  We show that a theory of complex scattering between many-body (Fock) states
can be constructed such that its classical limit is a canonical transformation
thus encoding quantum interference in the semiclassical form of the associated
unitary operator. Based on this idea, we study the different coherent effects
expected under different choices of the many-body states and provide different
representations of the associated transition probabilities. In this way, we
derive exact relations and representations of the scattering process that can
be used to attack timely problems related with Boson Sampling.
