Manuscript Em01-C@3 · Em01-C-v3.tex · section · f0953702

Transposition of the complete vertex

The complete contraction as a translation distribution › Transposition of the complete vertex · lines 1080–1120

Em01-C@3#eq:aPhi, Em01-C@3#eq:dPhi, Em01-C@3#eq:Lambda

Let $K$ be a fixed internal distribution and $Φ \in D (U; V_{out}^{*})$ . Define compact relative distributions

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

The pre-pushforward distributions have compact support. Their pushforward is defined by transposition against smooth relative functions.

Put

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] .

Displayed formulas in this passage

Em01-C@3#eq:aPhi · line 1085
$a_{I, ν} [Φ] d_{I} [Φ] = f_{*} (J \sum g (y) \partial_{y^{ν}} Φ_{J} (w) K^{I, J} (w)), = f_{*} (J \sum g (y) Φ_{J} (w) K^{I, J} (w)) .$
Em01-C@3#eq:dPhi · line 1085
$a_{I, ν} [Φ] d_{I} [Φ] = f_{*} (J \sum g (y) \partial_{y^{ν}} Φ_{J} (w) K^{I, J} (w)), = f_{*} (J \sum g (y) Φ_{J} (w) K^{I, J} (w)) .$
Em01-C@3#eq:Lambda · line 1106
$Λ_{I, μν} [Φ] = \frac{i}{2} (\partial_{2, μ} a_{I, ν} [Φ] + \partial_{2, ν} a_{I, μ} [Φ]) - \frac{i}{2} η_{μν} \partial_{2}^{λ} a_{I, λ} [Φ] - \frac{i m ^{2}}{2} η_{μν} d_{I} [Φ] .$