Expectation values of minimum-length Ricci scalar

Abstract

In this paper, we consider a specific model, implementing the existence of a fundamental limit distance [Formula: see text] between (space or time separated) points in spacetime, which in the recent past has exhibited the intriguing feature of having a minimum-length Ricci scalar [Formula: see text] that does not approach the ordinary Ricci scalar [Formula: see text] in the limit of vanishing [Formula: see text]. [Formula: see text] at a point has been found to depend on the direction along which the existence of minimum distance is implemented. Here, we point out that the convergence [Formula: see text] in the [Formula: see text] limit is anyway recovered in a relaxed or generalized sense, which is when we average over directions, this suggesting we might be taking the expectation value of [Formula: see text] promoted to be a quantum variable. It remains as intriguing as before the fact that we cannot identify (meaning this is much more than simply equating in the generalized sense above) [Formula: see text] with [Formula: see text] in the [Formula: see text] limit, namely, when we get ordinary spacetime. Thing is like if, even when [Formula: see text] (read here the Planck length) is far too small to have any direct detection of it feasible, the intrinsic quantum nature of spacetime might anyway be experimentally at reach, witnessed by the mentioned special feature of Ricci, not fading away with [Formula: see text] (i.e. persisting when taking the [Formula: see text] limit).