Manuscript X-F-ST@1 · X-F-ST.tex · section · e71f03ef

The first insertion kernel

F3: three response kernels, support, and admission › The first insertion kernel · lines 1360–1408

X-F-ST@1#eq:D, X-F-ST@1#eq:Dsupport, X-F-ST@1#eq:N

Set $F = E_{m}^{A} f$ , and let

a^{μν, α β} = \frac{1}{2} η^{μν} η^{α β} - \frac{1}{2} (η^{μα} η^{ν β} + η^{μ β} η^{ν α}) .

Then $a_{1}^{μν} [l] = a^{μν, α β} l_{α β}$ . The exact kernel $D$ of $d_{1}$ is

d_{1} [l] (x) D^{α β} (x, z) Λ^{μ, α β} (x, z) = \int D^{α β} (x, z) l_{α β} (z) d^{4} z, = \partial_{x^{μ}} (f (x) Λ^{μ, α β} (x, z)) - \partial_{x^{μ}} (g (x) a^{μν, α β} \partial_{ν} F (x) δ (x - z)) - \frac{1}{2} m^{2} g (x) F (x) η^{α β} δ (x - z), = - σ (z) a^{ν μ, α β} \partial_{x^{ν}} R (x - z), R = E_{0}^{R} .

It follows directly by transposing the first coordinate equation and expanding $- L_{1} [l] F$ .

In particular,

supp D \subset {(x, z) \in C \times D_{f} : z ⪯ x}, WF (D) \subset Z_{2} .

The local $g F$ -terms explain why the first support factor is $C$ , not generally $K_{f}$ .

Suppressing fixed tensor indices, define

N_{0} (x, y) = (D \circ Δ_{gr}^{R}) (x, y) = \int D (x, z) Δ_{gr}^{R} (z, y) d^{4} z .

It is covariant in the source-density argument $y$ .

Displayed formulas in this passage

X-F-ST@1#eq:D · line 1372
$d_{1} [l] (x) D^{α β} (x, z) Λ^{μ, α β} (x, z) = \int D^{α β} (x, z) l_{α β} (z) d^{4} z, = \partial_{x^{μ}} (f (x) Λ^{μ, α β} (x, z)) - \partial_{x^{μ}} (g (x) a^{μν, α β} \partial_{ν} F (x) δ (x - z)) - \frac{1}{2} m^{2} g (x) F (x) η^{α β} δ (x - z), = - σ (z) a^{ν μ, α β} \partial_{x^{ν}} R (x - z), R = E_{0}^{R} .$
X-F-ST@1#eq:Dsupport · line 1393
$supp D \subset {(x, z) \in C \times D_{f} : z ⪯ x}, WF (D) \subset Z_{2} .$
X-F-ST@1#eq:N · line 1402
$N_{0} (x, y) = (D \circ Δ_{gr}^{R}) (x, y) = \int D (x, z) Δ_{gr}^{R} (z, y) d^{4} z .$
unlabeled · line 1363
$a^{μν, α β} = \frac{1}{2} η^{μν} η^{α β} - \frac{1}{2} (η^{μα} η^{ν β} + η^{μ β} η^{ν α}) .$