Alejandro Perez

Spin foam models for quantum gravity

In this topical review, we review the present status of the spin foam formulation of non-perturbative (background-independent) quantum gravity. The topical review is divided into two parts. In the first part, we present a general introduction to the main ideas emphasizing their motivation from various perspectives. Riemannian three-dimensional gravity is used as a simple example to illustrate conceptual issues and the main goals of the approach. The main features of the various existing models for four-dimensional gravity are also presented here. We conclude with a discussion of important questions to be addressed in four dimensions (gauge invariance, discretization independence, etc). 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 and its Lorentzian variants.

Lorentz Invariance and Quantum Gravity: An Additional Fine-Tuning Problem?

Trying to combine standard quantum field theories with gravity leads to a breakdown of the usual structure of space time at around the Planck length, 1.6x10(-35) m, with possible violations of Lorentz invariance. Calculations of preferred-frame effects in quantum gravity have further motivated high precision searches for Lorentz violation. Here, we explain that combining known elementary particle interactions with a Planck-scale preferred frame gives rise to Lorentz violation at the percent level, some 20 orders of magnitude higher than earlier estimates, unless the bare parameters of the theory are unnaturally strongly fine tuned. Therefore an important task is not just the improvement of the precision of searches for violations of Lorentz invariance, but also the search for theoretical mechanisms for automatically preserving Lorentz invariance.

Three-dimensional loop quantum gravity: physical scalar product and spin-foam models

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 Rovellis 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 canonical quantization of three-dimensional gravity and spin-foam models. We emphasize two main properties of the result: first that no cut-off in the kinematical degrees of freedom of the theory is introduced (in contrast to standard lattice methods), and second that no ill-defined sum over spins (bubble divergences) are present in the spin-foam representation.

Black Hole Entropy and SU(2) Chern-Simons Theory

Black holes (BHs) in equilibrium can be defined locally in terms of the so-called isolated horizon boundary condition given on a null surface representing the event horizon. We show that this boundary condition can be treated in a manifestly SU(2) invariant manner. Upon quantization, state counting is expressed in terms of the dimension of Chern-Simons Hilbert spaces on a sphere with punctures. Remarkably, when considering an ensemble of fixed horizon area a(H), the counting can be mapped to simply counting the number of SU(2) intertwiners compatible with the spins labeling the punctures. The resulting BH entropy is proportional to a(H) with logarithmic corrections ΔS=-3/2 loga(H). Our treatment from first principles settles previous controversies concerning the counting of states.

Spin foam quantization and anomalies

The most common spin foam models of gravity are widely believed to be discrete path integral quantizations of the Plebanski action. However, their derivation in present formulations is incomplete and lower dimensional simplex amplitudes are left open to choice. Since their large-spin behavior determines the convergence properties of the state-sum, this gap has to be closed before any reliable conclusion about finiteness can be reached. It is shown that these amplitudes are directly related to the path integral measure and can in principle be derived from it, requiring detailed knowledge of the constraint algebra and gauge fixing. In a related manner, minimal requirements of background independence provide non trivial restrictions on the form of an anomaly free measure. Many models in the literature do not satisfy these requirements. A simple model satisfying the above consistency requirements is presented which can be thought of as a spin foam quantization of the Husain–Kuchař model.

On the quantum origin of the seeds of cosmic structure

The current understanding of the quantum origin of cosmic structure is discussed critically. We point out that in the existing treatments a transition from a symmetric quantum state to an (essentially classical) non-symmetric state is implicitly assumed, but not specified or analysed in any detail. In facing this issue, we are led to conclude that new physics is required to explain the apparent predictive power of the usual schemes. Furthermore, we show that the novel way of looking at the relevant issues opens new windows from where relevant information might be extracted regarding cosmological issues and perhaps even clues about aspects of quantum gravity.

Three-dimensional loop quantum gravity: coupling to point particles

We consider the coupling between three-dimensional gravity with zero cosmological constant and massive spinning point particles. First, we study the classical canonical analysis of the coupled system. Then, we go to the Hamiltonian quantization generalizing loop quantum gravity techniques. We give a complete description of the kinematical Hilbert space of the coupled system. Finally, we define the physical Hilbert space of the system of self-gravitating massive spinning point particles using Rovellis generalized projection operator which can be represented as a sum over spin-foam amplitudes. In addition we provide an explicit expression of the classical distance operator between two particles which is defined as an observable.

The SU(2) black hole entropy revisited

We study the state-counting problem that arises in the SU(2) black hole entropy calculation in loop quantum gravity. More precisely, we compute the leading term and the logarithmic correction of both the spherically symmetric and the distorted SU(2) black holes. Contrary to what has been done in previous works, we have to take into account “quantum corrections” in our framework in the sense that the level k of the Chern-Simons theory which describes the black hole is finite and not sent to infinity. Therefore, the new results presented here allow for the computation of the entropy in models where the quantum group corrections are important.

Spin foam diagrammatics and topological invariance

We provide a simple proof of the topological invariance of the Turaev–Viro model (corresponding to simplicial 3D pure Euclidean gravity with cosmological constant) by means of a novel diagrammatic formulation of the state sum models for quantum BF theories. Moreover, we prove the invariance under more general conditions allowing the state sum to be defined on arbitrary cellular decompositions of the underlying manifold. Invariance is governed by a set of identities corresponding to local gluing and rearrangement of cells in the complex. Due to the fully algebraic nature of these identities our results extend to a vast class of quantum groups. The techniques introduced here could be relevant for investigating the scaling properties of non-topological state sums, proposed as models of quantum gravity in 4D, under refinement of the cellular decomposition.

On the physical Hilbert space of loop quantum cosmology

In this paper we present a model of Riemannian loop quantum cosmology with a self-adjoint quantum scalar constraint. The physical Hilbert space is constructed using refined algebraic quantization. When matter is included in the form of a cosmological constant, the model is exactly solvable and we show explicitly that the physical Hilbert space is separable, consisting of a single physical state. We extend the model to the Lorentzian sector and discuss important implications for standard loop quantum cosmology.