Quantum Geometry I : Basics of Loop Quantum Gravity. The Quantum Polyhedra

TL;DR

This paper introduces the fundamental concepts of quantum geometry within loop quantum gravity, focusing on quantum polyhedra, aimed at bridging the gap between mathematics and quantum gravity research.

Contribution

It provides a basic, mathematically oriented introduction to quantum geometry and quantum polyhedra in the context of loop quantum gravity for graduate students.

Findings

Introduction of quantum polyhedra as fundamental geometric entities

Clarification of the mathematical structure of quantum geometry

Bridging the gap between mathematics and quantum gravity concepts

Abstract

General Relativity describes gravity in geometrical terms. This suggests that quantizing such theory is the same as quantizing geometry. The subject can therefore be called quantum geometry and one may think that mathematicians are responsible of this subject. Unfortunately, most mathematicians are not aware of this beautiful area of study. Here we give a basic introduction to what quantum geometry means to a community working in a theory known as loop quantum gravity. It is directed towards graduate or upper students of physics and mathematics. We do it from a point of view of a mathematician.