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.
Highlights
There have appeared two, rather elaborate formalisms for constructing the observables of a quantum field theory via a combination of the Batalin–Vilkovisky framework with renormalization methods
The primary goal in this paper is to examine in detail the case of free field theories, where renormalization plays no role and we can focus on comparing the local-to-global descriptions of observables
In the context of this free theory, we show how to relate the key structural features of algebraic quantum field theory (AQFT) and factorization algebras
Summary
Before delving into the constructions, we discuss field theory from a very high altitude, ignoring all but the broadest features, and explain how each formalism approaches observables. As Sol is a sheaf, O(Sol(−)) should be a cosheaf, meaning that it satisfies a gluing axiom so that the global observables are assembled from the local observables Nothing about this general story depends on the signature of the metric, and each formalism gives a detailed construction of a cosheaf of commutative algebras for a classical field theory ( some technical choices differ, e.g., with respect to functional analysis). The CG formalism provides a functor Obsq : Open(M) → Ch, which assigns a cochain complex (or differential graded (dg) vector space) of observables to each open set This cochain complex is a deformation of a commutative dg algebra Obscl , where H 0(Obscl (U )) = O(Sol(U )). We will organize our treatment of the free scalar field toward addressing these questions
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