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The Hilbert space formulation of interacting s=1 vector-potentials stands is
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 of 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 space-like string direction of its interpolating fields. In contrast to
gauge theory, whose direct physical range is limited to a gauge-invariant
perturbative S-matrix and local observables, its Hilbert space string-local
counterpart is a full-fledged quantum field theory (QFT). The new setting
reveals that the Lie structure of self-coupled vector mesons results from
perturbative implementation of the causal localization principles of QFT.
Dedicated to the memory of Raymond Stora and John Roberts
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