Definition: Free causal polynomial algebra

The algebra ${\mathcal{A}}_0$ is generated over $\mathbb{C}[[\hslash ]]$ by the compact interior smeared fields, subject to:

1. the auxiliary free equations

   $$A\gamma +2D^{\ast }b=0,b=D\gamma ,P_m\varphi =0,\square c=\square \bar{c}=0;$$

2. the graded commutation relations

   $$[{\Phi }_A(f),{\Phi }_B(k){]}_{\mathrm{gr}}=i\hslash ⟨f,{\Delta }_{AB}k⟩1.$$

Only finite polynomial degree is used at each formal $\hslash$ coefficient.