Manuscript V-B@1 · V-B-v1.tex · proposition · 3fadc1ad

Proposition: A free-null label can have nonzero response

The external first retarded coefficient › Switching and an off-shell-label witness · lines 1158–1185 · proof included or linked

V-B@1#prop:externalwitness

Back to manuscript

For every nonzero spatial-TT wave $h$ , there are real compact $f, g$ and a one-particle Schwartz vector $v$ such that

ϕ (f) = 0, ⟨ Ω, ϕ (q_{g, h, f}) a^{†} (v) Ω ⟩ \neq = 0.

Proof. Choose $x_{*}, p_{*}$ with $h^{ij} (x_{*}) p_{* i} p_{* j} \neq = 0$ . A small momentum bump, with phase centered at $x_{*}$ , gives a positive-frequency wave $w_{v}$ for which $Re (h^{ij} \partial_{i} \partial_{j} w_{v})$ has one nonzero sign on a small neighborhood.

Take a nonnegative real $ρ$ supported there, $f = P ρ$ , and compact $g = 1$ near $supp ρ$ . Then $ϕ (f) = 0$ , $u_{f} = - ρ$ , and $q_{g, h, f} = - Q_{h} ρ$ . Integration by parts gives

\int q_{g, h, f} w_{v} = \int ρ h^{ij} \partial_{i} \partial_{j} w_{v} \neq = 0.

Displayed formulas in this passage

unlabeled · line 1162
$ϕ (f) = 0, ⟨ Ω, ϕ (q_{g, h, f}) a^{†} (v) Ω ⟩ \neq = 0.$
unlabeled · line 1181
$\int q_{g, h, f} w_{v} = \int ρ h^{ij} \partial_{i} \partial_{j} w_{v} \neq = 0.$

Proposition: A free-null label can have nonzero response

Displayed formulas in this passage

Nearby in this section