Can the canonical quantization be accomplished within the intrinsic geometry?

TL;DR

This paper investigates whether intrinsic geometry allows for proper canonical quantization of particles constrained on curved surfaces, finding it feasible for catenoids but not for helicoids, and highlighting the necessity of embedding space for consistency.

Contribution

It demonstrates the limitations of intrinsic geometry for canonical quantization on certain minimal surfaces and emphasizes the role of embedding space in quantum theory formulation.

Findings

Canonical quantization is consistent on catenoids within intrinsic geometry.

Intrinsic geometry fails to provide a self-consistent quantum framework for helicoids.

Embedding space is essential for aligning geometric momentum and potential with Schrödinger theory.

Abstract

For particles constrained on a curved surface, how to perform quantization within Dirac's canonical quantization scheme is a long-standing problem. On one hand, Dirac stressed that the Cartesian coordinate system has fundamental importance in passing from the classical Hamiltonian to its quantum mechanical form while preserving the classical algebraic structure between positions, momenta and Hamiltonian to the extent possible. On the other, on the curved surface, we have no exact Cartesian coordinate system within intrinsic geometry. These two facts imply that the three-dimensional Euclidean space in which the curved surface is embedded must be invoked otherwise no proper canonical quantization is attainable. Since the minimum surfaces, catenoid and helicoid studied in this paper, have vanishing mean curvature, we explore whether the intrinsic geometry offers a proper framework in which the quantum theory can be established in a self-consistent way. Results show that it does for quantum motions on catenoid and it does not for that on helicoid, but neither is compatible with Schr\"odinger theory. In contrast, in three-dimensional Euclidean space, the geometric momentum and potential are then in agreement with those given by the Schr\"odinger theory.