The Quantum Mechanics Swampland

TL;DR

This paper explores the boundary between quantum mechanical potentials that can be embedded in a consistent quantum field theory (QFT) and those that cannot, proposing a 'Swampland' classification based on their singularity structure and UV completion potential.

Contribution

It introduces the concept of the Quantum Mechanics Swampland, classifying potentials based on their ability to be UV completed into a QFT, and provides criteria to distinguish between them.

Findings

Potentials generated by various operators are shown to be nonsingular despite singular terms.

A boundary condition matching procedure links quantum mechanics and QFT approximations.

Coulomb potentials remain nonsingular in arbitrary spacetime dimensions.

Abstract

We investigate non-relativistic quantum mechanical potentials between fermions generated by various classes of QFT operators and evaluate their singularity structure. These potentials can be generated either by four-fermion operators or by the exchange of a scalar or vector mediator coupled via renormalizable or non-renormalizable operators. In the non-relativistic regime, solving the Schr\"odinger equation with these potentials provides an accurate description of the scattering process. This procedure requires providing a set of boundary conditions. We first recapitulate the procedure for setting the boundary conditions by matching the first Born approximation in quantum mechanics to the tree-level QFT approximation. Using this procedure, we show that the potentials are nonsingular, despite the presence of terms proportional to $r^{-3}$ and $\nabla_{i}\nabla_{j}\delta^{3}(\vec{r})$. This surprising feature leads us to propose the \emph{Quantum Mechanics Swampland}, in which the Landscape consists of non-relativistic quantum mechanical potentials that can be UV completed to a QFT, and the Swampland consists of pathological potentials which cannot. We identify preliminary criteria for distinguishing potentials which reside in the Landscape from those that reside in the Swampland. We also consider extensions to potentials in higher dimensions and find that Coulomb potentials are nonsingular in an arbitrary number of spacetime dimensions.