Fields and Laplacians on Quantum Geometries

TL;DR

This paper develops a formalism for discrete calculus on quantum geometries, enabling analysis of spectral properties like the spectral dimension in quantum gravity models.

Contribution

It introduces a bra-ket formalism for functions on discrete geometries and applies it to quantum gravity, focusing on the quantum geometric Laplacian and spectral analysis.

Findings

Defined a bra-ket formalism for discrete geometries

Analyzed the quantum geometric Laplacian and heat kernel trace

Provided tools for spectral dimension computation in quantum gravity

Abstract

In fundamentally discrete approaches to quantum gravity such as loop quantum gravity, spin-foam models, group field theories or Regge calculus observables are functions on discrete geometries. We present a bra-ket formalism of function spaces and discrete calculus on abstract simplicial complexes equipped with geometry and apply it to the mentioned theories of quantum gravity. In particular we focus on the quantum geometric Laplacian and discuss as an example the expectation value of the heat kernel trace from which the spectral dimension follows.