Positive energy in quantum gravity

TL;DR

This paper explores extending Witten's classical positive energy proof to a non-perturbative quantum gravity context, proposing conditions for a positive Hamiltonian operator in loop quantum gravity.

Contribution

It introduces specific conditions, including a particular factor ordering, to establish positivity of the quantum Hamiltonian in a non-perturbative quantum gravity framework.

Findings

Identifies sufficient conditions for positive quantum energy

Proposes a specific factor ordering for quantum constraints

Builds on unpublished results with Artem Starodubtsev

Abstract

This paper addresses the question of whether Witten's proof of positive ADM energy for classical general relativity can be extended to give a proof of positive energy for a non-perturbative quantization of general relativity. To address this question, a set of conditions is shown to be sufficient for showing the positivity of a Hamiltonian operator corresponding to the ADM energy. One of these conditions is a particular factor ordering for the constraints of general relativity, in a representation where the states are functionals of the Ashtekar connection, and the auxiliary, Witten spinor. These developments are partly based on results derived with Artem Starodubtsev (unpublished notes, 2004).