Path Integral on Star Graph

TL;DR

This paper develops a new path integral formulation for a particle on a star graph, capturing boundary conditions at fixed points of boundary RG flow using a folding trick and group representations.

Contribution

It introduces a novel path integral approach that encodes boundary conditions via weight factors linked to Z_2 group representations, applicable to star graphs.

Findings

Path integral formulation correctly captures scale-invariant boundary conditions.

Weight factors are related to N-dimensional unitary representations of Z_2.

Framework can be extended to momentum-dependent weights and worldline formalism.

Abstract

In this paper we study path integral for a single spinless particle on a star graph with N edges, whose vertex is known to be described by U(N) family of boundary conditions. After carefully studying the free particle case, both at the critical and off-critical levels, we propose a new path integral formulation that correctly captures all the scale-invariant subfamily of boundary conditions realized at fixed points of boundary renormalization group flow. Our proposal is based on the folding trick, which maps a scalar-valued wave function on star graph to an N-component vector-valued wave function on half-line. All the parameters of scale-invariant subfamily of boundary conditions are encoded into the momentum independent weight factors, which appear to be associated with the two distinct path classes on half-line that form the cyclic group Z_2. We show that, when bulk interactions are edge-independent, these weight factors are generally given by an N-dimensional unitary representation of Z_2. Generalization to momentum dependent weight factors and applications to worldline formalism are briefly discussed.