Manuscript VII-D@1 · VII-D.tex · proposition · ef67c89c

Proposition: Normalization ledger

The reflected radial family: normalization and Lemma A › Normalization ledger · lines 867–1006 · proof included or linked

VII-D@1#prop:ledger, VII-D@1#ledger:1, VII-D@1#ledger:2, VII-D@1#ledger:3, VII-D@1#ledger:4, VII-D@1#ledger:5, VII-D@1#ledger:6, VII-D@1#eq:ledgerconclusion

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The leading reflected–reflected radial coefficient derived from (eq:Aintrinsic) equals the prescribed distribution (eq:rdef) on all joint tests. In particular, its constant is $c_{0} = 1/ (64 (2 π)^{3})$ , with the parameter density in (eq:nu).

Proof. The calculation is recorded in six identities.

1: exact pointwise kernel. Proposition prop:positive gives

T_{ε}^{ρ} = 2 K_{ε} (x, y) K_{ε} (x^{'}, y) D_{ε} (x - x^{'}) .

2: reflected amplitudes and conjugation. The reflected terms of the two mixed covariances have amplitudes

a_{R} (t, P) = - \frac{i}{8 P ^{2}}, a_{R} (t^{'}, P^{'}) = \frac{i}{8 P ^{'2}}, a_{R} (t, P) a_{R} (t^{'}, P^{'}) = \frac{1}{64 P ^{2} P ^{'2}} .

The corresponding time phases are, separately, $e^{i P (t + s)}$ and $e^{- i P^{'} (t^{'} + s)}$ . The right spatial phase is $e^{- i p^{'} \cdot (x^{'} - y)}$ . These are conjugations of the kernel, not of $Φ$ .

3: scalar density. The scalar factor is

D_{ε} (z) = \int \frac{d ^{3} q}{( 2 π ) ^{3}} \frac{ω _{m} ( q )}{2} M_{ε} (∣ q ∣)^{2} e^{- i ω_{m} (q) z^{0} + i q \cdot z} .

4: combined numerical prefactor. Consequently the reflected–reflected integrand has prefactor

2 \cdot \frac{1}{64 P ^{2} P ^{'2}} \cdot \frac{ω _{m} ( q )}{2} = \frac{ω _{m} ( q )}{64 P ^{2} P ^{'2}}, with measure \frac{d ^{3} q}{( 2 π ) ^{3}} .

The assignment factor in L1 has now been exhausted. No further factor of two occurs in $A (r)$ .

5: polar measure and radial normalization. With ordinary sphere area measure,

d^{3} q = r^{2} d r d Ω (ν), \frac{r ^{2} ω ( r )}{P ^{2} P ^{'2}} d r = \frac{r ^{3} ω ( r )}{P ^{2} P ^{'2}} \frac{d r}{r} .

This explains the extra $r^{3}$ in (eq:Aintrinsic), rather than adding a normalization by hand.

6: translation and Fourier inversion. For fixed $q$ , (eq:substitution) is a translation and has Jacobian one. The two mixed-covariance measures become exactly $d^{3} α d^{3} β / (2 π)^{6}$ . The Fourier inversion identity is

det \frac{\partial ( p , p ^{'} )}{\partial ( α , β )} = 1, \frac{d ^{3} p d ^{3} p ^{'}}{( 2 π ) ^{6}} = \frac{d ^{3} α d ^{3} β}{( 2 π ) ^{6}}, \int \frac{d ^{3} α d ^{3} β}{( 2 π ) ^{6}} e^{- ia ν \cdot α + ib ν \cdot β} H_{Φ} (t, s, t^{'}, y; α, β) = Φ (t, y + a ν; s, y; t^{'}, y + b ν) .

In particular, the two factors $(2 π)^{- 3}$ supplied by the mixed covariances are the normalization of the two Fourier inversions. They cancel the factors $(2 π)^{3}$ from the unnormalized Fourier delta identities; they are not canceled a second time against another Fourier measure.

