A simple background-independent hamiltonian quantum model

We study formulation and probabilistic interpretation of a simple general-relativistic hamiltonian quantum system. The system has no unitary evolution in background time. The quantum theory yields transition probabilities between measurable quantities (partial observables). These converge to the classical predictions in the $\hbar\to 0$ limit. Our main tool is the kernel of the projector on the solutions of Wheeler-deWitt equation, which we analyze in detail. It is a real quantity, which can be seen as a propagator that propagates "forward" as well as "backward" in a local parameter time. Individual quantum states, on the other hand, may contain only "forward propagating" components. The analysis sheds some light on the interpretation of background independent transition amplitudes in quantum gravity

The volume operator in covariant quantum gravity

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 particular, the geometrical observable giving the area of a surface has been shown to be the same as the one in loop quantum gravity. Here we discuss the volume observable. We derive the volume operator in the covariant theory, and show that it matches the one of loop quantum gravity, as does the area. We also reconsider the implementation of the constraints that defines the model: we derive in a simple way the boundary Hilbert space of the theory from a suitable form of the classical constraints, and show directly that all constraints vanish weakly on this space.Comment: 10 pages. Version 2: proof extended to gamma > 1

The century of the incomplete revolution: searching for general relativistic quantum field theory

In fundamental physics, this has been the century of quantum mechanics and general relativity. It has also been the century of the long search for a conceptual framework capable of embracing the astonishing features of the world that have been revealed by these two ``first pieces of a conceptual revolution''. I discuss the general requirements on the mathematics and some specific developments towards the construction of such a framework. Examples of covariant constructions of (simple) generally relativistic quantum field theories have been obtained as topological quantum field theories, in nonperturbative zero-dimensional string theory and its higher dimensional generalizations, and as spin foam models. A canonical construction of a general relativistic quantum field theory is provided by loop quantum gravity. Remarkably, all these diverse approaches have turn out to be related, suggesting an intriguing general picture of general relativistic quantum physics.Comment: To appear in the Journal of Mathematical Physics 2000 Special Issu

Physical boundary Hilbert space and volume operator in the Lorentzian new spin-foam theory

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 also generalize the definition of the volume operator in the spinfoam model to the Lorentzian signature, and show that it matches the one of loop quantum gravity, as does in the Euclidean case.Comment: 11 page

Second-order amplitudes in loop quantum gravity

We explore some second-order amplitudes in loop quantum gravity. In particular, we compute some second-order contributions to diagonal components of the graviton propagator in the large distance limit, using the old version of the Barrett-Crane vertex amplitude. We illustrate the geometry associated to these terms. We find some peculiar phenomena in the large distance behavior of these amplitudes, related with the geometry of the generalized triangulations dual to the Feynman graphs of the corresponding group field theory. In particular, we point out a possible further difficulty with the old Barrett-Crane vertex: it appears to lead to flatness instead of Ricci-flatness, at least in some situations. The observation raises the question whether this difficulty remains with the new version of the vertex.Comment: 22 pages, 18 figure

Spacetime states and covariant quantum theory

In it's usual presentation, classical mechanics appears to give time a very special role. But it is well known that mechanics can be formulated so as to treat the time variable on the same footing as the other variables in the extended configuration space. Such covariant formulations are natural for relativistic gravitational systems, where general covariance conflicts with the notion of a preferred physical-time variable. The standard presentation of quantum mechanics, in turns, gives again time a very special role, raising well known difficulties for quantum gravity. Is there a covariant form of (canonical) quantum mechanics? We observe that the preferred role of time in quantum theory is the consequence of an idealization: that measurements are instantaneous. Canonical quantum theory can be given a covariant form by dropping this idealization. States prepared by non-instantaneous measurements are described by "spacetime smeared states". The theory can be formulated in terms of these states, without making any reference to a special time variable. The quantum dynamics is expressed in terms of the propagator, an object covariantly defined on the extended configuration space.Comment: 20 pages, no figures. Revision: minor corrections and references adde

The projector on physical states in loop quantum gravity

We construct the operator that projects on the physical states in loop quantum gravity. To this aim, we consider a diffeomorphism invariant functional integral over scalar functions. The construction defines a covariant, Feynman-like, spacetime formalism for quantum gravity and relates this theory to the spin foam models. We also discuss how expectation values of physical quantity can be computed.Comment: 15 pages, 2 figures, substantially revised versio

The loop-quantum-gravity vertex-amplitude

Spinfoam theories are hoped to provide the dynamics of non-perturbative loop quantum gravity. But a number of their features remain elusive. The best studied one -the euclidean Barrett-Crane model- does not have the boundary state space needed for this, and there are recent indications that, consequently, it may fail to yield the correct low-energy $n$-point functions. These difficulties can be traced to the SO(4) -> SU(2) gauge fixing and the way certain second class constraints are imposed, arguably incorrectly, strongly. We present an alternative model, that can be derived as a bona fide quantization of a Regge discretization of euclidean general relativity, and where the constraints are imposed weakly. Its state space is a natural subspace of the SO(4) spin-network space and matches the SO(3) hamiltonian spin network space. The model provides a long sought SO(4)-covariant vertex amplitude for loop quantum gravity.Comment: 6page

Graviton propagator in loop quantum gravity

We compute some components of the graviton propagator in loop quantum gravity, using the spinfoam formalism, up to some second order terms in the expansion parameter.Comment: 41 pages, 6 figure