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

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