Fourier Transform of Scattering Amplitude and Channel Transfer Model

Abstract

The full scattering amplitude is transformed into the explicit Fourier transform. The inverse· transform involves also a Fourier·type integral form. The properties of the new amplitude closely resemble those of the partial wave scattering amplitude, and they also give rise to some physical quantities for the elastic and inelastic scattering cross sections over the entire range of energy. The Fourier·type scattering amplitude is extended to a complex integral, so that the full scattering amplitude is generated from poles of the new scattering amplitude. As an application, a response function is introduced between the s· and f·channel of the 7[+7[- scattering, and the differential cross sections are estimated at the energy region of the p·resonance. It is also shown that the response function is applicable to the evaluation of the p·exchange contribution in the 7[+7[­ scattering as well as in the 7[-P charge exchange scattering. The Fourier transform has contributed to the advancement of science and engi­ neering due to the application~ of its excellent formalism. This expression is formulated in terms of a newly designed variable different from the primary one. It is most applicable to investigate what behaviors give rise to a physical system with respect to the secondary parameter, and what method makes mathematical calcula­ tions easy. The former is seen in a study on the transfer function of an impulse response in the communication system, and the latter is observed in a mathematical calculation on the oscillating system and the electric circuit. The Fourier transform also is very important for the broad application of particle physics. In the scattering theory of particles, the scattering amplitude is divided into discrete partial waves by means of the Legendre polynomial. ; Although this expan­ sion holds a quantum mechanical nature, it has disturbed us because of its discrete property in some practical evaluations. In the high energy scattering, the scattering amplitude has often been expressed in a continuous integral form called impact parameter representation. It may be looked upon as the Laplace-type transform of a scattering amplitude, if we compare it with the formulation described in this article.I)-3) The other continuous spectrum representation was briefly formulated by the present author sixteen years ago. 4