Some New Aspects of Quantum Gravity

Abstract

We have proposed the quantization of the gravitational field in a synchronous reference frame taking as independent position fields, the six spatial components of the metric tensor. The Einstein-Hilbert Lagrangian is quadratic in the space time derivatives of these metric tensor components and hence in particular, the momentum fields become linear functions of the space-time derivatives of the position fields. It is this fact that gives a simple form to the Hamiltonian density of the gravitational field in a synchronous frame, this simple form of the Hamiltonian being a quadratic function of the momentum fields with a shift that is linear in the spatial derivatives of the metric, very much like the Hamiltonian of a non-relativistic particle moving in a vector potential. Differential equations for the gravitational field propagator are derived and we explain how approximations to this propagator can be derived and used to deduce the graviton propagator corrections caused by nonlinear interactions of the graviton field with itself. We explain how this corrected graviton propagator can be used to deduce how much mass the graviton acquires due to these self-interactions of cubic and higher order. We then consider the important problem involving the coupling of a nonlinear field theory described by its Lagrangian density to a quantum noisy bath and explain how the resulting Hamiltonian of the field plus bath can be used to derive the Hudson-Parthasarathy noisy Schrodinger equation (HPS) which is a quantum stochastic differential equation for the joint unitary evolution of the field interacting with the noisy bath. We explain this in the context of gravity coupled to a noisy bath like a noisy electromagnetic field. The HPS equation contains linear as well as quadratic terms in the white bath noise with the linear terms representing quantum annihilation and creation/quantum Brownian motion process differentials and the quadratic terms representing quantum conservation/Poisson processes differentials. Finally we explain how using Feynman path integrals for fields for evaluating the quantum effective action produced by higher order cumulants of the current field, we can calculate corrections to the quantum effective action produced by higher order cumulants of the current field and hence demonstrate how gauge symmetries of the classical action get broken when we pass over to the quantum effective action with additional symmetry breaking terms produced by the presence of higher order cumulants of the current. This kind of approximate symmetry breaking is known to give masses to massless particles or more generally, corrections to the masses of already massive particles and we illustrate this idea in the context of interactions of the gravitational field with a random electromagnetic field being regarded as the current. This interaction is the standard Maxwell action used in general relativity. The drawback of our approach to quantum gravity is that is its not diffeomorphic invariant since we have chosen our frame to be always synchronous. Further work on how one can incorporate interactions of the gravitational field with a random non-Abelian gauge field is in progress which becomes important because it generates non only quadratic but also cubic and fourth degree terms in the gauge field when it interacts with gravity.