Geodesic distance: A descriptor of geometry and correlator of pre-geometric density of spacetime events

TL;DR

This paper explores how geodesic distance can serve as an emergent quantum descriptor of spacetime geometry, linking it to a correlator of a pregeometric variable representing quantum spacetime density.

Contribution

It introduces a novel perspective where geodesic distance emerges as a correlator of a pregeometric variable, connecting quantum spacetime density with geometric descriptions.

Findings

Geodesic distance can be expressed as a correlator of a pregeometric variable.

Incorporating zero-point-length modifies the geodesic distance to include quantum effects.

Null surfaces are significant in the quantum-gravity interface.

Abstract

Classical geometry can be described either in terms of a metric tensor $g_{ab}(x)$ or in terms of the geodesic distance $\sigma^2(x,x')$. Recent work, however, has shown that the geodesic distance is better suited to describe the quantum structure of spacetime. This is because one can incorporate some of the key quantum effects by replacing $\sigma^2$ by another function $S[\sigma^2]$ such that $S[0]=L_0^2$ is non-zero. This allows one to introduce a zero-point-length in the spacetime. I show that the geodesic distance can be an emergent construct, arising in the form of a correlator $S[\sigma^2(x,y)]=\langle J(x)J(y)\rangle$, of a pregeometric variable $J(x)$, which, in turn, can be interpreted as the quantum density of spacetime events. This approach also shows why null surfaces play a special role in the interface of quantum theory and gravity. I describe several technical and conceptual aspects of this construction and discuss some of its implications.