Path integral on star graph Original Research Article

In this paper we study path integral for a single spinless particle on a star graph with image edges, whose vertex is known to be described by image 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 image-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 image. We show that, when bulk interactions are edge-independent, these weight factors are generally given by an image-dimensional unitary representation of image. Generalization to momentum dependent weight factors and applications to worldline formalism are briefly discussed.