Switching and the external gauge defect

The same coefficient construction applies to any smooth external symmetric tensor, because its product with $g$ is compact. For a smooth gauge parameter,

Using (eq:switch) and $P\varphi =0$,

Therefore

where

The proof uses $gP(X_{\xi }\varphi )=P(g{X}_{\xi }\varphi )-[P,g]X_{\xi }\varphi$ and $G_{R}P(g{X}_{\xi }\varphi )=g{X}_{\xi }\varphi$, valid on compactly supported domain matrix elements.

Smearing gives

Each term is a linearly smeared scalar field after integration by parts. It is generally nonzero even if $g=1$ near $\mathrm{supp}f$: switching terms in the causal past may remain, and the scalar coordinate variation is not a gauge-invariant observable by itself.

For fixed compact supports, $(g,f)\mapsto q_{g,h,f}$ is continuous into compact smooth tests, hence into Schwartz mass-shell vectors. This implies continuity in (eq:scalarbound). If $g_1-g_2$ vanishes on a neighborhood of $J^{-}(\mathrm{supp}f)\cap \mathrm{supp}h$, the coefficients coincide. That set need not be compact for a radiative $h$. No exhausting-switch limit follows.

Figure fig:external-witness-switch distinguishes the free-null insertion witness from the causal dependence on the interaction switch.