A measure for quantum paths, gravity and spacetime microstructure
The number of classical paths of a given length, connecting any two events in a (pseudo) Riemannian spacetime is, of course, infinite. It is, however, possible to define a useful, finite, measure [Formula: see text] for the effective number of quantum paths [of length [Formula: see text] connecting two events [Formula: see text]] in an arbitrary spacetime. When [Formula: see text], this reduces to [Formula: see text] giving the measure for closed quantum loops of length [Formula: see text] containing an event [Formula: see text]. Both [Formula: see text] and [Formula: see text] are well-defined and depend only on the geometry of the spacetime. Various other physical quantities like, for e.g. the effective Lagrangian, can be expressed in terms of [Formula: see text]. The corresponding measure for the total path length contributed by the closed loops, in a spacetime region [Formula: see text], is given by the integral of [Formula: see text] over [Formula: see text]. Remarkably enough [Formula: see text], the Ricci scalar; i.e. the measure for the total length contributed by infinitesimal closed loops in a region of spacetime gives us the Einstein–Hilbert action. Its variation, when we vary the metric, can provide a new route towards induced/emergent gravity descriptions. In the presence of a background electromagnetic field, the corresponding expressions for [Formula: see text] and [Formula: see text] can be related to the holonomies of the field. The measure [Formula: see text] can also be used to evaluate a wide class of path integrals for which the action and the measure are arbitrary functions of the path length. As an example, I compute a modified path integral which incorporates the zero-point-length in the spacetime. I also describe several other properties of [Formula: see text] and outline a few simple applications.