Theorem: Complete local Ward action and mixed products

Let $B_1,\dots ,B_n$ be any Wick fields, including $Q,L,t$ and their descendants. Let their compact insertion supports lie in $K_0$, and choose $q_X[a,\chi ]$ with the collar contract. Then, on ${\mathcal{D}}_E$,

and

The identity extends to every compact joint kernel with the same local support condition and every joint-Schwartz spectator vector.

The ordered vacuum Ward identities are

as joint tempered distributions.

Proof. The elementary action is Lemma lem:cauchyaction. Differentiating it gives (eq:jetaction).

To calculate the action on a normal product, first use separated point variables and subtract the vacuum contractions defining normal ordering. The commutator is a derivation on the ordinary separated field product. Any possible extra numerical term from the subtracted contractions is a derivative of $({d_X}^x+{d_X}^y)W(x-y)$, and therefore vanishes by Lemma lem:Wcov. Taking the normal-product limit gives (eq:prolong) and (eq:wickaction). The limits can be taken on ${\mathcal{D}}_E$: their Fourier kernels converge under the weighted fixed-sector bounds used in Lemma lem:wickest.

The common invariant domain makes the operator commutator Leibniz identity legitimate for every finite ordered product. This gives (eq:mixedaction).

For a compact joint kernel, cover its support by finitely many product patches and approximate in the compact-test topology by finite tensor sums with supports in those patches. Proposition prop:domain gives convergence both of the words and of their current insertions. The same argument applies to joint-Schwartz spectator vectors, using longer-word bounds. Thus the stated identities hold on the complete declared test class, not only on factorized tests.

Finally expand an ordered vacuum Wick correlation into its contractions. On each contraction the summed infinitesimal action vanishes by (eq:Wcov), including all differentiated contractions. Every contraction term, and hence the complete finite Wick sum, obeys (eq:vacward).