  We formulate a quantum waveguide theory of the Josephson effect in multiband
superconductors, with special emphasis on iron-based materials. By generalizing
the boundary conditions of the scattering problem, we first determine the
Andreev levels spectrum and then derive an explicit expression for the
Josephson current which generalizes the formula of the single band case. In
deriving the results, we provide a second quantization field theory allowing to
evaluate the current-phase relation and the Josephson current fluctuations in
multiband systems. We present results for two different order parameter
symmetries,namely $s_\pm$ and $s_{++}$, which are relevant in multiband
systems. The obtained results show that the $s_\pm$ symmetry can support $\pi$
states which are absent in the $s_{++}$ case. We also argue a certain fragility
of the Josephson current against phase fluctuations in the $s_{++}$ case. The
temperature dependence of the Josephson critical current is also analyzed and
we find, for both the order parameter symmetries, remarkable violations of the
Ambegaokar-Baratoff relation. The results are relevant in view of possible
experiments aimed at investigating the order parameter symmetry of multiband
superconductors using mesoscopic Josephson junctions.
