Path Integral Junctions

TL;DR

This paper develops a path integral framework for quantum systems on compact graphs with N segments, linking boundary conditions to matrix-valued weights represented by dihedral group symmetries.

Contribution

It introduces a novel path integral approach for quantum graphs with boundary conditions characterized by matrix weights related to dihedral group representations.

Findings

Boundary conditions correspond to N×N matrix weights

Weights are given by unitary representations of the infinite dihedral group

Provides a new link between quantum graph boundary conditions and group theory

Abstract

We propose path integral description for quantum mechanical systems on compact graphs consisting of N segments of the same length. Provided the bulk Hamiltonian is segment-independent, scale-invariant boundary conditions given by self-adjoint extension of a Hamiltonian operator turn out to be in one-to-one correspondence with N \times N matrix-valued weight factors on the path integral side. We show that these weight factors are given by N-dimensional unitary representations of the infinite dihedral group.