Absence of anomalous interactions in the quantum theory of constrained charged particles in presence of electrical currents

TL;DR

This paper clarifies that there are no anomalous interactions in the quantum theory of constrained charged particles on curved surfaces with electromagnetic fields, establishing a consistent framework for thin-wall quantization.

Contribution

It provides a rigorous framework for thin-wall quantization, resolving paradoxes about electromagnetic coupling on curved surfaces.

Findings

No anomalous coupling term linear in A_3 M exists.

Thin-wall quantization is valid for arbitrary constraints.

Separability of equations is unaffected by the choice of constraints.

Abstract

The experimental progress in synthesizing low-dimensional nanostructures where carriers are confined to bent surfaces has boosted the interest in the theory of quantum mechanics on curved two-dimensional manifolds. It was recently asserted that constrained electrically charged particles couple to a term linear in A_3 M, where A_3 is the transversal component of the electromagnetic vector potential and M the surface mean curvature, thereby making a dimensional reduction procedure impracticable in the presence of fields. Here we resolve this apparent paradox by providing a consistent general framework of the thin-wall quantization procedure. We also show that the separability of the equation of motions is not endangered by the particular choice of the constraint imposed on the transversal fluctuations of the wavefunction, which renders the thin-wall quantization procedure well-founded. It can be applied without restrictions.