Noncanonical Quantization of Gravity. I. Foundations of Affine Quantum Gravity

TL;DR

This paper develops a foundational approach to quantum gravity using affine commutation relations, constructing a kinematical representation and implementing constraints without gauge fixing, through reproducing kernel Hilbert spaces.

Contribution

It introduces a novel affine quantization framework for gravity, avoiding gauge fixing and auxiliary fields, and employs reproducing kernel methods for the physical Hilbert space.

Findings

Constructed a primary kinematical representation using reproducing kernels.

Applied the projection operator method to impose constraints without gauge fixing.

Illustrated the approach with quantum mechanical toy models.

Abstract

The nature of the classical canonical phase-space variables for gravity suggests that the associated quantum field operators should obey affine commutation relations rather than canonical commutation relations. Prior to the introduction of constraints, a primary kinematical representation is derived in the form of a reproducing kernel and its associated reproducing kernel Hilbert space. Constraints are introduced following the projection operator method which involves no gauge fixing, no complicated moduli space, nor any auxiliary fields. The result, which is only qualitatively sketched in the present paper, involves another reproducing kernel with which inner products are defined for the physical Hilbert space and which is obtained through a reduction of the original reproducing kernel. Several of the steps involved in this general analysis are illustrated by means of analogous steps applied to one-dimensional quantum mechanical models. These toy models help in motivating and understanding the analysis in the case of gravity.