Manuscript XIX-B@1 · XIX-B.tex · proposition · b5670709

Proposition: Classical retarded scalar-switch identity

The scalar-switch defect: a separate classical identity · lines 1187–1237 · proof included or linked

XIX-B@1#prop:switch, XIX-B@1#eq:switch-defect

For the displayed definitions,

A_{f, g}^{(1)} [h + K ξ; ϕ_{0}] - A_{f, g}^{(1)} [h; ϕ_{0}] = \int f [(g - 1) X_{ξ} ϕ_{0} + G_{m, 0} ((\partial_{μ} g) K ξ_{-}^{μν} \partial_{ν} ϕ_{0} - [P, g] X_{ξ} ϕ_{0})] d^{4} x,

where

[P, g] u = (□_{-} g) u + 2 (\partial^{μ} g) \partial_{μ} u .

This is a classical scalar-response identity. It is neither a loop anomaly nor an obstruction to the harmonic-coordinate Cauchy problem.

Proof. Set $u = X_{ξ} ϕ_{0}$ . Equations (eq:Qrelation)–(eq:puregauge) and $P ϕ_{0} = 0$ give $Q_{K ξ} ϕ_{0} = - P u$ . The density product rule gives

Q_{g K ξ} ϕ_{0} = g Q_{K ξ} ϕ_{0} - (\partial_{μ} g) K ξ_{-}^{μν} \partial_{ν} ϕ_{0} .

Therefore the scalar-response change is

G_{m, 0} (g P u + (\partial_{μ} g) K ξ_{-}^{μν} \partial_{ν} ϕ_{0}) .

Since $g P u = P (g u) - [P, g] u$ , and $g u$ has zero data, $G_{m, 0} P (g u) = g u$ . The coordinate change is $ξ$ , so the dressing subtracts $u$ . Smearing proves (eq:switch-defect).

The coordinate equation contained no $g$ . Its transformation was established before the scalar response was introduced.

Displayed formulas in this passage

XIX-B@1#eq:switch-defect · line 1190
$A_{f, g}^{(1)} [h + K ξ; ϕ_{0}] - A_{f, g}^{(1)} [h; ϕ_{0}] = \int f [(g - 1) X_{ξ} ϕ_{0} + G_{m, 0} ((\partial_{μ} g) K ξ_{-}^{μν} \partial_{ν} ϕ_{0} - [P, g] X_{ξ} ϕ_{0})] d^{4} x,$
unlabeled · line 1203
$[P, g] u = (□_{-} g) u + 2 (\partial^{μ} g) \partial_{μ} u .$
unlabeled · line 1217
$Q_{g K ξ} ϕ_{0} = g Q_{K ξ} ϕ_{0} - (\partial_{μ} g) K ξ_{-}^{μν} \partial_{ν} ϕ_{0} .$
unlabeled · line 1224
$G_{m, 0} (g P u + (\partial_{μ} g) K ξ_{-}^{μν} \partial_{ν} ϕ_{0}) .$

Proposition: Classical retarded scalar-switch identity

Displayed formulas in this passage

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