Laplacians on discrete and quantum geometries

TL;DR

This paper generalizes discrete calculus and Laplacian operators to abstract discrete geometries, compares dual constructions, and explores their applications in quantum gravity, including spectral analysis.

Contribution

It introduces a unified framework for defining discrete Laplacians on combinatorial complexes and analyzes duality choices, enhancing tools for quantum geometry research.

Findings

Barycentric dual yields more desirable Laplacian properties.

Derived Laplacian expressions suitable for quantum gravity models.

Explored position-momentum transformation for scalar fields in discrete geometries.

Abstract

We extend discrete calculus for arbitrary ($p$-form) fields on embedded lattices to abstract discrete geometries based on combinatorial complexes. We then provide a general definition of discrete Laplacian using both the primal cellular complex and its combinatorial dual. The precise implementation of geometric volume factors is not unique and, comparing the definition with a circumcentric and a barycentric dual, we argue that the latter is, in general, more appropriate because it induces a Laplacian with more desirable properties. We give the expression of the discrete Laplacian in several different sets of geometric variables, suitable for computations in different quantum gravity formalisms. Furthermore, we investigate the possibility of transforming from position to momentum space for scalar fields, thus setting the stage for the calculation of heat kernel and spectral dimension in discrete quantum geometries.