I consider a special atomic scattering problem where the target atom has distinct “soft” and “hard” excitation modes. I demonstrate that in this problem the integration out of “short-distance” (or “high-energy”) physics may occur automatically in the regular perturbative calculations, i.e., it may occur without any cut off and renormalization. Not only that, the soft inelastic processes happen already in the first Born approximation and the inclusive cross-sections become unavoidable from the very beginning. All that is possible because of correct physical and mathematical formulation of the problem. I propose to build QFT in a similar way.
P.S. I updated the article, mainly with writing an Appendix to address the mixed variables and the Hamiltonian splitting.
P.P.S. Now it's been published online: http://www.aiscience.org/journal/paperInfo/pj?paperId=2156