Canonical quantization of general relativity: the last 18 years in a nutshell

Abstract

This is a summary of the lectures presented at the Xth Brazilian school on cosmology and gravitation. The style of the text is that of a lightly written descriptive summary of ideas with almost no formulas, with pointers to the literature. We hope this style can encourage new people to take a look into these results. We discuss the variables that Ashtekar introduced 18 years ago that gave rise to new momentum in this field, the loop representation, spin networks, measures in the space of connections modulo gauge transformations, the Hamiltonian constraint, application to cosmology and the connection with potentially observable effects in gamma-ray bursts and conclude with a discussion of consistent discretizations of general relativity on the lattice.