Constrained Dynamics for Quantum Mechanics I. Restricting a Particle to a Surface

TL;DR

This paper develops a method for constraining quantum particles to curved surfaces, ensuring correct Hamiltonian dynamics, and aims to generalize the approach for broader quantum constraint systems.

Contribution

It introduces a new constraining method that adjusts quantum dynamics to match the correct Hamiltonian on submanifolds, advancing towards a Dirac algorithm for quantum constraints.

Findings

The method produces the correct Hamiltonian on curved surfaces.

It successfully constrains quantum particles in known examples.

The approach generalizes to selfadjoint operators with additional structures.

Abstract

We analyze constrained quantum systems where the dynamics do not preserve the constraints. This is done in particular for the restriction of a quantum particle in Euclidean n-space to a curved submanifold, and we propose a method of constraining and dynamics adjustment which produces the right Hamiltonian on the submanifold when tested on known examples. This method we hope will become the germ of a full Dirac algorithm for quantum constraints. We take a first step in generalising it to the situation where the constraint is a general selfadjoint operator with some additional structures.