Manuscript X-F-ST@1 · X-F-ST.tex · lemma · e71f03ef

Lemma: Inherited identity with its representative

Free polynomial carrier and the scalar background identity › Completed scalar identities as compact equation images · lines 1230–1270 · proof included or linked

X-F-ST@1#lem:scalaridentity, X-F-ST@1#eq:scalaridentity

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For fixed smooth backgrounds,

d_{1} [h^{'}] - d_{1} [h] d_{2} [h^{'}, k^{'}] - d_{2} [h, k] = P_{m} (X_{ξ} E_{m}^{A} f), = P_{m} (X_{ξ} E_{m}^{A} d_{1} [h] + Z_{ξ} E_{m}^{A} f) .

The displayed preimages are compactly supported.

Proof. Write

c = f + κ c_{1} + κ^{2} c_{2}, e = f + κ d_{1} + κ^{2} d_{2}, L = κ L_{1} + κ^{2} L_{2} .

Then $e = (1 + L E_{m}^{A})^{- 1} c$ . On the frozen plateau, naturality of the coordinate density and unchanged initial data give the formal coordinate intertwiner $X [q^{'}] = X [q] \circ F$ . Thus the inverse-coordinate smearing density satisfies $c [q^{'}] = F^{♯} c [q]$ .

Set $s = E_{m}^{A} e$ . It solves $(P_{m} + L [q]) s = c [q]$ with advanced support. Scalar-density covariance on the plateau gives

(P_{m} + L [q^{'}]) F^{*} = F^{♯} (P_{m} + L [q]) .

Advanced uniqueness therefore gives $s^{'} = F^{*} s$ . Applying $P_{m}$ yields $e^{'} - e = P_{m} ((F^{*} - 1) E_{m}^{A} e)$ . Expansion gives (eq:scalaridentity). The generator coefficients have compact support.

Displayed formulas in this passage

X-F-ST@1#eq:scalaridentity · line 1233
$d_{1} [h^{'}] - d_{1} [h] d_{2} [h^{'}, k^{'}] - d_{2} [h, k] = P_{m} (X_{ξ} E_{m}^{A} f), = P_{m} (X_{ξ} E_{m}^{A} d_{1} [h] + Z_{ξ} E_{m}^{A} f) .$
unlabeled · line 1246
$c = f + κ c_{1} + κ^{2} c_{2}, e = f + κ d_{1} + κ^{2} d_{2}, L = κ L_{1} + κ^{2} L_{2} .$
unlabeled · line 1262
$(P_{m} + L [q^{'}]) F^{*} = F^{♯} (P_{m} + L [q]) .$

Lemma: Inherited identity with its representative

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