A quantum theory of gravity

Abstract

We develop a covariant Hamiltonian formalism in which evolution is measured along the proper-time flow of an observer congruence, replacing the external coordinate-time derivative with the Lie derivative. Our formalism keeps the Hamiltonian split between evolution and configuration variables and expresses it through spacetime tensors projected relative to the chosen congruence. After quantization, the resulting observer-congruent brackets lead to relativistic Heisenberg and Schrödinger equations whose evolution parameter is proper time. We then apply this framework to gravity by taking the projected spatial metric and its conjugate momentum as canonical variables, obtaining a complete model of canonical quantum gravity. Among the predictions of this theory we find a resolution to the singularity problems in cosmology, black holes and black hole thermodynamics.