Quantum metric for null separated events and spacetime atoms

Abstract

Recently, a proposal has been made to figure out the expected discrete nature of spacetime at the smallest scales in terms of atoms of spacetime, capturing their effects through a scalar , related to their density, function of the point P and vector at P. This has been done in the Euclideanized space one obtains through analytic continuation from the Lorentzian sector at P. has been defined in terms of a peculiar ‘effective’ metric qab, of quantum origin, introduced for spacelike/timelike separated events. This metric stems from requiring that qab coincides with gab at large (space/time) distances, but gives finite distance in the coincidence limit, and implements directly this way one single, very basic aspect associated to any quantum description of spacetime: length quantization. Since the latter appears to be a quite common feature in the available quantum descriptions of gravity, this quantum metric qab can be suspected to have a rather general scope and to be re-derivable (and cross-checkable) in various specific quantum models of gravity, even markedly different one from the other.This work reports on an attempt to introduce a definition of not through the Euclidean but directly in the Lorentz sector. This turns out to be not a so trivial task, essentially because of the null case, meaning when is null, as in this case it seems we lack even a concept of qab. A notion for the quantum metric qab for null separated events is then proposed and an expression for it is derived. From it, a formula for is deduced, which turns out to coincide with what is obtained through analytic continuation. This virtually completes the task of having quantum expressions of any kind of spacetime intervals, with, moreover, defined directly in terms of them (not in the Euclideanized space).