Operads for algebraic quantum field theory

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.

v1: 64 pages - v2: 31 pages, presentation improved and shortened - v3: Final version to appear in Communications in Contemporary Mathematics

References in corpus (5)

• Wave Equations on Lorentzian Manifolds and Quantization
• Homotopy theory of algebraic quantum field theories
• Homotopical Quantum Field Theory
• Algebraic quantum field theory on spacetimes with timelike boundary
• Involutive categories, colored $\ast$-operads and quantum field theory

Cited by in corpus (17)

• Homotopical Quantum Field Theory
• Infinity Operads and Monoidal Categories with Group Equivariance
• Linear Yang-Mills theory as a homotopy AQFT
• Categorification of algebraic quantum field theories
• Snowmass White Paper: The Quest to Define QFT
• A skeletal model for 2d conformal AQFTs
• Wightman fields for two-dimensional conformal field theories with pointed representation category
• Quantization of Lorentzian free BV theories: factorization algebra vs algebraic quantum field theory
• Strictification theorems for the homotopy time-slice axiom
• Relative Cauchy evolution for linear homotopy AQFTs
• Involutive categories, colored $\ast$-operads and quantum field theory
• Spacetimes categories and disjointness for algebraic quantum field theory
• The linear CS/WZW bulk/boundary system in AQFT
• Lorentzian bordisms in algebraic quantum field theory
• On the equivalence of AQFTs and prefactorization algebras
• Gluing algebraic quantum field theories on manifolds
• $C^{\ast }$-categorical prefactorization algebras for superselection sectors and topological order