Generalized S-matrix in Mixed Representations
  A generalized scattering amplitude where momenta of incoming-particles and outgoing-particles as well as positions of incoming-particles and outgoing-particles are specified is formulated. Idealistic beams and idealistic measuring instruments where momenta and positions satisfy minimum uncertainty are studied with a use of minimum wave packets, coherent states. In the present work, we show general features of the generalized scattering amplitudes based on ${\phi}^4$ theory. We give a proof of completeness of many body states, asymptotic behaviors in the large distance region, and factorization of the amplitudes. Despite of the non-orthogonal properties of wave packets, we found that the probability interpretation is verified. A differential probability depends upon the wave packet size but a total probability that is integrated in the final states is independent from the size of final state wave packet and becomes universal. Few body amplitudes are studied as examples. 
Kenzo Ishikawa and Takashi Shimomura