An artificial boundary approach for short-ranged interactions

Abstract

Real physical systems are only understood, experimentally or theoretically, to a finite resolution so in their analysis there is generally an ignorance of possible short-range phenomena. It is also well-known that the boundary conditions of wavefunctions and fields can be used to model short-range interactions when those interactions are expected, a priori. Here, a real-space approach is described wherein an artificial boundary of ignorance is imposed to explicitly exclude from analysis the region of a system wherein short-distance effects may be obscure. The (artificial) boundary conditions encode those short-distance effects by parameterizing the possible UV completions of the wavefunction. Since measurable quantities, such as spectra and cross sections, must be independent of the position of the artificial boundary, the boundary conditions must evolve with the radius of the boundary in a particular way. As examples of this approach, an analysis is performed of the non-relativistic free particle, harmonic oscillator, and Coulomb potential, and some known results for point-like (contact) interactions are recovered, however from a novel perspective. Generically, observables differ from their canonical values and symmetries are anomalously broken compared to those of idealized models. Connections are made to well-studied physical systems, such as the binding of light nuclei and cold atomic systems. This method is arguably more physically transparent and mathematically easier to use than other techniques that require the regularization and renormalization of delta-function potentials, and may offer further generalizations of practical use.