Generalized S-Matrix in Mixed Representation

• Published in: Progress of theoretical and experimental physics 2005-12, Vol.114 (6), p.1201-1234
• Main Authors: Ishikawa, Kenzo, Shimomura, Takashi
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Physics
• ISSN: 0033-068X; 2050-3911; 1347-4081
• DOI: 10.1143/PTP.114.1201

We formulate a generalized scattering amplitude in which the momenta and positions of incoming particles and outgoing particles are specified. Ideal beams and ideal measuring instruments, for which the momenta and positions realize the minimum uncertainty, are studied using minimum wave packets, i.e., coherent states. In the present work, we elucidate the general features of the genelarized scattering amplitudes based on φ 4 theory. We prove the completeness of many body states and determine the asymptotic behavior in the large distance region and the factorization of the amplitudes. Despite the non-orthogonal properties of wave packets, we find that the probability interpretation is verified. The differential probability depends on the wave packet size, but the total probability that is integrated over the final states is independent of the size of the final state wave packet and is universal. Few-body amplitudes are studied as examples.