Manuscript IV-B@1 · IV-B-v1.tex · section · dfb66408

A nonzero record and its exact off-control

Model 1: the fixed TT plane and removable edge › A nonzero record and its exact off-control · lines 748–777

Translate so that $e^{ij} (t_{0}, 0) = E^{ij} \neq = 0$ . Choose real compact scalar data

φ_{*} (x) = - \frac{1}{2} E_{ij} x^{i} x^{j} η_{0} (x), ϖ_{*} = 0,

where $η_{0} = 1$ near zero. Then

\partial_{t}^{2} S_{t_{0}} (e) φ_{*} (t_{0}, 0) = ∣ E ∣^{2} > 0.

A small future neighborhood therefore contains a nonnegative compact test $f$ for which

d_{1} = \int f S_{t_{0}} (e) φ_{*} d^{4} x > 0.

Let $z_{*}$ be the scalar coherent label for these data. The states

Ψ_{0} = Coh (0) \otimes Coh (z_{*}), Ψ_{1} = Coh (1/ 2) \otimes Coh (z_{*})

have identical scalar factors, equal oscillator covariances, and $Δ ⟨ Q ⟩ = 1$ , $Δ ⟨ Π_{0} ⟩ = 0$ . Hence

Δ ⟨ A^{(1)} (λ)⟩ = λ d_{1} .

Their second moments are finite by Proposition prop:A1.

Displayed formulas in this passage

unlabeled · line 752
$φ_{*} (x) = - \frac{1}{2} E_{ij} x^{i} x^{j} η_{0} (x), ϖ_{*} = 0,$
unlabeled · line 757
$\partial_{t}^{2} S_{t_{0}} (e) φ_{*} (t_{0}, 0) = ∣ E ∣^{2} > 0.$
unlabeled · line 762
$d_{1} = \int f S_{t_{0}} (e) φ_{*} d^{4} x > 0.$
unlabeled · line 766
$Ψ_{0} = Coh (0) \otimes Coh (z_{*}), Ψ_{1} = Coh (1/ 2) \otimes Coh (z_{*})$
unlabeled · line 773
$Δ ⟨ A^{(1)} (λ)⟩ = λ d_{1} .$