Quantum gravity: unification of principles and interactions, and promises of spectral geometry

TL;DR

This paper explores the potential of spectral geometry in unifying quantum mechanics and general relativity, offering mathematical insights into quantum gravity and discussing its prospects for becoming a testable physical theory.

Contribution

It reviews spectral geometry's role in quantum gravity, challenging the view that mathematical complexity hinders unification efforts.

Findings

Spectral geometry provides valuable mathematical tools for quantum gravity.

Unification of fundamental interactions is advancing through spectral methods.

Mathematical complexity may not prevent the development of falsifiable theories.

Abstract

Quantum gravity was born as that branch of modern theoretical physics that tries to unify its guiding principles, i.e., quantum mechanics and general relativity. Nowadays it is providing new insight into the unification of all fundamental interactions, while giving rise to new developments in modern mathematics. It is however unclear whether it will ever become a falsifiable physical theory, since it deals with Planck-scale physics. Reviewing a wide range of spectral geometry from index theory to spectral triples, we hope to dismiss the general opinion that the mere mathematical complexity of the unification programme will obstruct that programme.