Quantum field theory on quantum graphs and application to their conductance

We construct a bosonic quantum field on a general quantum graph. Consistency of the construction leads to the calculation of the total scattering matrix of the graph. This matrix is equivalent to the one already proposed using generalized star product approach. We give several examples and show how they generalize some of the scattering matrices computed in the mathematical or condensed matter physics litterature. Then, we apply the construction for the calculation of the conductance of graphs, within a small distance approximation. The consistency of the approximation is proved by direct comparison with the exact calculation for the `tadpole’ graph.