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.
Highlights
Quantum-mechanical potentials are a low energy nonrelativistic description of a scattering process
This underlying correspondence between quantum field theory (QFT) and quantum mechanics naturally leads to the following criteria for determining which potentials can be consistently UV completed into a QFT: The space of consistent, nonsingular quantum-mechanical potentials consists of those arising from a well-defined QFT scattering process
In addition to setting the boundary conditions for the Schrödinger equation, this procedure provided us with an analytic cross-check for determining whether a potential is singular or not
Summary
Quantum-mechanical potentials are a low energy nonrelativistic description of a scattering process. In [5], we showed that at short distances, there was a match between the tree-level relativistic field theory description and the first Born approximation using the corresponding quantum-mechanical potential This underlying correspondence between QFT and quantum mechanics naturally leads to the following criteria for determining which potentials can be consistently UV completed into a QFT: The space of consistent, nonsingular quantum-mechanical potentials consists of those arising from a well-defined QFT scattering process. We showed that the operator accompanying the potential was critical in reproducing the physics, and it was incorrect to approximate it as a simple 1=r3 central potential With such a wide range of applications, isolating the space of consistent, nonsingular potentials becomes crucial for ensuring that the empirical description of a low-energy phenomenon can be consistently completed into a QFT.
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