Manuscript Em06-B@1 · Em06-B-v1.tex · lemma · 61e38859

Lemma: Mass-shell estimate and scalar equation descent

Independent classical audit and compact-test estimates › One-particle map and classical preparation · lines 526–553 · proof included or linked

Em06-B@1#lem:mass-shell

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On a fixed compact support $S \subset M$ ,

∥ k_{F} ∥_{h} \leq C_{m, S} p_{2} (F), k_{P_{m} ψ} = 0 (ψ \in C_{c}^{\infty} (M)) .

Proof. Spatial integration by parts gives

∣ k_{F} (p) ∣ \leq \frac{( 2 π ) ^{- 3/2}}{2 ω _{p} ( 1 + ∣ p ∣ ^{2} )} ∥ (1 - Δ_{x}) F ∥_{L^{1} (M)} .

The integral

\int_{R^{3}} \frac{d ^{3} p}{ω _{p} ( 1 + ∣ p ∣ ^{2} ) ^{2}}

is finite: $m > 0$ controls the origin, and the radial integrand decays as $r^{- 3}$ at infinity. The $L^{1}$ norm is bounded by a fixed-support constant times $p_{2} (F)$ .

The exponential in (eq:k) is annihilated by $P_{m}$ . Compact integration by parts proves $k_{P_{m} ψ} = 0$ .

Displayed formulas in this passage

unlabeled · line 529
$∥ k_{F} ∥_{h} \leq C_{m, S} p_{2} (F), k_{P_{m} ψ} = 0 (ψ \in C_{c}^{\infty} (M)) .$
unlabeled · line 537
$∣ k_{F} (p) ∣ \leq \frac{( 2 π ) ^{- 3/2}}{2 ω _{p} ( 1 + ∣ p ∣ ^{2} )} ∥ (1 - Δ_{x}) F ∥_{L^{1} (M)} .$
unlabeled · line 544
$\int_{R^{3}} \frac{d ^{3} p}{ω _{p} ( 1 + ∣ p ∣ ^{2} ) ^{2}}$

Lemma: Mass-shell estimate and scalar equation descent

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