The Affine Quantum Gravity Program

TL;DR

This paper proposes a new approach to quantum gravity based on affine variables, emphasizing positivity of the metric and using a projection operator method to handle constraints nonperturbatively.

Contribution

It introduces an affine quantum gravity framework with a novel functional integral formulation and a method to enforce constraints, advancing nonperturbative understanding of gravity.

Findings

Affine quantum gravity maintains metric positivity.

A new functional integral approach for constraints.

Potential nonperturbative insights into gravity.

Abstract

The central principle of affine quantum gravity is securing and maintaining the strict positivity of the matrix $\{\hg_{ab}(x)\}$ composed of the spatial components of the local metric operator. On spectral grounds, canonical commutation relations are incompatible with this principle, and they must 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 recently developed projection operator method, which treats all quantum constraints on an equal footing. Using this method, enforcement of regularized versions of the gravitational operator constraints is formulated quite naturally by means of a novel and relatively well-defined functional integral involving only the same set of variables that appears in the usual classical formulation. It is anticipated that skills and insight to study this formulation can be developed by studying special, reduced-variable models that still retain some basic characteristics of gravity, specifically a partial second-class constraint operator structure. Although perturbatively nonrenormalizable, gravity may possibly be understood nonperturbatively from a hard-core perspective that has proved valuable for specialized models. Finally, developing a procedure to pass to the genuine physical Hilbert space involves several interconnected steps that require careful coordination.