For the leading coefficient, replace the amplitude factors $P^{- 2}, P^{' - 2}$ by $r^{- 2}, r^{- 2}$ and $ω (r)$ by $r$ . In the phase, however, retain the first-order expansions

P = r - ν \cdot α, P^{'} = r - ν \cdot β, ω (r) = r .

Substituting $P = P^{'} = r$ into the phase itself would be incorrect. The retained time phase is exactly the inversion phase in L6.

The only remaining numerical factor is therefore

\frac{1}{( 2 π ) ^{3}} \frac{1}{64} = c_{0} .

After L6, the remaining measure is $d t d s d t^{'} d^{3} y d Ω (ν)$ . Changing $y$ to $z = y + s ν$ has unit Jacobian, so this is precisely the pushforward density (eq:nu). We have proved

A_{lead} = r in D^{'} (X), c_{0} = \frac{1}{64 ( 2 π ) ^{3}}, leading RR contribution = ⟨ r, Φ ⟩ \frac{d r}{r} .

The cutoff factors $M_{ε} (r)^{2} ϑ (εP) ϑ (ε P^{'})$ have not been used in this leading identification. They are treated by the replacement estimate below. Direct-branch amplitudes do not enter this calculation.

Displayed formulas in this passage

VII-D@1#ledger:1 · line 881
$T_{ε}^{ρ} = 2 K_{ε} (x, y) K_{ε} (x^{'}, y) D_{ε} (x - x^{'}) .$
VII-D@1#ledger:2 · line 890
$a_{R} (t, P) = - \frac{i}{8 P ^{2}}, a_{R} (t^{'}, P^{'}) = \frac{i}{8 P ^{'2}}, a_{R} (t, P) a_{R} (t^{'}, P^{'}) = \frac{1}{64 P ^{2} P ^{'2}} .$
VII-D@1#ledger:3 · line 904
$D_{ε} (z) = \int \frac{d ^{3} q}{( 2 π ) ^{3}} \frac{ω _{m} ( q )}{2} M_{ε} (∣ q ∣)^{2} e^{- i ω_{m} (q) z^{0} + i q \cdot z} .$
VII-D@1#ledger:4 · line 915
$2 \cdot \frac{1}{64 P ^{2} P ^{'2}} \cdot \frac{ω _{m} ( q )}{2} = \frac{ω _{m} ( q )}{64 P ^{2} P ^{'2}}, with measure \frac{d ^{3} q}{( 2 π ) ^{3}} .$
VII-D@1#ledger:5 · line 927
$d^{3} q = r^{2} d r d Ω (ν), \frac{r ^{2} ω ( r )}{P ^{2} P ^{'2}} d r = \frac{r ^{3} ω ( r )}{P ^{2} P ^{'2}} \frac{d r}{r} .$
VII-D@1#ledger:6 · line 942
$det \frac{\partial ( p , p ^{'} )}{\partial ( α , β )} = 1, \frac{d ^{3} p d ^{3} p ^{'}}{( 2 π ) ^{6}} = \frac{d ^{3} α d ^{3} β}{( 2 π ) ^{6}}, \int \frac{d ^{3} α d ^{3} β}{( 2 π ) ^{6}} e^{- ia ν \cdot α + ib ν \cdot β} H_{Φ} (t, s, t^{'}, y; α, β) = Φ (t, y + a ν; s, y; t^{'}, y + b ν) .$
VII-D@1#eq:ledgerconclusion · line 985
$A_{lead} = r in D^{'} (X), c_{0} = \frac{1}{64 ( 2 π ) ^{3}}, leading RR contribution = ⟨ r, Φ ⟩ \frac{d r}{r} .$
unlabeled · line 969
$P = r - ν \cdot α, P^{'} = r - ν \cdot β, ω (r) = r .$
unlabeled · line 977
$\frac{1}{( 2 π ) ^{3}} \frac{1}{64} = c_{0} .$

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section: Normalization ledger