# Quantum Smooth Boundary Forces from Constrained Geometries

## TL;DR

This paper applies a covariant integral quantization method to constrained geometries, resulting in quantum models with fuzzy boundaries, position-dependent mass, and additional potentials, and analyzes their semi-classical behavior.

## Contribution

It introduces a boundary-aware quantization approach that avoids operator ordering issues and provides a semi-classical analysis of constrained quantum systems.

## Key findings

- Derivation of a quantum model with fuzzy boundaries and PDM
- Regularization of classical dynamics through quantization
- Analysis of semi-classical phase space portraits

## Abstract

We implement the so-called Weyl-Heisenberg covariant integral quantization in the case of a classical system constrained by a bounded or semi-bounded geometry. The procedure, which is free of the ordering problem of operators, is illustrated with the basic example of the one-dimensional motion of a free particle in an interval, and yields a fuzzy boundary, a position-dependent mass (PDM), and an extra potential on the quantum level. The consistency of our quantization is discussed by analyzing the semi-classical phase space portrait of the derived quantum dynamics, which is obtained as a regularization of its original classical counterpart.