A Curvature Operator for a Regular Tetrahedron Shape in LQG

TL;DR

This paper introduces a new quantum operator for scalar curvature in Loop Quantum Gravity, specifically for regular tetrahedron shapes, revealing negative curvature states and semi-classical limits near Euclidean geometry.

Contribution

It proposes a novel 3D Ricci scalar curvature operator expressed via volume, area, and edge length operators, applied to regular tetrahedra in LQG.

Findings

Regular tetrahedron states exhibit negative scalar curvature.

Semi-classical limit spectrum approaches Euclidean regime.

New curvature operator links geometry with quantum states.

Abstract

An alternative approach introducing a 3 dimensional Ricci scalar curvature quantum operator given in terms of volume and area as well as new edge length operators is proposed. An example of monochromatic 4-valent node intertwiner state (equilateral tetrahedra) is studied and the scalar curvature measure for a regular tetrahedron shape is constructed. It is shown that all regular tetrahedron states are in the negative scalar curvature regime and for the semi-classical limit the spectrum is very close to the Euclidean regime.