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The experimental progress in synthesizing low-dimensional nanostructureswhere carriers are confined to bent surfaces has boosted the interest in thetheory of quantum mechanics on curved two-dimensional manifolds. It wasrecently asserted that constrained electrically charged particles couple to aterm linear in A_3 M, where A_3 is the transversal component of theelectromagnetic vector potential and M the surface mean curvature, therebymaking a dimensional reduction procedure impracticable in the presence offields. Here we resolve this apparent paradox by providing a consistent generalframework of the thin-wall quantization procedure. We also show that theseparability of the equation of motions is not endangered by the particularchoice of the constraint imposed on the transversal fluctuations of thewavefunction, which renders the thin-wall quantization procedure well-founded.It can be applied without restrictions.

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