Quantum metric for null separated events and spacetime atoms

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 $\rho$, related to their density, function of the point $P$ and vector $v^a$ at $P$. This has been done in the Euclideanized space one obtains through analytic continuation from Lorentzian sector at $P$. $\rho$ has been defined in terms of a peculiar `effective' metric $q_{ab}$, of quantum origin, introduced for spacelike/timelike separated events. This metric stems from requiring that $q_{ab}$ coincides with $g_{ab}$ 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 a quite common feature in the available quantum descriptions of gravity, this quantum metric $q_{ab}$ 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 $\rho$ 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 $v^a$ is null, as in this case it seems we lack even a concept of $q_{ab}$. A notion for the quantum metric $q_{ab}$ for null separated events is then proposed and an expression for it is derived. From it, a formula for $\rho$ is deduced, which turns out to coincide with what obtained through analytic continuation. This virtually completes the task of having quantum expressions of any kind of spacetime intervals, with, moreover, $\rho$ defined directly in terms of them (not in the Euclideanized space).