Quasi-Local Energy in Loop Quantum Gravity

TL;DR

This paper explores the quantization of various quasi-local energy notions within loop quantum gravity, constructing operators for energies like Brown-York and Hawking, and deriving a holographic entropy-area relation.

Contribution

It introduces quantum operators for multiple quasi-local energies in loop quantum gravity, enabling analysis of gravitational energy at the quantum level.

Findings

Quantized operators for Brown-York, Liu-Yau, Hawking, and Geroch energies.

Derivation of a general entropy-area relation from the Geroch energy operator.

Establishment of a holographic principle within loop quantum gravity.

Abstract

Although there is no known meaningful notion of the energy density of the gravitational field in general relativity, a few notions of quasi-local energy of gravity associated to extended but finite domains have been proposed. In this paper, the notions of quasi-local energy are studied in the framework of loop quantum gravity, in order to see whether these notions can be carried out at quantum level. Two basic quasi-local geometric quantities are quantized, which lead to well-defined operators in the kinematical Hilbert space of loop quantum gravity. We then use them as basic building blocks to construct different versions of quasi-local energy operators. The operators corresponding to Brown-York energy, Liu-Yau energy, Hawking energy, and Geroch energy are obtained respectively. The virtue of the Geroch energy operator is beneficial for us to derive a rather general entropy-area relation and thus a holographic principle from loop quantum gravity.