Manuscript X-G@1 · X-G.tex · proposition · 01566e14

Proposition: Auxiliary Green operators

The normalized free BRST multiplet › The metric–auxiliary Green matrix · lines 614–649 · proof included or linked

X-G@1#prop:Ghb, X-G@1#eq:Ghb

The retarded and advanced inverses are

G_{hb}^{\pm} = (Δ_{\pm}^{gr} \frac{1}{2} C Δ_{\pm}^{gr} \frac{1}{2} Δ_{\pm}^{gr} C^{*} \frac{1}{4} C Δ_{\pm}^{gr} C^{*} - \frac{1}{2} id) = (- 4 P E_{0}^{\pm} - 2 C P E_{0}^{\pm} K E_{0}^{\pm} 0) .

Proof. The auxiliary equation gives $b = (C h - j) /2$ , hence

P_{gr} h = J + \frac{1}{2} C^{*} j .

Apply its Green operator and substitute back. The identities

Δ_{\pm}^{gr} C^{*} = 2 K E_{0}^{\pm}, C Δ_{\pm}^{gr} C^{*} = 2 id

follow from (eq:Cstar), $P^{2} = id$ , and $C K = □$ . They give the second matrix. Source elimination proves both inverse identities on compact tests. All blocks have the stated causal support.

Displayed formulas in this passage

X-G@1#eq:Ghb · line 617
$G_{hb}^{\pm} = (Δ_{\pm}^{gr} \frac{1}{2} C Δ_{\pm}^{gr} \frac{1}{2} Δ_{\pm}^{gr} C^{*} \frac{1}{4} C Δ_{\pm}^{gr} C^{*} - \frac{1}{2} id) = (- 4 P E_{0}^{\pm} - 2 C P E_{0}^{\pm} K E_{0}^{\pm} 0) .$
unlabeled · line 636
$P_{gr} h = J + \frac{1}{2} C^{*} j .$
unlabeled · line 641
$Δ_{\pm}^{gr} C^{*} = 2 K E_{0}^{\pm}, C Δ_{\pm}^{gr} C^{*} = 2 id$

Proposition: Auxiliary Green operators

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