Manuscript Em01-D@2 · Em01-D-v2.tex · section · d3b49151

An intrinsic complete-sum translation representation

An intrinsic complete-sum translation representation · lines 1132–1177

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

For $K = K_{ret}$ and a compact joint test $Φ$ , define

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

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

The pre-pushforward distributions have compact support, so these pushforwards are defined by transposition.

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] .

Let $d (x, v) = x - v$ , and define

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

Displayed formulas in this passage

Em01-D@2#eq:Lambda · line 1154
$Λ_{I, μν} [Φ] = \frac{i}{2} (\partial_{2, μ} a_{I, ν} [Φ] + \partial_{2, ν} a_{I, μ} [Φ]) - \frac{i}{2} η_{μν} \partial_{2}^{λ} a_{I, λ} [Φ] - \frac{i m ^{2}}{2} η_{μν} d_{I} [Φ] .$
Em01-D@2#eq:Omega · line 1167
$Ω_{Φ} = I, μ, ν \sum d_{*} (Λ_{I, μν} [Φ] \otimes χ C_{I}^{μν}) .$
unlabeled · line 1136
$a_{I, ν} [Φ] = f_{*} (J \sum g (y) \partial_{y^{ν}} Φ_{J} (w) K^{I, J} (w)),$
unlabeled · line 1143
$d_{I} [Φ] = f_{*} (J \sum g (y) Φ_{J} (w) K^{I, J} (w)) .$