Lorentzian Spectral Geometry with Causal Sets

TL;DR

This paper investigates how spectral invariants can identify and classify causal sets in Lorentzian spectral geometry, introducing new methods that improve discrimination and have potential applications in quantum gravity.

Contribution

It introduces a novel spectral geometric method involving perturbations to better distinguish causal sets, advancing the understanding of discrete Lorentzian spectral geometry.

Findings

Two effective classification methods for causal sets

The new perturbation-based method fully distinguishes tested causal sets

Potential applications to quantum gravity path integrals

Abstract

We study discrete Lorentzian spectral geometry by investigating to what extent causal sets can be identified through a set of geometric invariants such as spectra. We build on previous work where it was shown that the spectra of certain operators derived from the causal matrix possess considerable but not complete power to distinguish causal sets. We find two especially successful methods for classifying causal sets and we computationally test them for all causal sets of up to $9$ elements. One of the spectral geometric methods that we study involves holding a given causal set fixed and collecting a growing set of its geometric invariants such as spectra (including the spectra of the commutator of certain operators). The second method involves obtaining a limited set of geometric invariants for a given causal set while also collecting these geometric invariants for small `perturbations' of the causal set, a novel method that may also be useful in other areas of spectral geometry. We show that with a suitably chosen set of geometric invariants, this new method fully resolves the causal sets we considered. Concretely, we consider for this purpose perturbations of the original causal set that are formed by adding one element and a link. We discuss potential applications to the path integral in quantum gravity.