We give a unified description of all recent spin foam models introduced by Engle, Livine, Pereira and Rovelli (ELPR) and by Freidel and Krasnov (FK). We show that the FK models are, for all values of the Immirzi parameter, 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 and Livine's SO(4) projected states. For 0 <= gamma < 1, the states can be restricted to SU(2) spin networks.
In this article we give a systematic definition of the recently introduced spin foam models for four dimensional quantum gravity reviewing the main results on their semiclassical limit on fixed discretizations.
We describe how the Barrett-Crane spin foam model defines transition amplitudes for quantum gravity states and how causality can be consistently implemented in it.
In this article we lay foundations for a formal relationship of spin foam models of gravity and BF theory to their continuum canonical formulations. First the derivation of the spin foam model of the BF theory from the discrete BF theory action in n dimensions is reviewed briefly. By foliating the underlying n dimensional simplicial manifold using n-1 dimensional simplicial hypersurfaces, the spin foam model is reformulated. Then it is shown that spin network functionals arise naturally on the foliations. The graphs of these spin network functionals are dual to the triangulations of the foliat…
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.
We present a comparison of the calculation of BTZ black hole entropy in loop quantum gravity and in spin foam models. We see that both give the same answer.
A spin-foam model is derived from the canonical model of Loop Quantum Gravity coupled to a massless scalar field. We generalized to the full theory the scheme first proposed in the context of Loop Quantum Cosmology by Ashtekar, Campiglia and Henderson, later developed by Henderson, Rovelli, Vidotto and Wilson-Ewing.
Starting from Plebanski formulation of gravity as a constrained BF theory we propose a new spin foam model for 4d Riemannian quantum gravity that generalises the well-known Barrett-Crane model and resolves the inherent to it ultra-locality problem. The BF formulation of 4d gravity possesses two sectors: gravitational and topological ones. The model presented here is shown to give a quantization of the gravitational sector, and is dual to the recently proposed spin foam model of Engle et al. which, we show, corresponds to the topological sector. Our methods allow us to introduce the Immirzi par…
We present a rigorous regularization of Rovellis's generalized projection operator in canonical 2+1 gravity. This work establishes a clear-cut connection between loop quantum gravity and the spin foam approach in this simplified setting. The point of view adopted here provides a new perspective to tackle the problem of dynamics in the physically relevant 3+1 case.
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 gra…
We discuss the notion of causality in Quantum Gravity in the context of sum-over-histories approaches, in the absence therefore of any background time parameter. In the spin foam formulation of Quantum Gravity, we identify the appropriate causal structure in the orientation of the spin foam 2-complex and the data that characterize it; we construct a generalised version of spin foam models introducing an extra variable with the interpretation of proper time and show that different ranges of integration for this proper time give two separate classes of spin foam models: one corresponds to the sp…
I develop a phenomenological approach to the description of the noise levels that the space-time foam of quantum gravity could induce in modern gravity-wave detectors. Various possibilities are considered, including white noise and random-walk noise. In particular, I find that the sensitivity level expected for the planned LIGO and VIRGO interferometers and for the next upgrade of the NAUTILUS resonant-bar detector corresponds to a white-noise level which can be naturally associated with the Planck length.
In the search for a covariant formulation for Loop Quantum Gravity, spin foams have arised as the corresponding discrete space-time structure and, among the different models, the Barrett-Crane model seems the most promising. Here, we study its boundary states and introduce cylindrical functions on both the Lorentz connection and the time normal to the studied hypersurface. We call them projected cylindrical functions and we explain how they would naturally arise in a covariant formulation of Loop Quantum Gravity.
This paper is twofold. First of all a complete unified picture of $n$-dimensional quantum gravity is proposed in the following sense: In spin foam models of quantum gravity the evaluation of spin networks play a very important role. These evaluations correspond to amplitudes which contribute in a state sum model of quantum gravity. In \cite{fk}, the evaluation of spin networks as integrals over internal spaces was described. This evaluation was restricted to evaluations of spin networks in $n$-dimensional Euclidean quantum gravity. Here we propose that a similar method can be considered to inc…
Spin foam models for gravity or BF theory can be constructed by path integral formulation of the classical discrete models formulated on simplicial manifolds. Using this, we discuss the rigorous construction of Lorentzian spin foam models for gravity and BF theory based on the Gelfand-Naimark theory of the representations of SL(2,C). First we construct the simplex amplitude for the BF SL(2,C) model. Next we discuss the implementation of the Barrett-Crane constraints on this model to derive the spin foam model for gravity. The non-trivial constraints are the cross simplicity constraints which s…
We reveal an underlying flaw in Reginald T. Cahill's recently promoted quantum foam inflow theory of gravity. It appears to arise from a confusion of the idea of the Galilean invariance of the acceleration of an individual flow with what is obtained as an acceleration when a homogeneous flow is superposed with an inhomogeneous flow. We also point out that the General Relativistic covering theory he creates by substituting a generalized Painleve-Gullstrand metric into Einstein's field equations leads to absurd results.
