Construction of Field Algebras with Quantum Symmetry from Local Observables

TL;DR

This paper develops a method to construct field algebras with quantum symmetries from local observables, ensuring fields act on a positive Hilbert space and obey braid relations instead of traditional Bose/Fermi statistics.

Contribution

It introduces a reconstruction approach for quantum symmetries and field operators from observable algebras, extending the framework of local quantum theory.

Findings

Quantum symmetries are reconstructed from observable algebras.

Field operators act on positive Hilbert spaces and transform covariantly.

Fields obey local braid relations instead of Bose/Fermi commutation.

Abstract

It has been discussed earlier that ( weak quasi-) quantum groups allow for conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid-) statistics and locality was established. This work addresses to the reconstruction of quantum symmetries and algebras of field operators. For every algebra $\A$ of observables satisfying certain standard assumptions, an appropriate quantum symmetry is found. Field operators are obtained which act on a positive definite Hilbert space of states and transform covariantly under the quantum symmetry. As a substitute for Bose/Fermi (anti-) commutation relations, these fields are demonstrated to obey local braid relation.