A Primer of Group Theory for Loop Quantum Gravity and Spin-foams

TL;DR

This paper provides a comprehensive, accessible overview of the group theory tools essential for Loop Quantum Gravity and spin-foam models, aiming to unify and clarify the mathematical foundations used in the field.

Contribution

It compiles and standardizes the key group theory concepts and conventions used in LQG and spin-foams, serving as both an educational resource and a reference guide.

Findings

Clarifies the mathematical tools used in LQG and spin-foams

Provides a unified notation and conventions

Serves as a bridge between physics and mathematics

Abstract

Calculations in Loop Quantum Gravity (LQG) and spin-foams theory rely heavily on group theory of SU(2) and SL(2,C). Even though many monographs exist devoted to this theory, the different tools needed (e.g. representation theory, harmonic analysis, recoupling theory...) are often dispersed in different books, with different conventions and notations. This was the initial motivation for the compilation of the present document. Generally, these notes can serve three main purposes: a concise introduction for students to the essential mathematical tools of LQG, a convenient compendium for researchers, a translational hub between the conventions of the main references. These notes are aimed both at physicists, caring about their tools being mathematically well-grounded, and at mathematicians, curious about how some of their familiar abstract structures can reveal the beauty of quantum gravity.