We propose a way to describe the origin of black hole entropy in the spin foam models of quantum gravity. This stimulates a new way to study the relation of spin foam models and loop quantum gravity.
Spin-foam models are hoped to provide a dynamics for loop quantum gravity. These start from the Plebanski formulation of gravity, in which gravity is obtained from a topological field theory, BF theory, through constraints, which, however, select more than one gravitational sector, as well as an unphysical degenerate sector. We show this is why terms beyond the needed Feynman-prescribed one appear in the semiclassical limit of the EPRL spin-foam amplitude. By quantum mechanically isolating a single gravitational sector, we modify this amplitude, yielding a spin-foam amplitude for loop quantum …
We numerically study Barrett-Crane models of Riemannian quantum gravity. We have extended the existing numerical techniques to handle q-deformed models and arbitrary space-time triangulations. We present and interpret expectation values of a few selected observables for each model, including a spin-spin correlation function which gives insight into the behaviour of the models. We find the surprising result that, as the deformation parameter q goes to 1 through roots of unity, the limit is discontinuous.
In this article we consider physical states in the hypercuboidal truncation of the EPRL-FK spin foam quantum gravity model. In particular, these states are defined on graphs which allow considering the entanglement entropy (EE) associated to the bipartition of space. We compute the EE numerically for some examples, and find that it depends on the coupling constants within the theory. We also find that there appears a maximum of the EE within the region of the coupling constant containing the non-Gaussian fixed point of the RG flow of the truncated model. We discuss the relation of this behavio…
Using numerical calculations, we compare three versions of the Barrett-Crane model of 4-dimensional Riemannian quantum gravity. In the version with face and edge amplitudes as described by De Pietri, Freidel, Krasnov, and Rovelli, we show the partition function diverges very rapidly for many triangulated 4-manifolds. In the version with modified face and edge amplitudes due to Perez and Rovelli, we show the partition function converges so rapidly that the sum is dominated by spin foams where all the spins labelling faces are zero except for small, widely separated islands of higher spin. We al…
We compute metric correlations in loop quantum gravity with the dynamics defined by the new spin foam models. The analysis is done at the lowest order in a vertex expansion and at the leading order in a large spin expansion. The result is compared to the graviton propagator of perturbative quantum gravity.
The graphical calculus method is generalized to study the relation between covariant and canonical dynamics of loop quantum gravity. On one hand, a graphical derivation of the partition function of the generalized Euclidean Engle-Pereira-Rovelli-Livine (EPRL) spin-foam model is presented. On the other hand, the action of a Euclidean Hamiltonian constraint operator on certain spin network states is calculated by graphical method. It turns out that the EPRL model can provide a rigging map such that the Hamiltonian constraint operator is weakly satisfied on certain physical states for the Immirzi…
Spin foam models for quantum gravity are derived from lattice path integrals. The setting involves variables from both lattice BF theory and Regge calculus. The action consists in a Regge action, which depends on areas, dihedral angles and includes the Immirzi parameter. In addition, a measure is inserted to ensure a consistent gluing of simplices, so that the amplitude is dominated by configurations which satisfy the parallel transport relations. We explicitly compute the path integral as a sum over spin foams for a generic measure. The Freidel-Krasnov and Engle-Pereira-Rovelli models corresp…
We construct a canonical formulation of general relativity for the case of a timelike foliation of spacetime. The formulation possesses explicit covariance with respect to Lorentz transformations in the tangent space. Applying the loop approach to quantize the theory we derive the spectrum of the area operator of a two-dimensional surface. Its different branches are naturally associated to spacelike and timelike surfaces. The results are compared with the predictions of Lorentzian spin foam models. A restriction of the representations labeling spin networks leads to perfect agreement between t…
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 Plank scale; 2) a microscopical derivation of the Bekenstein-Hawking black-hole entropy formula. Unfortunately, despite recent results, the dynamical aspect of the t…
Building on recent advances in defining Wilsonian RG flows, and in particular the notion of scales, for background-independent theories, we present a first investigation of the renormalization of the 4d spin foam path integral for quantum gravity, both analytically and numerically. Focussing on a specific truncation of the model using a hypercubic lattice, we compute the RG flow and find strong indications for a phase transition, as well as an interesting interplay between the different observed phases and the (broken) diffeomorphism symmetry of the model. Most notably, it appears that the cri…
We propose a spin foam model of four-dimensional quantum gravity which is based on the integration of the tetrads in the path integral for the Palatini action of General Relativity. In the Euclidian gravity case we show that the model can be understood as a modification of the Barrett-Crane spin foam model. Fermionic matter can be coupled by using the path integral with sources for the tetrads and the spin connection, and the corresponding state sum is based on a spin foam where both the edges and the faces are colored independently with the irreducible representations of the spacetime rotatio…
We give an introductory account to the renormalization of models without metric background. We sketch the application to certain discrete models of quantum gravity such as spin foam models.
