Probing the Shape of Quantum Surfaces: the Quadrupole Moment Operator

TL;DR

This paper introduces quadrupole moment operators to probe the shape of quantum surfaces in loop quantum gravity, providing a new tool to analyze shape fluctuations and the semi-classical regime of quantum geometry.

Contribution

It defines dual multipole moments for quantum surfaces and demonstrates the quadrupole moment as the Hessian of the large spin approximation, enhancing shape analysis in loop quantum gravity.

Findings

Quadrupole moments probe the shape of quantum surfaces.

Quadrupole appears as the Hessian in semi-classical approximations.

Improves modeling of gravitational wave-like shape fluctuations.

Abstract

The standard toolkit of operators to probe quanta of geometry in loop quantum gravity consists in area and volume operators as well as holonomy operators. New operators have been defined, in the U(N) framework for intertwiners, which allow to explore the finer structure of quanta of geometry. However these operators do not carry information on the global shape of the intertwiners. Here we introduce dual multipole moments for continuous and discrete surfaces, defined through the normal vector to the surface, taking special care to maintain parametrization invariance. These are raised to multipole operators probing the shape of quantum surfaces. Further focusing on the quadrupole moment, we show that it appears as the Hessian matrix of the large spin Gaussian approximation of coherent intertwiners, which is the standard method for extracting the semi-classical regime of spinfoam transition amplitudes. This offers an improvement on the usual loop quantum gravity techniques, which mostly focus on the volume operator, in the perspective of modeling (quantum) gravitational waves as shape fluctuations waves propagating on spin network states.