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

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.