Manuscript III-B@1 · III-B-v1.tex · proposition · d2c98f62

Proposition: Adjoint-exact zero family

Compact coupled Ward observables › An exact family that vanishes · lines 1402–1427 · proof included or linked

III-B@1#prop:exact-zero

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

O_{J_{ψ}, P ψ} (h, φ) = 0

whenever $P φ = - L_{h} Φ$ .

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

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

Also integration by parts in (eq:Lh) gives

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

Formal self-adjointness of $P$ yields

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

Displayed formulas in this passage

unlabeled · line 1405
$O_{J_{ψ}, P ψ} (h, φ) = 0$
unlabeled · line 1413
$\partial_{μ} J_{ψ}^{μν} = \frac{1}{2} (P ψ) \partial^{ν} Φ.$
unlabeled · line 1418
$\int h_{μν} J_{ψ}^{μν} d^{4} x = \int ψ L_{h} Φ d^{4} x .$
unlabeled · line 1423
$O_{J_{ψ}, P ψ} = \int ψ (L_{h} Φ + P φ) d^{4} x = 0.$