# Geometric Potential Resulting from Dirac Quantization

## TL;DR

This paper addresses the discrepancy in geometric potential formulations in Dirac quantization of particles on curved surfaces, proposing a scheme that yields the expected potential form by simultaneous quantization of key operators.

## Contribution

It introduces a scheme for Dirac quantization that achieves the correct geometric potential by quantizing positions, momenta, and Hamiltonian simultaneously.

## Key findings

- Resolves the discrepancy in geometric potential forms.
- Identifies an operator-ordering-free quantization scheme.
- Achieves the expected geometric potential form.

## Abstract

A fundamental problem regarding the Dirac quantization of a free particle on an $N-1$ curved hypersurface embedded in $N$($\geq 2$) flat space is the impossibility to give the same form of the curvature-induced quantum potential, the geometric potential as commonly called, as that given by the Schr\"{o}dinger equation method where the particle moves in a region confined by a thin-layer sandwiching the surface. We resolve this problem by means of previously proposed scheme that hypothesizes a simultaneous quantization of positions, momenta, and Hamiltonian, among which the operator-ordering-free section is identified and is then found sufficient to lead to the expected form of geometric potential.