Manuscript IV@1 · IV.tex · section · 351cb2d7

The exact mask family

Model 2: dense vacuum TT form and the actual cutoffs › The exact mask family · lines 1578–1610

IV@1#eq:masks

For $R \geq 2$ , take smooth real even masks satisfying

0 \leq α_{R}, β_{R} \leq 1, α_{R} = 1 (R^{- 1} \leq ∣ p ∣ \leq R), supp α_{R} \subset {(2 R)^{- 1} < ∣ p ∣ < 2 R}, β_{R} = 1 (∣ q ∣ \leq R), supp β_{R} \subset {∣ q ∣ < 2 R}, H_{R}^{ϵδ} = α_{R} (p) β_{R} (q) H^{ϵδ} .

No derivative bounds or monotonicity in $R$ are assumed. These are gravitational IR and UV masks and a scalar UV mask.

Put $L_{R} = H_{R}^{-+} : h_{ϕ} \to h_{g}$ and $P_{R} = H_{R}^{--} \in h_{g} \otimes h_{ϕ}$ . On $h = h_{g} \oplus h_{ϕ}$ , set

T_{R} = (0 L_{R}^{*} L_{R} 0), K_{R} = (i_{g} \otimes i_{ϕ}) P_{R} + F (i_{g} \otimes i_{ϕ}) P_{R} .

For a symmetric pair tensor $K$ , use

a^{†} (K) Ψ_{n} = (n + 1) (n + 2) S_{n + 2} (K \otimes Ψ_{n}) .

Displayed formulas in this passage

IV@1#eq:masks · line 1581
$0 \leq α_{R}, β_{R} \leq 1, α_{R} = 1 (R^{- 1} \leq ∣ p ∣ \leq R), supp α_{R} \subset {(2 R)^{- 1} < ∣ p ∣ < 2 R}, β_{R} = 1 (∣ q ∣ \leq R), supp β_{R} \subset {∣ q ∣ < 2 R}, H_{R}^{ϵδ} = α_{R} (p) β_{R} (q) H^{ϵδ} .$
unlabeled · line 1599
$T_{R} = (0 L_{R}^{*} L_{R} 0), K_{R} = (i_{g} \otimes i_{ϕ}) P_{R} + F (i_{g} \otimes i_{ϕ}) P_{R} .$
unlabeled · line 1606
$a^{†} (K) Ψ_{n} = (n + 1) (n + 2) S_{n + 2} (K \otimes Ψ_{n}) .$