Manuscript X-L-ST@1 · X-L-ST.tex · proposition · 20a32a22
Proposition: Auxiliary Green matrix with the retained signs
The normalized free auxiliary–ghost package › The complete free Green matrix · lines 482–534 · proof included or linked
X-L-ST@1#prop:Ghb, X-L-ST@1#eq:Ghb
For either support choice,
\mathcal G_{rb}^{\mathrm R/\mathrm A}
&=
\begin{pmatrix}
\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}
&\frac12\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}\mathsf C^*\\
\frac12\mathsf C\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}
&\frac14\mathsf C\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}
\mathsf C^*-\frac12I
\end{pmatrix}
\nonumber\\
&=
\begin{pmatrix}
-4\mathsf P_{\mathrm{tr}}\mathsf E_0^{\mathrm R/\mathrm A}
&K\mathsf E_0^{\mathrm R/\mathrm A}\\
-2\mathsf C\mathsf P_{\mathrm{tr}}\mathsf E_0^{\mathrm R/\mathrm A}&0
\end{pmatrix},
\label{eq:Ghb}where
Both inverse identities hold on compact tests.
Proof. The sourced equations are
Eliminate . Then
Apply the metric Green inverse and substitute back.
The simplification follows from
These identities use , , and the compact-source Green identities. The same elimination on compact field pairs proves the left inverse. Differentiation preserves causal support.
Displayed formulas in this passage
X-L-ST@1#eq:Ghb · line 485
\mathcal G_{rb}^{\mathrm R/\mathrm A} &= \begin{pmatrix} \Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A} &\frac12\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}\mathsf C^*\\ \frac12\mathsf C\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A} &\frac14\mathsf C\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A} \mathsf C^*-\frac12I \end{pmatrix} \nonumber\\ &= \begin{pmatrix} -4\mathsf P_{\mathrm{tr}}\mathsf E_0^{\mathrm R/\mathrm A} &K\mathsf E_0^{\mathrm R/\mathrm A}\\ -2\mathsf C\mathsf P_{\mathrm{tr}}\mathsf E_0^{\mathrm R/\mathrm A}&0 \end{pmatrix}, \label{eq:Ghb}unlabeled · line 505
unlabeled · line 514
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