Analytic Extrapolation Techniques and Stability Problems in Dispersion Relation Theory
The point we try to make is that in an indirect science like elementary particle physics, it is not sufficient to have a specific description of the world brought by some happy inspiration, but rather it is necessary to optimize among large classes of (preferably among all) possible logically equivalent “revelations”. Indeed, although the leading concepts of which every description of nature makes use should bear a very close relation to the experimentally accessible data, in those situations when the basic laws are inherited from other fields, their concepts may prove to be very remote from experiment, and to “measure” them one might have to go through wildly unstable inverse problems (ill posed problems in the Hadamard sense). Moreover, the instabilities of the inverse laws become especially dangerous when the corresponding “direct laws” are too smooth, as it happens in particle physics whenever we try to cling to classical concepts (Lagrangians, interaction terms, etc.) which were purposedly chosen to produce “good” classical physics laws.