2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/148295The application of the Tomita-Takesaki modular theory to the Haag-Kastler net approach in QFT yields external (space-time) symmetries as well as internal ones (internal ``gauge para-groups") and their dual counterparts (the ``super selection para-group"). An attempt is made to develop a (speculative) picture on ``quantum symmetry" which links space-time symmetries in an inexorable way with internal symmetries. In the course of this attempt, we present several theorems and in particular derive the Kac-Wakimoto formula which links Jones inclusion indices with the asymptotics of expectation values in physical temperature states. This formula is a special case of a new asymptotic Gibbs-state representation of mapping class group matrices (in a Haag-Kastler net indexed by intervals on the circle!) as well as braid group matrices.35 pages, LaTeXHigh Energy Physics - Theorytext