Completion Green operators and causal compatibility

Let ${E_0}^{\mathrm{R}/\mathrm{A}}$ and ${E_m}^{\mathrm{R}/\mathrm{A}}$ be the positive equation inverses of $\square$ and $P_m$ on $\hat{M}$. Their existence and causal support are standard on this globally hyperbolic completion BGP.

The metric–auxiliary inverse is

The scalar action-Hessian inverse is $-{E_m}^{\mathrm{R}/\mathrm{A}}$.

For clarity about odd signs, place Grassmann variations on the left. The ghost Euler derivatives are

In this source-dual convention their inverse is

Changing the order or parity convention for the odd source dual changes printed inverse matrices. It does not license changing the action or the causal brackets.

Put ${\delta }_{\mu }={E_{\mu }}^{\mathrm{R}}-{E_{\mu }}^{\mathrm{A}}$. The causal brackets used here have blocks

\begin{array}{c|c@{\qquad}c|c}
 AB&\Delta_{AB}&AB&\Delta_{AB}\\ \hline
 rr&-4\mathsf P_{\mathrm{tr}}\delta_0&\phi\phi&-\delta_m\\
 rb&K\delta_0&br&-2\mathsf C\mathsf P_{\mathrm{tr}}\delta_0\\
 c\bar c&-\delta_0&\bar c c&\delta_0
\end{array}
 \label{eq:causalblocks}

and ${\Delta }_{bb}=0$. Thus the scalar sign and the odd signs are not inferred from an unsigned array.