Speaker
Description
Quantum Geometrodynamics represents an early attempt at the canonical quantization of General Relativity. In a seminal paper, DeWitt proposed a formal Hamiltonian constraint operator by substituting canonical momenta with variational derivative operators. However, the rigorous interpretation of this operator has remained elusive due to the highly nonlinear nature of the constraint and the associated issues arising from the multiplication of distributions. Consequently, a well-defined Hilbert space for the theory could not be specified. In this talk, we propose a novel approach to overcome the difficulties faced by Quantum Geometrodynamics. We employ a lattice discretization that adheres as closely as possible to the original formulation, and examine the lattice corrections to the theory. We discuss the steps necessary for obtaining a well-defined continuum limit and explain how to perform calculations in the effective theory.
Moreover, we establish Hilbert spaces for the lattice theories that allow for well-defined lattice approximations of all continuum quantities. These Hilbert spaces exhibit a non-standard representation of the canonical commutation relations between the matrix elements of the spatial metric and the conjugate momenta. This approach ensures that states are exclusively supported on positive definite symmetric matrices.
