Smooth 1-dimensional algebraic quantum field theories

TL;DR

This paper introduces a refined concept of algebraic quantum field theories that emphasizes smoothness over families of spacetimes, providing concrete 1-dimensional examples and discussing challenges for higher dimensions.

Contribution

It develops a framework for smooth AQFTs using stacks of categories and constructs explicit examples in 1D, advancing the mathematical foundation of AQFTs.

Findings

Constructed concrete examples of smooth 1D AQFTs

Developed a stack-based formalism for smooth AQFTs

Identified challenges for extending to higher dimensions and gauge theories

Abstract

This paper proposes a refinement of the usual concept of algebraic quantum field theories (AQFTs) to theories that are smooth in the sense that they assign to every smooth family of spacetimes a smooth family of observable algebras. Using stacks of categories, this proposal is realized concretely for the simplest case of 1-dimensional spacetimes, leading to a stack of smooth 1-dimensional AQFTs. Concrete examples of smooth AQFTs, of smooth families of smooth AQFTs and of equivariant smooth AQFTs are constructed. The main open problems that arise in upgrading this approach to higher dimensions and gauge theories are identified and discussed.