In this lectures, I will introduce the basics of loop quantum gravity and sketch its developments over the last years, focusing on the covariant definition of the theory, up to its group field theory formulation.
I will first describe the first steps of the canonical quantization of Einstein's gravity in terms of the Ashtekar connection, leading to the definition of a kinematical Hilbert space in which states of quantum spacetime are associated to graphs embedded in the canonical manifold and are labelled by algebraic data from the representation theory of the Lorentz groups. The main features of this Hilbert space and of geometric operators acting on it will be sketched.
Next, the re-interpretation of the same quantum states in terms of abstract cellular structures will be explained, and with it, a new path toward the covariant definition of the quantum dynamics: spin foam models, discrete and algebraic analogues of a sum over geometries. I will show both their definition in terms of state sums and their equivalence to discrete gravity path integrals. Some of the many results obtained in recent years will be presented as well, in particular concerning their semi-classical limit.
In the last part of the lectures, I will present a possible completion of the spin foam formulation of loop quantum gravity: group field theories. These are quantum field theories on group manifolds that can be understood as a 2nd quantized reformulation of loop quantum gravity. They define the quantum dynamics of spacetime via a sum over cellular complexes, weighted by spin foam amplitudes or, equivalently, by discrete gravity path integrals. In doing so, they generalize matrix models for 2d quantum gravity, in the spirit of tensor models, of which they represent a group-theoretic enrichment. I will conclude with an overview of recent results obtained in this group field theory setting, concerning phase transitions, renormalization and effective continuum dynamics.
General introduction and reviews on loop quantum gravity, spin foam models, group field theories and tensor models, from various perspectives: