AQFT from n-functorial QFT
arXiv:0806.1079 · doi:10.1007/s00220-009-0840-2
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.
39 pages; further examples added: Hopf spin chains and asymptotic inclusion of subfactors; references added
References in corpus (6)
- Higher gauge theory I: 2-Bundles
- Remarks on Chern-Simons Theory
- 2-C*-categories with non-simple units
- Categorical and combinatorial aspects of descent theory
- Uniqueness of open/closed rational CFT with given algebra of open states
- Classification of operator algebraic conformal field theories in dimensions one and two