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 $\mathcal{V}$, is given by the integral of $L(\sigma;x) \equiv\sigma C(\sigma;x)$ over $\mathcal{V}$. Remarkably enough $L(0;x) \propto R(x)$, 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.