The exact meaning of free recovery

At $\kappa =0$, the field rules are precisely (eq:s0), and the normalized quadratic action and Green matrix are (eq:freeaction) and (eq:fullGreen). The free causal presentation therefore recovers ${\mathfrak{A}}_0$ by Theorem thm:freeBRST.

This recovery has three explicit steps:

It is not an algebra homomorphism from a commutative classical field algebra to the noncommutative causal algebra.