Manuscript Em01-G@5 · Em01-G-v5.tex · section · 3fbfe951

Positive-scale vertex, local equations, and cutoffs

Positive-scale vertex, local equations, and cutoffs · lines 2404–2472

The relative map is

f (a, b, z, y) = (y - a, y - b) .

For smooth $H$ , define

V (K, H)^{J} = \partial_{y^{ν}} [g (y) I \sum K^{I, J} Z_{I}^{ν} (H)] - \frac{m ^{2}}{2} g (y) I \sum K^{I, J} M_{I} (H),

Z_{I}^{ν} (H) = f^{*} (\partial_{2, μ} H_{I}^{μν} - \frac{1}{2} η_{μα} \partial_{2}^{ν} H_{I}^{μα}), M_{I} (H) = f^{*} (η_{μν} H_{I}^{μν}) .

On $x_{1}^{0} > 0, x_{2}^{0} < 0$ ,

D_{T, 0} (x_{1}) = W_{0} (x_{1}), D_{T, m} (x_{2}) = W_{m} (- x_{2}),

so the crossed array is $i ℏ^{2} N (W_{0} \otimes Δ_{m})$ . Also

N_{I}^{μν} = (e^{I})^{μν} - \frac{1}{2} η^{μν} ϑ_{I}, η_{μν} N_{I}^{μν} = - ϑ_{I} .

The trace-kinetic terms cancel. The two factors $i$ multiply to $- 1$ ; transposition of the output derivative yields

t_{δ}^{app} (Φ) = λ^{3} ℏ^{2} I, J \sum ⟨ K_{ret}^{I, J}, U_{δ}^{I, J} [Φ]⟩ .

At positive scale the convolution is smooth, so all products with the internal distributions are defined. If two relative cutoffs equal one near the required compact image, their difference vanishes on a fixed neighborhood. For small $δ$ , every translated smoothing test and every differentiated-profile test centered near that image stays in the neighborhood. The crossed jets and complete family therefore agree exactly at small scales.

The same support argument applies to local differential identities. For instance,

(□_{η, 2} + m^{2}) (χD) = [(□_{η, 2} + m^{2}), χ] D

has support away from the required relative neighborhood. Its convolution is zero there at sufficiently small scales, proving the local Klein–Gordon identity used in Lemma lem:positive-reduction.

All operations act on reconstructed kernels; the identity $q δ_{D}^{'} (q) + δ_{D} (q) = 0$ is therefore preserved. No profile factorization or unrestricted-component positivity is used.

Displayed formulas in this passage

unlabeled · line 2408
$f (a, b, z, y) = (y - a, y - b) .$
unlabeled · line 2412
$V (K, H)^{J} = \partial_{y^{ν}} [g (y) I \sum K^{I, J} Z_{I}^{ν} (H)] - \frac{m ^{2}}{2} g (y) I \sum K^{I, J} M_{I} (H),$
unlabeled · line 2418
$Z_{I}^{ν} (H) = f^{*} (\partial_{2, μ} H_{I}^{μν} - \frac{1}{2} η_{μα} \partial_{2}^{ν} H_{I}^{μα}), M_{I} (H) = f^{*} (η_{μν} H_{I}^{μν}) .$
unlabeled · line 2428
$D_{T, 0} (x_{1}) = W_{0} (x_{1}), D_{T, m} (x_{2}) = W_{m} (- x_{2}),$
unlabeled · line 2433
$N_{I}^{μν} = (e^{I})^{μν} - \frac{1}{2} η^{μν} ϑ_{I}, η_{μν} N_{I}^{μν} = - ϑ_{I} .$
unlabeled · line 2441
$t_{δ}^{app} (Φ) = λ^{3} ℏ^{2} I, J \sum ⟨ K_{ret}^{I, J}, U_{δ}^{I, J} [Φ]⟩ .$
unlabeled · line 2459
$(□_{η, 2} + m^{2}) (χD) = [(□_{η, 2} + m^{2}), χ] D$