A Hilbert Space Setting for Interacting Higher Spin Fields and the Higgs Issue
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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.
KeywordsString-localized massive vectormesons and Hilbert space positivity Coupling massive vectormesons to Hermitian scalar fields Induced H-selfinteraction versus symmetry-breaking Higgs mechanism.
I am indebted to Jens Mund for numerous discussions and for making his notes available prior to publication and last not least for reading my manuscript and suggesting improvements. Although for the time being we have split the problems of the \(s\ge 1\) interactions in Hilbert space into mathematical/conceptual and historical/conceptual part, it is our intention to unite them in a future project and in this way to lessen the present deep schism between LQP and theoretical problems of Standard Model particle theory. I also thank Raymond Stora for having accompanied this new development with great interest and challenging questions which led to better formulations of some important points.
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