Operads for algebraic quantum field theory

TL;DR

This paper introduces a colored operad framework to systematically describe various algebraic quantum field theories, enabling new constructions and comparisons across different types of theories.

Contribution

It constructs a functorial colored operad capturing algebraic quantum field theories, allowing for new adjunctions and extensions between different theory categories.

Findings

Unified operad framework for quantum field theories

New adjunctions between different theory categories

Comparison with Fredenhagen's universal algebra

Abstract

We construct a colored operad whose category of algebras is the category of algebraic quantum field theories. This is achieved by a construction that depends on the choice of a category, whose objects provide the operad colors, equipped with an additional structure that we call an orthogonality relation. This allows us to describe different types of quantum field theories, including theories on a fixed Lorentzian manifold, locally covariant theories and also chiral conformal and Euclidean theories. Moreover, because the colored operad depends functorially on the orthogonal category, we obtain adjunctions between categories of different types of quantum field theories. These include novel and interesting constructions, such as time-slicification and local-to-global extensions of quantum field theories. We compare the latter to Fredenhagen's universal algebra.