Quantum dynamics of non-relativistic particles and isometric embeddings

TL;DR

This paper investigates the quantum behavior of particles constrained to curved manifolds embedded in higher-dimensional spaces, revealing that existing constrained systems theory does not resolve ambiguities in quantization methods.

Contribution

It generalizes previous work by considering arbitrary codimension embeddings and clarifies that constrained systems theory does not eliminate quantization ambiguities.

Findings

Constrained systems theory does not resolve quantization ambiguities.

Generalization to arbitrary codimension embeddings.

Discussion of discrepancies with recent claims.

Abstract

It is considered, in the framework of constrained systems, the quantum dynamics of non-relativistic particles moving on a d-dimensional Riemannian manifold M isometrically embedded in $R^{d+n}$. This generalizes recent investigations where M has been assumed to be a hypersurface of $R^{d+1}$. We show, contrary to recent claims, that constrained systems theory does not contribute to the elimination of the ambiguities present in the canonical and path integral formulations of the problem. These discrepancies with recent works are discussed.