Relating nets and factorization algebras of observables: free field theories

TL;DR

This paper compares two rigorous mathematical frameworks for perturbative quantum field theory, demonstrating their equivalence for free theories using the free scalar field as a key example.

Contribution

It establishes a natural transformation linking perturbative algebraic QFT and factorization algebras, showing their equivalence under the time-slice axiom for free field theories.

Findings

Both frameworks encode equivalent information for free theories.

Time-ordered products serve as an intermediate step in the comparison.

The results apply to any field theory with Green-hyperbolic equations of motion.

Abstract

In this paper we relate two mathematical frameworks that make perturbative quantum field theory rigorous: perturbative algebraic quantum field theory (pAQFT) and the factorization algebras framework developed by Costello and Gwilliam. To make the comparison as explicit as possible, we use the free scalar field as our running example, while giving proofs that apply to any field theory whose equations of motion are Green-hyperbolic (which includes, for instance, free fermions). The main claim is that for such free theories, there is a natural transformation intertwining the two constructions. In fact, both approaches encode equivalent information if one assumes the time-slice axiom. The key technical ingredient is to use time-ordered products as an intermediate step between a net of associative algebras and a factorization algebra.