On ambiguities of sign determination of the S-matrix from energy levels in a finite box

TL;DR

This paper investigates the ambiguity in determining the sign of the S-matrix from finite-volume energy levels, confirming the ambiguity exists but does not hinder extracting physical observables.

Contribution

It provides a straightforward derivation confirming the sign ambiguity and argues that this does not restrict the determination of physical observables from finite-volume spectra.

Findings

Sign ambiguity in off-diagonal S-matrix elements confirmed

Ambiguity does not prevent extracting physical observables

Simplified derivation supports previous findings

Abstract

In a recent paper the authors make a study on the determination of the S-matrix elements for scattering of particles in the infinite volume from the energy levels in the finite box for the case of multiple channels. The study is done with a toy model in 1+1 dimension and the authors find that there is some ambiguity in the sign of nondiagonal matrix elements, casting doubts on whether the needed observables in the infinite volume can be obtained from the energy levels of the box. In this paper I present an easy derivation, confirming the ambiguity of the sign and argue that this, however, does not put restrictions in the determination of observables.