Manuscript Psy02-B@2 · Psy02-B-v2.tex · section · cfd06e2e

Both K 5 placements and the other retained terms

Track B internal kernels, trees, and the grouped target › Both K 5 placements and the other retained terms · lines 1880–1985

Psy02-B@2#eq:K5, Psy02-B@2#eq:K47, Psy02-B@2#eq:K6, Psy02-B@2#eq:K8, Psy02-B@2#eq:K9

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Let

a_{1}^{α β} [h] = A^{α β, I} h_{I}, a_{2} ˙^{α μ} [G; h] = D^{α μ, I, J} h_{I} G_{J},

with fixed constant fiber maps from the inherited coefficient sheet. The supplied retained kernels are

K_{5}^{I, J} (a, b, z) = \partial_{b^{μ}} [+ f (b) σ_{ε} (a) \partial_{a^{α}} R_{ret} (b - a) A^{α β, I} \partial_{a^{β}} L_{coord}^{μ, J} (a, z) f (b) σ_{ε} (z) \partial_{z^{α}} R_{ret} (b - z) A^{α β, J} \partial_{z^{β}} L_{coord}^{μ, I} (z, a)],

K_{47}^{I, J} K_{6}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L_{coord}^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L_{coord}^{ρ, J} (b, z))) L_{coord}^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R_{ret} (b - a) δ^{(4)} (a - z)] .

They are source records with the types now printed.

Both $K_{5}$ products are internally defined:

R_{ret} (b - a) R_{ret} (a - z), R_{ret} (b - z) R_{ret} (z - a) .

The respective coordinate changes to root plus independent differences are diffeomorphisms. Thus they are pullbacks of tensor products, including at $a = z$ . Differentiation and smooth multiplication preserve existence. No quantum cross-line has yet been multiplied.

On the chronological separated patch, the supplied coordinate response term is

K_{8}^{I, J} (a, b, z) = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B_{ret} (b - v) Q^{M, I, J} (v; a, z) d^{4} v] .

Lemma lem:retarded explains its retarded convolution and cone coefficient. Its source-specific finite order still depends on $Q$ .

A type-complete scalar-leg definition can be given without inventing its absent component expansion:

⟨ K_{9}, h \otimes ϕ \otimes G ⟩ = - \int f E_{m}^{R} (V_{1, g} [P E_{0}^{R} J_{D}^{(2)} [h, G]] ϕ) d^{4} x .

For fixed coefficients and compact input tests, this is a continuous trilinear functional, by local smooth wave propagation and compact support control. The Schwartz kernel theorem therefore defines a distribution. Equation (eq:K9) matches the scalar-leg structure, but its identity with every inherited $K_{9}$ component requires the exact $V_{1, g}$ and source coefficients. It is not a numerical or microlocal completion of the full target.

Set

K_{ret} = K_{47} + K_{5} + K_{6} + K_{8} + K_{9}, K_{supp} = K_{6} + K_{8} + K_{9} .

The first is the retained six-branch sum, because $K_{47}$ groups branches $4, 7$ and $K_{5}$ contains both placements.

For an inserted smooth metric $G$ , the complete term retained in $K_{47}$ is represented by

(C_{G})^{μ}_{ν} = f \partial_{ν} X_{1}^{μ} [G] - δ^{μ}_{ν} \partial_{ρ} (f X_{1}^{ρ} [G]) .

Deleting its second contribution changes the grouped expression.

Displayed formulas in this passage

Psy02-B@2#eq:K5 · line 1891
$K_{5}^{I, J} (a, b, z) = \partial_{b^{μ}} [+ f (b) σ_{ε} (a) \partial_{a^{α}} R_{ret} (b - a) A^{α β, I} \partial_{a^{β}} L_{coord}^{μ, J} (a, z) f (b) σ_{ε} (z) \partial_{z^{α}} R_{ret} (b - z) A^{α β, J} \partial_{z^{β}} L_{coord}^{μ, I} (z, a)],$
Psy02-B@2#eq:K47 · line 1906
$K_{47}^{I, J} K_{6}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L_{coord}^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L_{coord}^{ρ, J} (b, z))) L_{coord}^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R_{ret} (b - a) δ^{(4)} (a - z)] .$
Psy02-B@2#eq:K6 · line 1906
$K_{47}^{I, J} K_{6}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L_{coord}^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L_{coord}^{ρ, J} (b, z))) L_{coord}^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R_{ret} (b - a) δ^{(4)} (a - z)] .$
Psy02-B@2#eq:K8 · line 1935
$K_{8}^{I, J} (a, b, z) = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B_{ret} (b - v) Q^{M, I, J} (v; a, z) d^{4} v] .$
Psy02-B@2#eq:K9 · line 1949
$⟨ K_{9}, h \otimes ϕ \otimes G ⟩ = - \int f E_{m}^{R} (V_{1, g} [P E_{0}^{R} J_{D}^{(2)} [h, G]] ϕ) d^{4} x .$
unlabeled · line 1883
$a_{1}^{α β} [h] = A^{α β, I} h_{I}, a_{2} ˙^{α μ} [G; h] = D^{α μ, I, J} h_{I} G_{J},$
unlabeled · line 1923
$R_{ret} (b - a) R_{ret} (a - z), R_{ret} (b - z) R_{ret} (z - a) .$
unlabeled · line 1970
$K_{ret} = K_{47} + K_{5} + K_{6} + K_{8} + K_{9}, K_{supp} = K_{6} + K_{8} + K_{9} .$
unlabeled · line 1979
$(C_{G})^{μ}_{ν} = f \partial_{ν} X_{1}^{μ} [G] - δ^{μ}_{ν} \partial_{ρ} (f X_{1}^{ρ} [G]) .$