Manuscript XVI-J@1 · XVI-J.tex · assumption · 6c1f306d

Assumption: Preparation, source, density, and root

Conventions, inherited inputs, and the selected sector › Checklist of external interfaces · lines 267–335

XVI-J@1#input:all, XVI-J@1#hyp:prep, XVI-J@1#hyp:source, XVI-J@1#hyp:polar, XVI-J@1#hyp:root, XVI-J@1#eq:rootinput

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The preparation is fixed, with
$σ_{ε} (t) = 0 (t \leq 0), σ_{ε} (t) = 1 (t \geq ε), σ_{ε} \in C^{\infty} (R) .$
The independent interaction switch is $χ_{I} \in C_{c}^{\infty} (M; R)$ .
The supplied XII-C, version 1, source formula is
$J_{D}^{μν} = σ_{ε} J_{3}^{μν} + 2 C_{2}^{μν α β} \partial_{α} \partial_{β} σ_{ε} - 2 M_{2}^{μν ρ} \partial_{ρ} σ_{ε} .$
Only its $C_{2}$ summand is retained. The finite definitions of $C_{2}$ and of the unintegrated cubic density are given below. They are mathematical definitions in this manuscript, as well as matching the supplied XII-C representative.
The matching off-shell insertion has the symmetric quadratic normalization
$F_{t} (h, φ) = ⟨ t_{D}, h \otimes h \otimes φ ⟩, D_{h}^{2} F_{t} [A, C] = 2 ⟨ t_{D}, A \otimes C \otimes φ ⟩ .$
The source polarization uses
$s (X, Y) = \frac{1}{2} (Q (X + Y) - Q (X) - Q (Y)) .$
These are the supplied XVI-H representation and the revised XVI-I normalization. They are used only to identify the selected-sector graph factor.
The root is a covariant tensor-valued distribution
$B_{f, g, μν} \in E^{'} (M_{y} \times M_{x}) .$
Each component is compactly supported and satisfies
$WF (B_{f, g, μν}) \subset Z_{2} := {(y, x; ξ, - ξ) : ξ \neq = 0} .$
This is the root-kernel interface attributed in the supplied XVI-J to XII-D, version 1, lem:root-kernel. Its internal advanced-kernel composition is not reopened. Compactness and the wavefront inclusion are assumed here, not proved.

Displayed formulas in this passage

XVI-J@1#eq:rootinput · line 321
$WF (B_{f, g, μν}) \subset Z_{2} := {(y, x; ξ, - ξ) : ξ \neq = 0} .$
unlabeled · line 272
$σ_{ε} (t) = 0 (t \leq 0), σ_{ε} (t) = 1 (t \geq ε), σ_{ε} \in C^{\infty} (R) .$
unlabeled · line 284
$J_{D}^{μν} = σ_{ε} J_{3}^{μν} + 2 C_{2}^{μν α β} \partial_{α} \partial_{β} σ_{ε} - 2 M_{2}^{μν ρ} \partial_{ρ} σ_{ε} .$
unlabeled · line 299
$F_{t} (h, φ) = ⟨ t_{D}, h \otimes h \otimes φ ⟩, D_{h}^{2} F_{t} [A, C] = 2 ⟨ t_{D}, A \otimes C \otimes φ ⟩ .$
unlabeled · line 307
$s (X, Y) = \frac{1}{2} (Q (X + Y) - Q (X) - Q (Y)) .$
unlabeled · line 317
$B_{f, g, μν} \in E^{'} (M_{y} \times M_{x}) .$

Assumption: Preparation, source, density, and root

Displayed formulas in this passage

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