# Operads for algebraic quantum field theory

September 25, 2017

We construct a colored operad whose category of algebras is canonically
isomorphic to 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 physically very interesting
constructions, such as time-slicification and local-to-global extensions of
quantum field theories. We compare the latter to Fredenhagen's universal
algebra.

open access link

@article{Benini:2017fnn,
author = "Benini, Marco and Schenkel, Alexander and Woike, Lukas",
title = "{Operads for algebraic quantum field theory}",
doi = "10.1142/S0219199720500078",
year = "2017",
eprint = "1709.08657",
archivePrefix = "arXiv",
primaryClass = "math-ph",
reportNumber = "ZMP-HH/17-26, Hamburger Beitraege zur Mathematik Nr. 682,
ZMP-HH-17-26",
SLACcitation = "%%CITATION = ARXIV:1709.08657;%%"
}

Keywords:

algebraic quantum field theory, locally covariant quantum field theory, colored operads, change of color adjunctions, Fredenhagenâ€™s universal algebra