On the equivalence of AQFTs and prefactorization algebras

TL;DR

This paper introduces a new approach to establish the equivalence between algebraic quantum field theories and prefactorization algebras on globally hyperbolic Lorentzian manifolds, simplifying the problem into manageable spacetime-wise components.

Contribution

It presents a novel structural method for the additivity property and reduces the global equivalence problem to simpler local problems, extending previous theorems to more complex categorical settings.

Findings

New structural implementation of additivity property.

Reduction of global equivalence to spacetime-wise problems.

Extension of equivalence theorem to symmetric monoidal 1-categories.

Abstract

This paper revisits the equivalence problem between algebraic quantum field theories and prefactorization algebras defined over globally hyperbolic Lorentzian manifolds. We develop a radically new approach whose main innovative features are 1.) a structural implementation of the additivity property used in earlier approaches and 2.) a reduction of the global equivalence problem to a family of simpler spacetime-wise problems. When applied to the case where the target category is a symmetric monoidal $1$-category, this yields a generalization of the equivalence theorem from [Commun. Math. Phys. 377, 971 (2019)]. In the case where the target is the symmetric monoidal $\infty$-category of cochain complexes, we obtain a reduction of the global $\infty$-categorical equivalence problem to simpler, but still challenging, spacetime-wise problems. The latter would be solved by showing that certain functors between $1$-categories exhibit $\infty$-localizations, however the available detection criteria are inconclusive in our case.