Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2011-04-28
Physics
High Energy Physics
High Energy Physics - Theory
19 pages, 15 figures; typos corrected
Scientific paper
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 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.
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