Hamiltonian approach to GR - Part 2: covariant theory of quantum gravity

TL;DR

This paper develops a non-perturbative, covariant quantum gravity theory based on Hamiltonian structures, deriving a quantum wave equation, and exploring its implications for graviton mass and the cosmological constant.

Contribution

It introduces a novel covariant quantum gravity framework using Hamiltonian and Hamilton-Jacobi formalisms, with a new quantum wave equation and perturbative approximation scheme.

Findings

Derivation of a covariant quantum wave equation for gravity.

Identification of conditions for a discrete energy spectrum.

Estimation of graviton mass and quantum origin of the cosmological constant.

Abstract

A non-perturbative quantum field theory of General Relativity is presented which leads to a new realization of the theory of Covariant Quantum-Gravity (CQG-theory). The treatment is founded on the recently-identified Hamiltonian structure associated with the classical space-time, i.e., the corresponding manifestly-covariant Hamilton equations and the related Hamilton-Jacobi theory. The quantum Hamiltonian operator and the CQG-wave equation for the corresponding CQG-state and wave-function are realized in $% 4-$scalar form. The new quantum wave equation is shown to be equivalent to a set of quantum hydrodynamic equations which warrant the consistency with the classical GR Hamilton-Jacobi equation in the semiclassical limit. A perturbative approximation scheme is developed, which permits the adoption of the harmonic oscillator approximation for the treatment of the Hamiltonian potential. As an application of the theory, the stationary vacuum CQG-wave equation is studied, yielding a stationary equation for the CQG-state in terms of the $4-$scalar invariant-energy eigenvalue associated with the corresponding approximate quantum Hamiltonian operator. The conditions for the existence of a discrete invariant-energy spectrum are pointed out. This yields a possible estimate for the graviton mass together with a new interpretation about the quantum origin of the cosmological constant.