An Affinity for Affine Quantum Gravity

TL;DR

This paper explores affine quantum gravity, emphasizing the positivity of the metric operator, and introduces a projection operator method to handle quantum constraints nonperturbatively, offering a new perspective on quantum gravity.

Contribution

It proposes a novel approach using affine commutation relations and the projection operator method to formulate quantum gravity constraints nonperturbatively.

Findings

Formulation of gravitational constraints as a functional integral

Use of affine commutation relations for metric positivity

Potential nonperturbative understanding of gravity

Abstract

The main principle of affine quantum gravity is the strict positivity of the matrix \{\hat g_{ab}(x)\} composed of the spatial components of the local metric operator. Canonical commutation relations are incompatible with this principle, and they can be replaced by noncanonical, affine commutation relations. Due to the partial second-class nature of the quantum gravitational constraints, it is advantageous to use the projection operator method, which treats all quantum constraints on an equal footing. Using this method, enforcement of regularized versions of the gravitational constraint operators is formulated quite naturally as a novel and relatively well-defined functional integral involving only the same set of variables that appears in the usual classical formulation. Although perturbatively nonrenormalizable, gravity may possibly be understood nonperturbatively from a hard-core perspective that has proved valuable for specialized models.