Symmetry Under $\alpha \rightarrow \alpha + 1$ is Forbidden by Helicity Conservation

TL;DR

This paper investigates whether helicity conservation alone can determine the wave function form in spin-half Aharonov-Bohm scattering, revealing the existence of singular solutions that break a key symmetry and challenge anyon interpretation.

Contribution

It demonstrates that helicity conservation permits a family of solutions but cannot exclude singular solutions that violate the symmetry $oldsymbol{ ext{α} ightarrow ext{α} + 1}$, impacting the understanding of anyon models.

Findings

Helicity conservation allows a one-parameter family of solutions.

Singular solutions breaking the symmetry exist despite conservation laws.

The symmetry breaking affects the interpretation of anyons in this context.

Abstract

The question as to whether helicity conservation in spin one-half Aharonov Bohm scattering is sufficient in itself to determine uniquely the form of the spinor wave function near the origin is examined. Although it is found that a one parameter family of solutions is compatible with this conservation law, there must nonetheless be singular solutions which break the symmetry $\alpha \rightarrow \alpha +1$ required for an anyon interpretation. Thus the free parameter which occurs does not allow one to eliminate the singular solutions even though it does in principle mean that they can be transferred at will between the spin up and spin down configurations.