Manuscript III@2 · III.tex · proposition · 82dad3e5

Proposition: Exact compact repair

Compact coupled Ward observables › The adjoint-exact repair is on-shell zero · lines 2106–2142 · proof included or linked

III@2#prop:zero-repair

The pair $(J_{ψ}, P ψ)$ satisfies (eq:Ward), but

O_{J_{ψ}, P ψ} (h, φ) = 0

for every smooth solution of $P φ = - L_{h} Φ$ .

Proof. Using $P Φ = 0$ , direct differentiation gives

\partial_{μ} J_{ψ}^{μν} = \frac{1}{2} (P ψ) \partial^{ν} Φ.

The mixed Hessian terms cancel; the remaining expression is

\frac{1}{2} (□_{η} ψ) \partial^{ν} Φ - \frac{1}{2} ψ \partial^{ν} □_{η} Φ = \frac{1}{2} (P ψ) \partial^{ν} Φ.

Integration by parts in (eq:Lh) gives

\int_{M} h_{μν} J_{ψ}^{μν} d^{4} x = \int_{M} ψ L_{h} Φ d^{4} x .

The Hessian contribution cancels after moving the derivative from the $D h$ term. Formal self-adjointness of $P$ then gives

O_{J_{ψ}, P ψ} = \int_{M} ψ (L_{h} Φ + P φ) d^{4} x = 0.

There is no Cauchy boundary term because $ψ$ is compact in spacetime.

Displayed formulas in this passage

unlabeled · line 2109
$O_{J_{ψ}, P ψ} (h, φ) = 0$
unlabeled · line 2117
$\partial_{μ} J_{ψ}^{μν} = \frac{1}{2} (P ψ) \partial^{ν} Φ.$
unlabeled · line 2122
$\frac{1}{2} (□_{η} ψ) \partial^{ν} Φ - \frac{1}{2} ψ \partial^{ν} □_{η} Φ = \frac{1}{2} (P ψ) \partial^{ν} Φ.$
unlabeled · line 2129
$\int_{M} h_{μν} J_{ψ}^{μν} d^{4} x = \int_{M} ψ L_{h} Φ d^{4} x .$
unlabeled · line 2136
$O_{J_{ψ}, P ψ} = \int_{M} ψ (L_{h} Φ + P φ) d^{4} x = 0.$