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 N(x(2), x(1); sigma) for the effective number of quantum paths [of length sigma connecting two events (x(1), x(2))] in an arbitrary spacetime. When x(2) = x(1), this reduces to C(x, sigma) giving the measure for closed quantum loops of length sigma containing an event x. Both N(x(2), x(1); sigma) and C(x, sigma) 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 N(x(2), x(1); sigma). The corresponding measure for the total path length contributed by the closed loops, in a spacetime region V, is given by the integral of L(sigma; x) sigma C (sigma; x) over V. Remarkably enough L(0; x) proportional to R(x), 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 N(x(2), x(1); sigma) and C(x, sigma) can be related to the holonomies of the field. The measure N(x(2), x(1); sigma) 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 N(x(2), x(1); sigma) and outline a few simple applications.