam models of 4-dimensional general relativity. We argue that the usual simplicity constraint must be supplemented by a constraint on holonomies and together they must be inserted explicitly into the discretized path integral. At the same time, the closure constraint must be relaxed so that the new constraint expresses covariance of intertwiners assigned to tetrahedra by spin foam quantization. As a result, the spin foam boundary states are shown to be realized in terms of projected spin networks of the covariant loop approach to quantum gravity.

BF description of higher-dimensional gravity theories

It is well known that, in the first-order formalism, pure three-dimensional gravity is just the BF theory. Similarly, four-dimensional general relativity can be formulated as BF theory with an additional constraint term added to the Lagrangian. In this paper we show that the same is true also for higherdimensional Einstein gravity: in any dimension gravity can be described as a constrained BF theory. Moreover, in any dimension these constraints are quadratic in the B field. After describing in details the structure of these constraints, we scketch the "spin foam" quantization of these theories, which proves to be quite similar to the spin foam quantization of general relativity in three and four dimensions. In particular, in any dimension, we solve the quantum constraints and find the so-called simple representations and intertwiners. These exhibit a simple and beautiful structure that is common to all dimensions. * In three spacetime dimensions Einstein's general relativity becomes a beautiful and simple theory. There are no local degrees of freedom, and gravity is an example of topological field theory. Owing to this fact, a variety of techniques from TQFT can be used, and a great deal is known about quantization of the theory. More precisely, when written in the first order formalism, three-dimensional gravity is just the BF theory, whose action is given by: