Optimal paths across potentials on scalar field space

Motivated by the Swampland Distance Conjecture, we study distances in field space using the framework of Optimal Transport. The associated optimisation problem naturally leads to a notion of distance in terms of a (generalised) Wasserstein distance between probability distributions over field space. In the absence of dynamical gravity, we relate the transport problem to Hamilton-Jacobi and continuity equations arising from a WKB expansion of a Schr\"odinger equation associated with the physical configuration. We then formulate an extension in the presence of dynamical gravity. Using the ADM formalism, we establish the corresponding transport problem through the Wheeler-DeWitt equation, giving rise to different possible choices of cost functions. The resulting notions of distances are naturally defined on the full configuration space, while an interpretation in terms of a genuine scalar field distance requires additional modifications. We further discuss several applications and examples, and indicate possible implications for different themes within the Swampland program.