Lemma: Auxiliary transposition and normalization

For the translation-invariant data (eq:h),

The matrix $H$ solves the full free equations in both slots. Its graded antisymmetric part is

where $\Delta$ is exactly XI-B's causal matrix.

Proof. Differentiating $-\mathcal{P}{h}_0$ in its second tensor slot and using ${\partial }_y{h}_0=-{\partial }_x{h}_0$ gives ${\mathcal{R}}_x{h}_0$. In contrast, differentiation in the first slot gives the displayed $D_x$ formula. Applying both divergences produces a constant multiple of ${\square }_x{h}_0$, hence zero.

The auxiliary equations hold because $b=D\gamma$ in each slot and $-\square \mathcal{P}{H}_{\gamma \gamma }=0$. The scalar and ghost equations follow from the corresponding bisolution equations.

Equation (eq:h) gives

Differentiation gives the mixed auxiliary blocks $i\mathcal{R}{\delta }_0$ and $iD(-\mathcal{P}{\delta }_0)$. For the odd pair,

which is $i{\Delta }_{c\bar{c}}$ since ${\Delta }_{c\bar{c}}=-i{\delta }_0$. The reversed pair similarly gives $-{\delta }_0$.