Manuscript XX-B@1 · XX-B.tex · theorem · 56d708a5
Theorem: Full free-multiplet Green matrices
Retarded and advanced Green distributions › Complete auxiliary and ghost inverse · lines 1090–1133 · proof included or linked
XX-B@1#thm:full-green, XX-B@1#eq:even-green, XX-B@1#eq:odd-green
The complete retarded and advanced inverses are
\mathbb G_{\mathrm{ev}}^\pm
&=\begin{pmatrix}
G_{\mathcal B}^\pm&K G_Q^\pm\\
G_Q^\pm K^\dagger&0
\end{pmatrix},
\label{eq:even-green}\\
\mathbb G_{\mathrm{odd}}^\pm
&=\begin{pmatrix}
0&-G_Q^\pm\\
G_Q^\pm&0
\end{pmatrix}.
\label{eq:odd-green}They are left and right inverses on compact tests, with causal support in every component.
Proof. The Schur complement of the lower-right even block is . Its formal block inverse is
Proposition prop:intertwining converts the off-diagonal blocks to those in (eq:even-green). The lower-right block equals
Direct multiplication verifies both inverse identities. The odd inverse is verified by matrix multiplication. Differential operators are local, so all blocks retain the corresponding causal support.
Displayed formulas in this passage
XX-B@1#eq:even-green · line 1093
\mathbb G_{\mathrm{ev}}^\pm &=\begin{pmatrix} G_{\mathcal B}^\pm&K G_Q^\pm\\ G_Q^\pm K^\dagger&0 \end{pmatrix}, \label{eq:even-green}\\ \mathbb G_{\mathrm{odd}}^\pm &=\begin{pmatrix} 0&-G_Q^\pm\\ G_Q^\pm&0 \end{pmatrix}. \label{eq:odd-green}XX-B@1#eq:odd-green · line 1093
\mathbb G_{\mathrm{ev}}^\pm &=\begin{pmatrix} G_{\mathcal B}^\pm&K G_Q^\pm\\ G_Q^\pm K^\dagger&0 \end{pmatrix}, \label{eq:even-green}\\ \mathbb G_{\mathrm{odd}}^\pm &=\begin{pmatrix} 0&-G_Q^\pm\\ G_Q^\pm&0 \end{pmatrix}. \label{eq:odd-green}unlabeled · line 1115
unlabeled · line 1125