Generalized S-Matrix in Mixed Representation

A generalized scattering amplitude where momenta of incoming-particles andoutgoing-particles as well as positions of incoming-particles andoutgoing-particles are specified is formulated. Idealistic beams and idealisticmeasuring instruments where momenta and positions satisfy minimum uncertaintyare studied with a use of minimum wave packets, coherent states. In the presentwork, we show general features of the generalized scattering amplitudes basedon ${\phi}^4$ theory. We give a proof of completeness of many body states,asymptotic behaviors in the large distance region, and factorization of theamplitudes. Despite of the non-orthogonal properties of wave packets, we foundthat the probability interpretation is verified. A differential probabilitydepends upon the wave packet size but a total probability that is integrated inthe final states is independent from the size of final state wave packet andbecomes universal. Few body amplitudes are studied as examples.Comment: 33 pages, 1 figur