Running Boundary Condition

TL;DR

This paper explores how boundary conditions in quantum mechanics can evolve with energy scale, analyzing their renormalization group flow and fixed points, with implications for duality and generalizations to other systems.

Contribution

It provides an exact analysis of the RG flow of U(2) boundary conditions in quantum mechanics and discusses their fixed points and dualities.

Findings

Scale-independent boundary conditions are fixed points.

The RG flow of boundary conditions can be exactly determined.

Duality between boundary conditions is explained via RG analysis.

Abstract

In this paper we argue that boundary condition may run with energy scale. As an illustrative example, we consider one-dimensional quantum mechanics for a spinless particle that freely propagates in the bulk yet interacts only at the origin. In this setting we find the renormalization group flow of U(2) family of boundary conditions exactly. We show that the well-known scale-independent subfamily of boundary conditions are realized as fixed points. We also discuss the duality between two distinct boundary conditions from the renormalization group point of view. Generalizations to conformal mechanics and quantum graph are also discussed.