Proposition: Existence of the algebra

Definition def:algebra gives a nonzero associative graded polynomial algebra. All pairings in (eq:CCR) exist.

Proof. Smooth compact tests can be paired with the displayed causal distributions. Transposition of the Green operators implies graded antisymmetry: the even blocks are antisymmetric, while the odd–odd block is symmetric under simultaneous exchange of its slots.

The equations annihilate the causal matrix on both sides, so the pairing descends to the test space modulo equation tests. It is a constant even graded antisymmetric form on that quotient. The associated central graded Lie algebra satisfies the Jacobi identity because every bracket is central. Its enveloping algebra, with the central generator set to the unit and scaled by $i\hslash$, supplies the required algebra. Equivalently, symmetrization gives the constant graded Moyal product on polynomials. This establishes associativity and the polynomial vector-space presentation without choosing a state FR.