Manuscript XVIII-B@2 · XVIII-B-v2.tex · theorem · 27ed4f4b

Theorem: Distributional linear Ward identities

Both linear Ward identities and the nonlocal invariant · lines 1097–1130 · proof included or linked

XVIII-B@2#thm:ward, XVIII-B@2#eq:ward

Every extension in Theorem thm:contacts satisfies

\partial^{a} Π^{R}_{ab c d} = 0, \partial^{c} Π^{R}_{ab c d} = 0 in S^{'} (M) .

Let

(K ξ)^{ab} = \partial^{a} ξ^{b} + \partial^{b} ξ^{a}

for a compact smooth vector field $ξ$ . The unprepared bilinear response satisfies

R (j, K ξ) = 0, R (K ξ, h) = 0.

These are complete linear identities for the printed kernel, all compact tests, and the stated contact family. They are not interacting gravitational BRST/BV identities.

Proof. Lemma lem:transverse annihilates each polynomial tensor in the reference and contact formulas. Multiplication by the scalar boundary values preserves the polynomial zero identities. Fourier inversion proves (eq:ward). Compact integration by parts in either source slot gives the two bilinear identities.

Displayed formulas in this passage

XVIII-B@2#eq:ward · line 1100
$\partial^{a} Π^{R}_{ab c d} = 0, \partial^{c} Π^{R}_{ab c d} = 0 in S^{'} (M) .$
unlabeled · line 1108
$(K ξ)^{ab} = \partial^{a} ξ^{b} + \partial^{b} ξ^{a}$
unlabeled · line 1113
$R (j, K ξ) = 0, R (K ξ, h) = 0.$

Theorem: Distributional linear Ward identities

Displayed formulas in this passage

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