Semiclassical analysis of loop quantum gravity

Abstract

In this PhD thesis I discuss various aspects of the semiclassical dynamics of Loop Quantum Gravity (LQG) as defined by Spin Foam models (the covariant version of canonical LQG). In particular I consider the ”new spin foam models” as candidates for the LQG vertex amplitude. I introduce a technique for testing the semiclassical behaviour which is the study of the propagation of semiclassical wave-packets, obtaining some preliminary good indications. Then I study the asymptotics of a building block of the spin foam amplitude which are the fusion coefficients; these are combinatorial symbols that realize the equivalence between the LQG and the SF kinematical state space. Their asymptotics shows nice properties in the semiclassical sector. One of the most important test is the comparison of the n-point functions computed in LQG with the ones of standard perturbative quantum gravity. I compute the connected 2-point function of metric operators, and compare it with the graviton propagator of standard QFT, finding a complete agreement (scaling and tensorial structure) for a suitable choice of the few free parameters. This is an important test for the ”new SF models” since the previous major model, the Barrett-Crane model, failed to yield the correct graviton propagator. The computation of the propagator is based on a rather particular choice of the boundary state (the one representing the semiclassical geometry over which the gravitons propagate), which is dictated by geometrical intuition. The robustness of this formalism is strengthened in my recent work ”Coherent spin-networks”. Here I define coherent states for full LQG from a heat-kernel over phase-space (like in ordinary QM) and find that in the semiclassical limit their asymptotics reproduce exactly the states used in SF models. The importance of coherent spin-networks, defined over a three-dimensional hypersurface, is that we have a clear geometrical interpretation of the classical geometry (intrinsic and extrinsic) they are peaked on; hence we can in principle construct quantum states having minimal uncertainty in conjugate quantities that represent a given (e.g. Minkowski or deSitter) classical space-time.