Coalescence constraints of many-body systems in one dimension

TL;DR

This paper derives the coalescence constraints for one-dimensional many-body quantum systems with Coulomb and delta interactions, revealing that particles cannot coalesce or cross, and are inherently distinguishable regardless of their statistics.

Contribution

It provides rigorous coalescence conditions for particles in 1D Coulomb systems and extends the analysis to short-range delta interactions, clarifying particle distinguishability and ordering.

Findings

Particles cannot coalesce or cross in Coulomb systems.

Particles are inherently distinguishable regardless of charge or statistics.

Noninteracting systems cannot serve as perturbative bases for Coulomb interactions.

Abstract

For one-dimensional many-body systems interacting via the \textit{Coulomb force} and with \textit{arbitrary} external potential energy, we derive (\textit{i}) the \textit{node coalescence condition} for the wave function. This condition rigorously proves the following: (\textit{ii}) that the particles satisfy \textit{only} a node coalescence condition; (\textit{iii}) that irrespective of their charge or statistics, the particles cannot coalesce; (\textit{iv}) that the particles cannot cross each other, and must be ordered; (\textit{v}) the particles are therefore distinguishable; (\textit{vi}) as such their statistics are not significant; (\textit{vii}) conclusions similar to those of the spin-statistics theorem of quantum field theory are arrived at via non-relativistic quantum mechanics; (\textit{viii}) the noninteracting system \textit{cannot} be employed as the lowest-order in a perturbation theory of the interacting system. (\textit{ix}) Finally, the coalescence condition for particles with the short-ranged delta-function interaction and \textit{arbitrary} external potential energy, is also derived. These particles can coalesce and cross each other.