The free causal algebra

Let ${\mathfrak{A}}_0$ be the unital associative graded algebra over $\mathbb{C}[[\hslash ]]$ generated by compactly smeared linear fields, modulo the free equations

and the causal relations

No inverse of $\hslash$ is adjoined.

The Green identities imply that the causal form vanishes on equation tests, so it descends to their linear quotient. Its central graded Lie algebra has an enveloping algebra with the prescribed central relation. This gives the associative polynomial causal carrier. It does not require a state or a time-ordered product.