Optimal paths across potentials on scalar field space

TL;DR

This paper introduces a novel approach to measuring distances in scalar field space using Optimal Transport theory, connecting it to quantum and gravitational formalisms, with implications for the Swampland conjectures.

Contribution

It formulates a new framework for defining distances in field space via Wasserstein metrics, incorporating gravity through the Wheeler-DeWitt equation and exploring various applications.

Findings

Established a link between optimal transport and Hamilton-Jacobi equations in field theory.

Extended the transport problem to include dynamical gravity using the Wheeler-DeWitt equation.

Proposed new notions of scalar field distances with potential implications for the Swampland program.

Abstract

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.