Compositional Quantum Field Theory: An axiomatic presentation

TL;DR

This paper presents Compositional Quantum Field Theory (CQFT), an axiomatic framework that emphasizes locality and compositionality, refining existing models and enabling formalization through category theory, with applications to 2D quantum theories.

Contribution

It introduces CQFT as a new axiomatic approach based on categorical and operadic structures, extending the General Boundary Quantum Field Theory framework.

Findings

CQFT formalizes the compositional structure of quantum field theories.

The framework is applied to 2D quantum theories, including quantum Yang-Mills.

It demonstrates equivariance and algebraic structures on strings within CQFT.

Abstract

We introduce Compositional Quantum Field Theory (CQFT) as an axiomatic model of Quantum Field Theory, based on the principles of locality and compositionality. Our model is a refinement of the axioms of General Boundary Quantum Field Theory, and is phrased in terms of correspondences between certain commuting diagrams of gluing identifications between manifolds and corresponding commuting diagrams of state-spaces and linear maps, thus making it amenable to formalization in terms of involutive symmetric monoidal functors and operad algebras. The underlying novel framework for gluing leads to equivariance of CQFT. We study CQFTs in dimension 2 and the algebraic structure they define on open and closed strings. We also consider, as a particular case, the compositional structure of 2-dimensional pure quantum Yang-Mills theory.