We show from one-loop quantum gravity and statistical thermodynamics that the thermodynamics of quantum foam in flat space-time and Schwarzschild space-time is exactly the same as that of Hawking-Unruh radiation in thermal equilibrium. This means we show unambiguously that Hawking-Unruh thermal radiation should contain thermal gravitons or the contribution of quantum space-time foam. As a by-product, we give also the quantum gravity correction in one-loop approximation to the classical black hole thermodynamics.
We construct a spin foam model of Yang-Mills theory coupled to gravity by using a discretized path integral of the BF theory with polynomial interactions and the Barret-Crane ansatz. In the Euclidian gravity case we obtain a vertex amplitude which is determined by a vertex operator acting on a simple spin network function. The Euclidian gravity results can be straightforwardly extended to the Lorentzian case, so that we propose a Lorentzian spin foam model of Yang-Mills theory coupled to gravity.
Spin foam models are a new approach to a formulation of quantum gravity which is fully background independent, non-perturbative, and covariant, in the spirit of path integral formulations of quantum field theory. In this thesis we describe in details the general ideas and formalism of spin foam models, and review many of the results obtained recently in this approach. We concentrate, for the case of 3-dimensional quantum gravity, on the Turaev-Viro model, and, in the 4-dimensional case, which is our main concern, on the Barrett-Crane model. In particular, for the Barrett-Crane model: we descri…
Spin foam vertex amplitudes are the key ingredient of spin foam models for quantum gravity. These fall into the realm of discretized path integral, and can be seen as generalized lattice gauge theories. They can be seen as an attempt at a 4-dimensional generalization of the Ponzano-Regge model for 3d quantum gravity. We motivate and review the construction of the vertex amplitudes of recent spin foam models, giving two different and complementary perspectives of this construction. The first proceeds by extracting geometric configurations from a topological theory of the BF type, and can be see…
The problem of background independent quantum gravity is the problem of defining a quantum field theory of matter and gravity in the absence of an underlying background geometry. Loop quantum gravity (LQG) is a promising proposal for addressing this difficult task. Despite the steady progress of the field, dynamics remains to a large extend an open issue in LQG. Here we present the main ideas behind a series of proposals for addressing the issue of dynamics. We refer to these constructions as the {\em spin foam representation} of LQG. This set of ideas can be viewed as a systematic attempt at …
We describe a class of spin foam models of four-dimensional quantum gravity which is based on the integration of the tetrad one-forms in the path integral for the Palatini action of General Relativity. In the Euclidian gravity case this class of models can be understood as a modification of the Barrett-Crane spin foam model. Fermionic matter can be coupled by using the path integral with sources for the tetrads and the spin connection, and the corresponding state sum is based on a spin foam where both the edges and the faces are colored independently with the irreducible representations of the…
In this paper we tackle the issue of causality in quantum gravity, in the context of 3d spin foam models. We identify the correct procedure for implementing the causality/orientation dependence restriction that reduces the path integral for BF theory to that of quantum gravity in first order form. We construct explicitly the resulting causal spin foam model. We then add matter degrees of freedom to it and construct a causal spin foam model for 3d quantum gravity coupled to matter fields. Finally, we show that the corresponding spin foam amplitudes admit a natural approximation as the Feynman a…
A number of approaches to four-dimensional quantum gravity, such as loop quantum gravity and holography, situate areas as their fundamental variables. However, this choice of kinematics can easily lead to gravitational dynamics peaked on flat spacetimes. We show that this is due to how regions are glued in the gravitational path integral via a discrete spin foam model. We introduce a family of `effective' spin foam models that incorporate a quantum area spectrum, impose gluing constraints as strongly as possible, and leverage the discrete general relativity action to specify amplitudes. These …
We study a generalized action for gravity as a constrained BF theory, and its relationship with the Plebanski action. We analyse the discretization of the constraints and the spin foam quantization of the theory, showing that it leads naturally to the Barrett-Crane spin foam model for quantum gravity. Our analysis holds true in both the Euclidean and Lorentzian formulation.
