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

Classical geometry can be described either in terms of a metric tensor $g_{ab}(x)$ or in terms of the geodesic distance ${\sigma }^2(x,x^{\mathrm{\prime }})$. 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 $S[{\sigma }^2(x,y)]=⟨J(x)J(y)⟩$, 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.