Canonical Loop Quantum Gravity

This article presents a comprehensive and self-contained review of Loop Quantum Gravity (LQG), focusing on both its canonical (Hamiltonian) formulation and covariant (spin foam) path integral approach. Beginning with the reformulation of General Relativity using Ashtekar–Barbero variables, it systematically constructs the kinematical Hilbert space, details the spectra of geometric operators (areas and volumes), and rigorously examines the Hamiltonian constraint and its quantization via Thiemann’s regularization. The work further develops the connection to spin foam models, with a detailed exposition of the EPRL-FK model, semiclassical limits, and the emergence of Regge calculus. Throughout, emphasis is placed on technical derivations, including the algebra of constraints, the action of the Hamiltonian on spin networks, and the role of spin foams as projectors onto physical states. The article concludes by discussing open problems such as renormalization, the continuum limit, and the integration of matter fields, offering a forward-looking perspective on the role of LQG in understanding the quantum structure of spacetime.