This is an introduction to spin foam models for non-perturbative quantum gravity, an approach that lies at the point of convergence of many different research areas, including loop quantum gravity, topological quantum field theories, path integral quantum gravity, lattice field theory, matrix models, category theory, statistical mechanics. We describe the general formalism and ideas of spin foam models, the picture of quantum geometry emerging from them, and give a review of the results obtained so far, in both the Euclidean and Lorentzian case. We focus in particular on the Barrett-Crane mode…
The expectation that it should not be possible to gain experimental insight on the structure of space-time at Planckian distance scales has been recently challenged by several studies. With respect to space-time fluctuations, one of the conjectured features of quantum-gravity foam, the experiments that have the best sensitivity are the ones which were originally devised for searches of the classical-physics phenomenon of gravity waves. In experiments searching for classical gravity waves the presence of space-time fluctuations would introduce a source of noise just like the ordinary (non-gravi…
We link the notion causality with the orientation of the 2-complex on which spin foam models are based. We show that all current spin foam models are orientation-independent, pointing out the mathematical structure behind this independence. Using the technology of evolution kernels for quantum fields/particles on Lie groups/homogeneous spaces, we construct a generalised version of spin foam models, introducing an extra proper time variable and prove that different ranges of integration for this variable lead to different classes of spin foam models: the usual ones, interpreted as the quantum g…
This article reviews the present status of the spin foam approach to the quantization of gravity. Special attention is payed to the pedagogical presentation of the recently introduced new models for four dimensional quantum gravity. The models are motivated by a suitable implementation of the path integral quantization of the Plebanski formulation of gravity on a simplicial regularization. The article also includes a self-contained treatment of the 2+1 gravity. The simple nature of the latter provides the basis and a perspective for the analysis of both conceptual and technical issues that rem…
This paper aims at clarifying the link between Loop Quantum Gravity and Spin-Foam models in four dimensions. Starting from the canonical framework, we construct an operator P acting on the space of cylindrical functions Cyl($Γ$), where $Γ$ is the 4-simplex graph, such that its ma- trix elements are, up to some normalization factors, the vertex amplitude of Spin-Foam models. The Spin-Foam models we are considering are the topological model, the Barrett-Crane model and the Engle-Pereira-Rovelli model. The operator P is usually called the "projector" into physical states and its matrix elements g…
Spin foam models of quantum gravity are based on Plebanski's formulation of general relativity as a constrained BF theory. We give an alternative formulation of gravity as BF theory plus a certain potential term for the B-field. When the potential is taken to be infinitely steep one recovers general relativity. For a generic potential the theory still describes gravity in that it propagates just two graviton polarizations. The arising class of theories is of the type amenable to spin foam quantization methods, and, we argue, may allow one to come to terms with renormalization in the spin foam …
The simplicial framework of Engle-Pereira-Rovelli-Livine spin-foam models is generalized to match the diffeomorphism invariant framework of loop quantum gravity. The simplicial spin-foams are generalized to arbitrary linear 2-cell spin-foams. The resulting framework admits all the spin-network states of loop quantum gravity, not only those defined by triangulations (or cubulations). In particular the notion of embedded spin-foam we use allows to consider knotting or linking spin-foam histories. Also the main tools as the vertex structure and the vertex amplitude are naturally generalized to ar…
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.
A model of space-time foam, made by $N$ wormholes is considered. The Casimir energy leading to such a model is computed by means of the phase shift method which is in agreement with the variational approach used in Refs.[9-14]. The collection of Schwarzschild and Reissner-Nordström wormholes are separately considered to represent the foam. The Casimir energy shows that the Reissner-Nordström wormholes cannot be used to represent the foam.
Spinorial tools have recently come back to fashion in loop gravity and spin foams. They provide an elegant tool relating the standard holonomy-flux algebra to the twisted geometry picture of the classical phase space on a fixed graph, and to twistors. In these lectures we provide a brief and technical introduction to the formalism and some of its applications.
