We investigate the propagation of a massless scalar field on a star graph, modeling the junction of $n$ quantum wires. The vertex of the graph is represented by a point-like impurity (defect), characterized by a one-body scattering matrix. The general case of off-critical scattering matrix with bound and/or antibound states is considered. We demonstrate that the contribution of these states to the scalar field is fixed by causality (local commutativity), which is the key point of our investigation. Two different regimes of the theory emerge at this stage. If bound sates are absent, the energy is conserved and the theory admits unitary time evolution. The behavior changes if bound states are present, because each such state generates a kind of damped harmonic oscillator in the spectrum of the field. These oscillators lead to the breakdown of time translation invariance. We study in both regimes the electromagnetic conductance of the Luttinger liquid on the quantum wire junction. We derive an explicit expression for the conductance in terms of the scattering matrix and show that antibound and bound states have a different impact, giving raise to oscillations with exponentially damped and growing amplitudes respectively.
Download