Boundaries without boundaries

TL;DR

This paper explores how boundaries in quantum systems can emerge from boundaryless manifolds through group actions and folding procedures, leading to various boundary conditions and potential non-local effects.

Contribution

It introduces methods to generate quantum boundaries from boundaryless spaces using group quotients and folding, including the emergence of different boundary conditions and ancillary degrees of freedom.

Findings

Boundaries can be generated via group actions with fixed points.

Different boundary conditions (Neumann, Dirichlet, Robin) can be realized.

A folding procedure with ancillary spins offers an alternative boundary generation method.

Abstract

Starting with a quantum particle on a closed manifold without boundary, we consider the process of generating boundaries by modding out by a group action with fixed points, and we study the emergent quantum dynamics on the quotient manifold. As an illustrative example, we consider a free nonrelativistic quantum particle on the circle and generate the interval via parity reduction. A free particle with Neumann and Dirichlet boundary conditions on the interval is obtained, and, by changing the metric near the boundary, Robin boundary conditions can also be accommodated. We also indicate a possible method of generating non-local boundary conditions. Then, we explore an alternative generation mechanism which makes use of a folding procedure and is applicable to a generic Hamiltonian through the emergence of an ancillary spin degree of freedom.