Quantization of Einstein Gravity: Extraction of Hilbert Space and Constraints

TL;DR

This paper develops a novel approach to quantizing Einstein gravity by deriving quantum constraints directly from the functional integral, avoiding classical guesswork and clarifying the Hilbert space structure.

Contribution

It introduces a method to extract operator constraints directly from the functional integral, bypassing classical quantization ambiguities in Einstein gravity.

Findings

Diffeomorphism-invariant measure is constructed using the metric of metrics.

Operator constraints are obtained directly from the functional integral.

The formalism clarifies the structure of the quantum Hilbert space in gravity.

Abstract

In this paper, a careful treatment of extraction of the Hilbert space and constraints from the formal functional integral with the Einstein-Hilbert action is given. The diffeomorphism inavariant measure is worked out using the metric of metrics. The procedure of quantizing the classical constraints is bypassed. Instead the relevant operator constraints are directly obtained from the functional integral, removing the ambiguities and uncertainties involved in guessing them from the classical theory. The novelties of the resulting formalism are briefly discussed.