AQFT from n-functorial QFT

TL;DR

This paper shows how 2-dimensional extended Minkowskian functorial QFTs can naturally produce local nets of observables, linking two major axiomatizations of quantum field theory through a functorial perspective.

Contribution

It establishes a natural relation between algebraic and functorial QFT approaches by deriving local nets from 2-functorial QFT under mild assumptions.

Findings

2-dimensional extended Minkowskian QFTs yield local nets

The construction mimics the Schroedinger to Heisenberg transition

Method generalizes to higher dimensions and pseudo-Riemannian structures

Abstract

There are essentially two different approaches to the axiomatization of quantum field theory (QFT): algebraic QFT, going back to Haag and Kastler, and functorial QFT, going back to Atiyah and Segal. More recently, based on ideas by Baez and Dolan, the latter is being refined to "extended" functorial QFT by Freed, Hopkins, Lurie and others. The first approach uses local nets of operator algebras which assign to each patch an algebra "of observables", the latter uses n-functors which assign to each patch a "propagator of states". In this note we present an observation about how these two axiom systems are naturally related: we demonstrate under mild assumptions that every 2-dimensional extended Minkowskian QFT 2-functor ("parallel surface transport") naturally yields a local net. This is obtained by postcomposing the propagation 2-functor with an operation that mimics the passage from the Schroedinger picture to the Heisenberg picture in quantum mechanics. The argument has a straightforward generalization to general pseudo-Riemannian structure and higher dimensions.