Building on recent significant results of Frodden, Ghosh and Perez (FGP) and Bianchi, I present a quantum version of Jacobson's argument that the Einstein equations emerge as the equation of state of a quantum gravitational system. I give three criteria a quantum theory of gravity must satisfy if it is to allow Jacobson's argument to be run. I then show that the results of FGP and Bianchi provide evidence that loop quantum gravity satisfies two of these criteria and argue that the third should also be satisfied in loop quantum gravity. I also show that the energy defined by FGP is the canonica…
A covariant spin-foam formulation of quantum gravity has been recently developed, characterized by a kinematics which appears to match well the one of canonical loop quantum gravity. In this paper we reconsider the implementation of the constraints that defines the model. We define in a simple way the boundary Hilbert space of the theory, introducing a slight modification of the embedding of the SU(2) representations into the SL(2,C) ones. We then show directly that all constraints vanish on this space in a weak sense. The vanishing is exact (and not just in the large quantum number limit.) We…
We report on the numerical analysis of the area correlations in spin foam gravity on a single 4-simplex considered by Rovelli in PRL 97 (2006) 151301. We compare the asymptotics and confirm the inverse squared distance leading behaviour at large scales. This supports the recent advances on testing the semiclassical limit of the theory. Furthermore, we show that the microscopic discreteness of the theory dynamically suppresses and regularizes the correlations at the Planck scale.
We derive the the Barrett-Crane spin foam model for Euclidean 4 dimensional quantum gravity from a discretized BF theory, imposing the constraints that reduce it to gravity at the quantum level. We obtain in this way a precise prescription of the form of the Barrett-Crane state sum, in the general case of an arbitrary manifold with boundary. In particular we derive the amplitude for the edges of the spin foam from a natural procedure of gluing different 4-simplices along a common tetrahedron. The generalization of our results to higher dimensions is also shown.
We show that N = 1 supersymmetric BF theory in 3d leads to a supersymmetric spin foam amplitude via a lattice discretisation. Furthermore, by analysing the supersymmetric quantum amplitudes, we show that they can be re-interpreted as 3d gravity coupled to embedded fermionic Feynman diagrams.
In quantum gravity, we envision renormalization as the key tool for bridging the gap between microscopic models and observable scales. For spin foam quantum gravity, which is defined on a discretisation akin to lattice gauge theories, the goal is to derive an effective theory on a coarser discretisation from the dynamics on the finer one, coarse graining the system in the process and thus relating physics at different scales. In this review I will discuss the motivation for studying renormalization in spin foam quantum gravity, e.g. to restore diffeomorphism symmetry, and explain how to define…
It is shown that 3D part of a spherically symmetric solution in conformal Weyl gravity interacting with Maxwell electrodynamics is a Yamabe flow as well. The Yamabe flow describes the transition from a horn of an initial wormhole to a 3D Euclidean space both filled with a radial electric field. It is supposed that such transition may describe appearing/disappearing quantum wormholes in a spacetime foam.
In loop quantum gravity we now have a clear picture of the quantum geometry of space, thanks in part to the theory of spin networks. The concept of `spin foam' is intended to serve as a similar picture for the quantum geometry of spacetime. In general, a spin network is a graph with edges labelled by representations and vertices labelled by intertwining operators. Similarly, a spin foam is a 2-dimensional complex with faces labelled by representations and edges labelled by intertwining operators. In a `spin foam model' we describe states as linear combinations of spin networks and compute tran…
An example illustrating a continuum spin foam framework is presented. This covariant framework induces the kinematics of canonical loop quantization, and its dynamics is generated by a {\em renormalized} sum over colored polyhedra. Physically the example corresponds to 3d gravity with cosmological constant. Starting from a kinematical structure that accommodates local degrees of freedom and does not involve the choice of any background structure (e. g. triangulation), the dynamics reduces the field theory to have only global degrees of freedom. The result is {\em projectively} equivalent to th…
Spacetime undergoes quantum fluctuations, giving rise to spacetime foam, a.k.a. quantum foam. We discuss some properties of spacetime foam, and point out the conceptual interconnections in the physics of quantum foam, black holes, and quantum computation. We also discuss the phenomenology of quantum foam, and conclude that it may be difficult, but by no means impossible, to detect its tiny effects in the not-too-distant future.
In this paper, we address the problem of the dynamics in three dimensional loop quantum gravity with zero cosmological constant. We construct a rigorous definition of Rovelli's generalized projection operator from the kinematical Hilbert space--corresponding to the quantization of the infinite dimensional kinematical configuration space of the theory--to the physical Hilbert space. In particular, we provide the definition of the physical scalar product which can be represented in terms of a sum over (finite) spin-foam amplitudes. Therefore, we establish a clear-cut connection between the canon…
