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

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.