Towards Spectral Geometry for Causal Sets

TL;DR

This paper demonstrates that spectral properties of the Feynman propagator and related operators encode comprehensive geometric information of causal sets, offering a basis-independent and gauge-invariant approach to measure their geometric similarity.

Contribution

It introduces spectral methods to extract and compare geometric information from causal sets using the Feynman propagator and d'Alembertian spectra, advancing causal set geometry analysis.

Findings

Spectral data contains complete causal set geometry information.

Spectral distance correlates with geometric similarity between causal sets.

Spectral invariants are basis and gauge invariant.

Abstract

We show that the Feynman propagator (or the d'Alembertian) of a causal set contains the complete information about the causal set. Intuitively, this is because the Feynman propagator, being a correlator that decays with distance, provides a measure for the invariant distance between pairs of events. Further, we show that even the spectra alone (of the self-adjoint and anti-self-adjoint parts) of the propagator(s) and d'Alembertian already carry large amounts of geometric information about their causal set. This geometric information is basis independent and also gauge invariant in the sense that it is relabeling invariant (which is analogue to diffeomorphism invariance). We provide numerical evidence that the associated spectral distance between causal sets can serve as a measure for the geometric similarity between causal sets.