Quantum Symmetry

TL;DR

This paper explores the connection between quantum field theories and algebraic structures, proposing that every such theory with certain axioms has an associated weak quasi Hopf algebra as gauge symmetry, aiding in understanding gauge invariance.

Contribution

It establishes a method to construct a weak quasi Hopf algebra from the representation category of a quantum field theory, generalizing the concept of gauge symmetry.

Findings

Construction of a functor from representation categories to vector spaces.

Use of a generalized reconstruction theorem to identify the symmetry algebra.

Application of the symmetry algebra to build gauge covariant field algebras.

Abstract

The representations of the observable algebra of a low dimensional quantum field theory form the objects of a braided tensor category. The search for gauge symmetry in the theory amounts to finding an algebra which has the same representation category. In this paper we try to establish that every quantum field theory satisfying some basic axioms posseses a weak quasi Hopf algebra as gauge symmetry. The first step is to construct a functor from the representation category to the category of finite dimensional vector spaces. Given such a functor we can use a generalized reconstruction theorem to find the symmetry algebra. It is shown how this symmetry algebra is used to build a gauge covariant field algebra and we investigate the question why this generality is necessary.