Beyond gauge theory: Hilbert space positivity and causal localization in the presence of vector mesons

Abstract

The Hilbert space formulation of interacting $s=1$ vector-potentials stands in an interesting contrast with the point-local Krein space setting of gauge theory. Already in the absence of interactions the Wilson loop in a Hilbert space setting has a "topological property" which is missing in the gauge theoretic description (Haag duality, Aharonov-Bohm effect); the conceptual differences increase in the presence of interactions. The Hilbert space positivity weakens the causal localization properties if interacting fields which results in the replacement of the gauge-variant point-local matter fields in Krein space by string-local physical fields in Hilbert space. The gauge invariance of the perturbative S-matrix corresponds to its independence of the spacelike string direction of its interpolating.fields. In contrast to gauge theory, whose physical range is limited to gauge invariant perturbative S-matrix and local observables, its Hilbert space string-local counterpart in is a full-fledged quantum field theory. The new setting reveals that the Lie-structure of self-coupled vector mesons results from perturbative implementation of the causal localization principles of QFT.