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

Quantum Theory from Quantum Gravity

We provide a mechanism by which, from a background independent model with no quantum mechanics, quantum theory arises in the same limit in which spatial properties appear. Starting with an arbitrary abstract graph as the microscopic model of spacetime, our ansatz is that the microscopic dynamics can be chosen so that 1) the model has a low low energy limit which reproduces the non-relativistic classical dynamics of a system of N particles in flat spacetime, 2) there is a minimum length, and 3) some of the particles are in a thermal bath or otherwise evolve stochastically. We then construct simple functions of the degrees of freedom of the theory and show that their probability distributions evolve according to the Schroedinger equation. The non-local hidden variables required to satisfy the conditions of Bell's theorem are the links in the fundamental graph that connect nodes adjacent in the graph but distant in the approximate metric of the low energy limit. In the presence of these links, distant stochastic fluctuations are transferred into universal quantum fluctuations.Comment: 17 pages, 2 eps figure

Physical boundary state for the quantum tetrahedron

We consider stability under evolution as a criterion to select a physical boundary state for the spinfoam formalism. As an example, we apply it to the simplest spinfoam defined by a single quantum tetrahedron and solve the associated eigenvalue problem at leading order in the large spin limit. We show that this fixes uniquely the free parameters entering the boundary state. Remarkably, the state obtained this way gives a correlation between edges which runs at leading order with the inverse distance between the edges, in agreement with the linearized continuum theory. Finally, we give an argument why this correlator represents the propagation of a pure gauge, consistently with the absence of physical degrees of freedom in 3d general relativity.Comment: 20 pages, 6 figure

Discrete Symmetries in Covariant LQG

We study time-reversal and parity ---on the physical manifold and in internal space--- in covariant loop gravity. We consider a minor modification of the Holst action which makes it transform coherently under such transformations. The classical theory is not affected but the quantum theory is slightly different. In particular, the simplicity constraints are slightly modified and this restricts orientation flips in a spinfoam to occur only across degenerate regions, thus reducing the sources of potential divergences.Comment: 8 pages, v2: Minor change

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

Numerical indications on the semiclassical limit of the flipped vertex

We introduce a technique for testing the semiclassical limit of a quantum gravity vertex amplitude. The technique is based on the propagation of a semiclassical wave packet. We apply this technique to the newly introduced "flipped" vertex in loop quantum gravity, in order to test the intertwiner dependence of the vertex. Under some drastic simplifications, we find very preliminary, but surprisingly good numerical evidence for the correct classical limit.Comment: 4 pages, 8 figure

On the perturbative expansion of a quantum field theory around a topological sector

The idea of treating general relativistic theories in a perturbative expansion around a topological theory has been recently put forward in the quantum gravity literature. Here we investigate the viability of this idea, by applying it to conventional Yang--Mills theory on flat spacetime. We find that the expansion around the topological theory coincides with the usual expansion around the abelian theory, though the equivalence is non-trivial. In this context, the technique appears therefore to be viable, but not to bring particularly new insights. Some implications for gravity are discussed.Comment: 7 page

Coherent State Approach to Quantum Clocks

The ``problem of time'' has been a pressing issue in quantum gravity for some time. To help understand this problem, Rovelli proposed a model of a two harmonic oscillators system where one of the oscillators can be thought of as a ``clock'' for the other oscillator thus giving a natural time reference frame for the system. Recently, the author has constructed an explicit form for the coherent states on the reduced phase space of this system in terms of Klauder's projection operator approach. In this paper, by using coherent state representations and other tools from coherent state quantization, I investigate the construction of gauge invariant operators on this reduced phase space, and the ability to use a quantum oscillator as a ``clock.''Comment: 13 pages, Late

A semiclassical tetrahedron

We construct a macroscopic semiclassical state state for a quantum tetrahedron. The expectation values of the geometrical operators representing the volume, areas and dihedral angles are peaked around assigned classical values, with vanishing relative uncertainties.Comment: 10 pages; v2 revised versio

On knottings in the physical Hilbert space of LQG as given by the EPRL model

We consider the EPRL spin foam amplitude for arbitrary embedded two-complexes. Choosing a definition of the face- and edge amplitudes which lead to spin foam amplitudes invariant under trivial subdivisions, we investigate invariance properties of the amplitude under consistent deformations, which are deformations of the embedded two-complex where faces are allowed to pass through each other in a controlled way. Using this surprising invariance, we are able to show that in the physical Hilbert space as defined by the sum over all spin foams contains no knotting classes of graphs anymore.Comment: 22 pages, 14 figure