  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.
