Modular Theory and Symmetry in QFT

TL;DR

This paper explores how modular theory in quantum field theory (QFT) reveals connections between space-time and internal symmetries, proposing a unified view of quantum symmetries with mathematical theorems linking indices and expectation values.

Contribution

It develops a speculative framework linking space-time and internal symmetries in QFT using modular theory, deriving new formulas like the Kac-Wakimoto formula.

Findings

Derived the Kac-Wakimoto formula relating Jones indices to expectation value asymptotics.

Presented a new asymptotic Gibbs-state representation of mapping class group matrices.

Linked braid group matrices with quantum symmetry structures.

Abstract

The 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.