Introduction to Modern Canonical Quantum General Relativity
Thomas Thiemann
TL;DR
This article surveys the status and foundations of canonical quantum general relativity (loop quantum gravity), emphasizing a connection-dynamics approach that preserves background independence in four-dimensional spacetime. It outlines the transition from ADM geometrodynamics to Ashtekar’s connection variables, and then to the rigorous loop- and projective-limit framework built on distributional connections, holonomies, and spin networks. A core focus is the construction of a diffeomorphism-invariant kinematical Hilbert space via the Ashtekar–Lewandowski uniform measure, the spin-network basis, and the rigorous treatment of Gauss and diffeomorphism constraints, laying groundwork for nonperturbative quantum gravity. The work analyzes both the mathematical scaffolding (C*-algebras, projective limits, measures) and the physical content (quantum kinematics, constraint implementation), while highlighting remaining challenges such as the full Hamiltonian constraint and semiclassical limit. Overall, the paper provides a detailed, technically precise roadmap for building a background-independent quantum theory of gravity based on connections and loop variables, with clear implications for discreteness of geometry and UV finiteness arising from quantum gravity dynamics.
Abstract
This is an introduction to the by now fifteen years old research field of canonical quantum general relativity, sometimes called "loop quantum gravity". The term "modern" in the title refers to the fact that the quantum theory is based on formulating classical general relativity as a theory of connections rather than metrics as compared to in original version due to Arnowitt, Deser and Misner. Canonical quantum general relativity is an attempt to define a mathematically rigorous, non-perturbative, background independent theory of Lorentzian quantum gravity in four spacetime dimensions in the continuum. The approach is minimal in that one simply analyzes the logical consequences of combining the principles of general relativity with the principles of quantum mechanics. The requirement to preserve background independence has lead to new, fascinating mathematical structures which one does not see in perturbative approaches, e.g. a fundamental discreteness of spacetime seems to be a prediction of the theory providing a first substantial evidence for a theory in which the gravitational field acts as a natural UV cut-off. An effort has been made to provide a self-contained exposition of a restricted amount of material at the appropriate level of rigour which at the same time is accessible to graduate students with only basic knowledge of general relativity and quantum field theory on Minkowski space.