Nonlinear quantum gravity on the constant mean curvature foliation

TL;DR

This paper introduces a nonlinear quantum gravity framework using constant mean curvature foliation, transforming the Hamiltonian constraint into an expectation value equation, and applies it to Bianchi cosmological models.

Contribution

It develops a nonlinear, unitary quantum gravity theory based on a novel quantization scheme and employs the constant mean curvature foliation to simplify the constraints in canonical general relativity.

Findings

The theory is inherently nonlinear but maintains unitarity.

The Hamiltonian constraint is reformulated as an expectation value equation.

Application to Bianchi models demonstrates the framework's versatility.

Abstract

A new approach to quantum gravity is presented based on a nonlinear quantization scheme for canonical field theories with an implicitly defined Hamiltonian. The constant mean curvature foliation is employed to eliminate the momentum constraints in canonical general relativity. It is, however, argued that the Hamiltonian constraint may be advantageously retained in the reduced classical system to be quantized. This permits the Hamiltonian constraint equation to be consistently turned into an expectation value equation on quantization that describes the scale factor on each spatial hypersurface characterized by a constant mean exterior curvature. This expectation value equation augments the dynamical quantum evolution of the unconstrained conformal three-geometry with a transverse traceless momentum tensor density. The resulting quantum theory is inherently nonlinear. Nonetheless, it is unitary and free from a nonlocal and implicit description of the Hamiltonian operator. Finally, by imposing additional homogeneity symmetries, a broad class of Bianchi cosmological models are analyzed as nonlinear quantum minisuperspaces in the context of the proposed theory.