Manuscript X-E@1 · X-E.tex · section · 293fae6a

Kernel and support

E4: one tree › Kernel and support · lines 1225–1245

X-E@1#eq:D1

Write $d_{1} [k] (x) = \int D_{1}^{α β} (x, z) k_{α β} (z) d z$ , with

D_{1}^{α β} (x, z) Λ^{μ, α β} (x, z) = \partial^{x}_{μ} (f (x) Λ^{μ, α β} (x, z)) - \partial^{x}_{μ} (g (x) a^{μν, α β} \partial_{ν} (E_{m}^{A} f) (x) δ (x - z)) - \frac{1}{2} m^{2} g (x) (E_{m}^{A} f) (x) η^{α β} δ (x - z), = - \partial^{x}_{ν} E_{0}^{R} (x, z) σ_{ε} (z) a^{ν μ, α β}, a^{ν μ, α β} = \frac{1}{2} η^{ν μ} η^{α β} - \frac{1}{2} (η^{ν α} η^{μ β} + η^{ν β} η^{μα}),

so that $a^{ν μ} [k] = a^{ν μ, α β} k_{α β}$ .

Displayed formulas in this passage

X-E@1#eq:D1 · line 1229
$D_{1}^{α β} (x, z) Λ^{μ, α β} (x, z) = \partial^{x}_{μ} (f (x) Λ^{μ, α β} (x, z)) - \partial^{x}_{μ} (g (x) a^{μν, α β} \partial_{ν} (E_{m}^{A} f) (x) δ (x - z)) - \frac{1}{2} m^{2} g (x) (E_{m}^{A} f) (x) η^{α β} δ (x - z), = - \partial^{x}_{ν} E_{0}^{R} (x, z) σ_{ε} (z) a^{ν μ, α β}, a^{ν μ, α β} = \frac{1}{2} η^{ν μ} η^{α β} - \frac{1}{2} (η^{ν α} η^{μ β} + η^{ν β} η^{μα}),$