Canonical Loop Quantum Gravity

Beginning with the reformulation of General Relativity using <b>Ashtekar–Barbero variables</b>, it systematically constructs the <b>kinematical Hilbert space</b>, details the <b>spectra of geometric operators</b> (areas and volumes), and rigorously examines the <b>Hamiltonian constraint</b> and its quantization via <b>Thiemann’s regularization</b>.

The work further develops the connection to <b>spin foam models</b>, with a detailed exposition of the <b>EPRL-FK model</b>, semiclassical limits, and the emergence of Regge calculus. Throughout, emphasis is placed on <b>technical derivations</b>, 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.</p>