Manuscript Em01-E@2 · Em01-E-v2.tex · section · 54f8f8c2

Route C: an exact translation-origin certificate

Three sufficient verification routes › Route C: an exact translation-origin certificate · lines 973–1016

Em01-E@2#eq:Lambda, Em01-E@2#eq:Omega

For a compact joint test, define compact relative distributions

a_{I, ν} [Φ] = f_{*} (J \sum g (y) \partial_{y^{ν}} Φ_{J} K^{I, J}), d_{I} [Φ] = f_{*} (J \sum g (y) Φ_{J} K^{I, J}) .

Their pushforwards are defined because the distributions being pushed forward are compactly supported. Set

Λ_{I, μν} [Φ] = \frac{i}{2} (\partial_{2, μ} a_{I, ν} [Φ] + \partial_{2, ν} a_{I, μ} [Φ]) - \frac{i}{2} η_{μν} \partial_{2}^{λ} a_{I, λ} [Φ] - \frac{i m ^{2}}{2} η_{μν} d_{I} [Φ] .

For $d (x, v) = x - v$ , put

Ω_{Φ} = I, μ, ν \sum d_{*} (Λ_{I, μν} [Φ] \otimes χ C_{I}^{μν}) .

Displayed formulas in this passage

Em01-E@2#eq:Lambda · line 991
$Λ_{I, μν} [Φ] = \frac{i}{2} (\partial_{2, μ} a_{I, ν} [Φ] + \partial_{2, ν} a_{I, μ} [Φ]) - \frac{i}{2} η_{μν} \partial_{2}^{λ} a_{I, λ} [Φ] - \frac{i m ^{2}}{2} η_{μν} d_{I} [Φ] .$
Em01-E@2#eq:Omega · line 1006
$Ω_{Φ} = I, μ, ν \sum d_{*} (Λ_{I, μν} [Φ] \otimes χ C_{I}^{μν}) .$
unlabeled · line 976
$a_{I, ν} [Φ] = f_{*} (J \sum g (y) \partial_{y^{ν}} Φ_{J} K^{I, J}), d_{I} [Φ] = f_{*} (J \sum g (y) Φ_{J} K^{I, J}) .$