The adaptation of the Kramers-Kronig dispersion relations to the causal localization structure of QFT led to an important project in particle physics, the only one with a successful closure. The same cannot be said about the subsequent attempts to formulate particle physics as a pure S-matrix project.The feasibility of a pure S-matrix approach are critically analyzed and their serious shortcomings are highlighted. Whereas the conceptual/mathematical demands of renormalized perturbation theory are modest and misunderstandings could easily be corrected, the correct understanding (...) about the origin of the crossing property requires the use of the mathematical theory of modular localization and its relation to the thermal KMS condition. These new concepts, which combine localization, vacuum polarization and thermal properties under the roof of modular theory, will be explained and their potential use in a new constructive (nonperturbative) approach to QFT will be indicated. The S-matrix still plays a predominant role but, different from Heisenberg’s and Mandelstam’s proposals, the new project is not a pure S-matrix approach. The S-matrix plays a new role as a “relative modular invariant”. (shrink)

Wigner’s famous 1939 classification of positive energy representations, combined with the more recent modular localization principle, has led to a significant conceptual and computational extension of renormalized perturbation theory to interactions involving fields of higher spin. Traditionally the clash between pointlike localization and the the Hilbert space was resolved by passing to a Krein space setting which resulted in the well-known BRST gauge formulation. Recently it turned out that maintaining a Hilbert space formulation for interacting higher spin fields requires a (...) weakening of localization from point- to string-like fields for which the d = s + 1 short distance scaling dimension for integer spins is reduced to d = 1 and and renormalizable couplings in the sense of power-counting exist for any spin. This new setting leads to a significant conceptual change of the relation of massless couplings with their massless counterpart. Whereas e.g. the renormalizable interactions of s = 1 massive vectormesons with s \ 1 matter falls within the standard field-particle setting, their zero mass limits lead to much less understood phenomena as “infraparticles” and gluon/quark confinement. It is not surprising that such drastic conceptual changes in the area of gauge theories also lead to a radical change concerning the Higgs issue. (shrink)