Small-scale structure of spacetime and its implications

If there exists a lower bound l0 to spacetime intervals which is Lorentz-invariant, then the effective description of spacetime that incorporates such a lower bound must necessarily be nonlocal. Such a nonlocal description can be derived using standard tools of differential geometry, but using as basic variables certain bi-tensors instead of the conventional metric tensor gab(x). This allows one to construct a qmetric qab(x; y), using the Synge's world function ?(x,y) and the van Vleck determinant ?(x,y), that incorporates the lower bound on spacetime intervals. The same nonanalytic structure of the reconstructed spacetime which renders a perturbative expansion in l0 meaningless, will then also generically leave a non-trivial "relic" in the limit l0 ? 0. We present specific results derived from qab(x; y) where such a relic term manifests, and discuss several implications of the same. Specifically, we will discuss how these results: (i) suggest a description of gravitational dynamics different from the conventional one based on the Einstein-Hilbert Lagrangian, (ii) imply a dimensional reduction to 2 at small scales and (iii) can be significant for the idea that the cosmological constant itself might be related to some nonlocal vestige of the small-scale structure of spacetime. We will conclude by discussing the ramifications of these ideas in the context of quantum gravity.