Manuscript XX-A@1 · XX-A.tex · corollary · 3a586110

Corollary: Coupled classical Ward identity

Classical gauge identity and rod-detection tests › The linear observable and Hessian must be coupled · lines 843–869 · proof included or linked

XX-A@1#cor:coupled, XX-A@1#eq:linear-ward, XX-A@1#eq:quadratic-ward

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For every admitted $ξ$ and every classical $z$ ,

F_{1, f} (K ξ) F_{2, f} (z, K ξ) + F_{1, f} (L_{ξ} z) = 0, = 0.

The second equation does not assert $F_{2, f} (z, K ξ) = 0$ separately.

Proof. Use the infinitesimal pullback generated by $s ξ$ . The variation of the first jet is $K ξ$ , and the variation of the second jet is $2 L_{ξ} z$ . Differentiate (eq:observable-jet) and use Theorem thm:classical-gauge. The coefficients of $s$ and $s^{2}$ give the two equations.

As a direct first-order check,

(K ξ)_{00} = - 2 ξ ˙^{0}, L_{ξ} ϕ ˉ = \frac{A}{t} ξ^{0} .

Substitution in (eq:n1-explicit) gives zero.

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XX-A@1#eq:linear-ward · line 846
$F_{1, f} (K ξ) F_{2, f} (z, K ξ) + F_{1, f} (L_{ξ} z) = 0, = 0.$
XX-A@1#eq:quadratic-ward · line 846
$F_{1, f} (K ξ) F_{2, f} (z, K ξ) + F_{1, f} (L_{ξ} z) = 0, = 0.$
unlabeled · line 864
$(K ξ)_{00} = - 2 ξ ˙^{0}, L_{ξ} ϕ ˉ = \frac{A}{t} ξ^{0} .$

Corollary: Coupled classical Ward identity

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