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

TL;DR

This paper explores a Hilbert space approach to interacting vector mesons, highlighting differences from gauge theory, especially regarding localization, topological properties, and the structure of the S-matrix.

Contribution

It introduces a string-local Hilbert space formulation for vector mesons that preserves positivity and causal localization, contrasting with traditional gauge theory.

Findings

Hilbert space formulation exhibits topological properties like Haag duality absent in gauge theory.

String-local physical fields replace gauge-variant point-local matter fields in Hilbert space.

Lie-structure of vector mesons arises from causal localization principles in QFT.

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.