Theorem: Normalized complete free contraction matrix

The matrices (eq:Wfull):

1. are bisolutions of the full even and odd free operators;

2. have the graded canonical antisymmetric part $i({\mathbb{G}}^{+}-{\mathbb{G}}^{-})$;

3. obey the positive-frequency wavefront upper bound componentwise.

They therefore define normalized contractions for compactly smeared free fields.

Proof. For the even matrix, the lower row equation uses $B=-r\mathcal{F}z$, and the upper row reduces to $\mathcal{B}{W}_{\mathcal{B}}=0$. Right equations follow by adjunction. The odd field equations follow from $Q{W}_Q=W_{Q}Q=0$.

The $zz$ antisymmetric part is $i{E}_{\mathcal{B}}$. The $zB$ and $Bz$ parts follow from

The $BB$ part vanishes because

For odd variables, graded antisymmetrization is ordinary symmetrization across the two odd slots. The upper cross entry gives

and the lower cross entry gives $i{E}_Q$, exactly as in (eq:odd-green).

Differential operators in either slot do not enlarge the wavefront set.