One‐dimensional collisions between two isotropic solids, in which the equations of physics lead to ‘‘multiplications of distributions,’’ are considered. It can inherit from the algebraic structure of ℬ some good properties with respect to nonlinear operations. The microlocal asymptotic analysis is a new spectral study of singularities. The frequential microlocal analysis based on the Fourier transform leads to the study of propagation of singularities under only linear (including pseudodifferential) operators in the theories described here, but has been extended to some non linear cases in classical theories involving Sobolev techniques. We show that all previously considered cases of local analysis of generalized functions (defined from duality or algebraic techniques) can be interpretated as the ℬ-local analysis of sections ofīut the microlocal analysis of the sections of sheaves or presheaves under consideration is dissociated into a “frequential microlocal analysis” and into a “microlocal asymptotic analysis”. We introduce a general context involving a presheaf It is an attempt to use nonlinear generalized functions in QFT, starting directly from the calculations made by physicists, in the same way as they have already been used in classical mechanics and general relativity. The aim of this paper is to present, on the grounds of a standard mathematical model of QFT (a self interacting scalar boson field), a basis for improvement without significant prerequisites in mathematics and physics. (i) Domains of the interacting field operators: a priori the H-P calculations give time dependent dense domains, what is not very convenient (ii) Calculations of the resulting matrix elements of the S operator: from the unitarity of the S operator as a whole there are no longer ``infinities,'' but a priori there is no other hope than heavy computer calculations (iii) Connection with renormalization theory: it should provide an approximation when the coupling constant is small. By providing an appropriate mathematical setting, nonlinear generalized functions open doors for their understanding but there remains presumably very hard technical problems. We show how the unmodified Heisenberg-Pauli calculations make sense mathematically by using a theory of generalized functions adapted to nonlinear operations. This general theory of quantized fields has remained undisputed up to now.
In 1929 Heisenberg and Pauli laid the foundations of QFT by quantizing the fields (method of canonical quantization).
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