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We construct a bosonic quantum field on a general quantum graph. Consistencyof the construction leads to the calculation of the total scattering matrix ofthe graph. This matrix is equivalent to the one already proposed usinggeneralized star product approach. We give several examples and show how theygeneralize some of the scattering matrices computed in the mathematical orcondensed matter physics litterature.  Then, we apply the construction for the calculation of the conductance ofgraphs, within a small distance approximation. The consistency of theapproximation is proved by direct comparison with the exact calculation for the`tadpole' graph.

Quantum field theory on quantum graphs and application to their conductance