Manuscript Em02-D@4 · Em02-D-v4.tex · proposition · fcf2fac7

Proposition: Exact 5z decomposition

Exact signed internal convolution for 5z › Boundary/interior decomposition · lines 1582–1668 · proof included or linked

Em02-D@4#prop:5z-decomposition, Em02-D@4#eq:5z-boundary, Em02-D@4#eq:5z-interior

At every sufficiently small positive scale,

t_{δ, 5 z}^{ζ} = t_{δ, 5 z}^{bdry} + t_{δ, 5 z}^{int},

where

t_{δ, 5 z}^{bdry} = i \int d^{4} a d^{3} s {G_{0}^{r} [D^{2} U_{δ}] - G_{1}^{r} [D U_{δ}] + 2 G_{0}^{r} [\partial_{r^{0}} D U_{δ}] + G_{2}^{r} [U_{δ}] - 2 G_{1}^{r} [\partial_{r^{0}} U_{δ}] + G_{0}^{r} [\partial_{r^{0}}^{2} U_{δ}]}_{S = s},

and

t_{δ, 5 z}^{int} = i \int_{d (S) > 0} d^{4} a d^{4} S {Γ_{3}^{r} (S; U_{δ}) - 2 Γ_{2}^{r} (S; \partial_{r^{0}} U_{δ}) + Γ_{1}^{r} (S; \partial_{r^{0}}^{2} U_{δ})} .

Every internal coefficient derivative freezes its $r$ -weight. The $D$ derivatives on $U_{δ}$ are total temporal derivatives at fixed $a, r$ .

Proof. Apply Lemma lem:internal-identity at each fixed $a, S$ , with its actual parameter-dependent weight $U_{δ} (a, S, \cdot)$ . For temporal derivatives, $D d = 1$ , $D^{2} d = 0$ , so

D_{p}^{3} (Θ (d) Γ) D_{p}^{2} (Θ (d) Γ) D_{p} (Θ (d) Γ) = Θ (d) Γ_{3} + 3 δ_{D} (d) Γ_{2} + 3 δ_{D}^{'} (d) Γ_{1} + δ_{D}^{''} (d) Γ_{0}, = Θ (d) Γ_{2} + 2 δ_{D} (d) Γ_{1} + δ_{D}^{'} (d) Γ_{0}, = Θ (d) Γ_{1} + δ_{D} (d) Γ_{0} .

Integrate $S^{0}$ , with $d = S^{0} - ∣ S ∣$ . For example,

\int δ_{D}^{''} (d) Γ_{0} (S; U_{δ} (S)) d S^{0} = G_{2}^{r} [U_{δ}] + 2 G_{1}^{r} [D U_{δ}] + G_{0}^{r} [D^{2} U_{δ}],

whereas

3 \int δ_{D}^{'} (d) Γ_{1} (S; U_{δ} (S)) d S^{0} = - 3 G_{2}^{r} [U_{δ}] - 3 G_{1}^{r} [D U_{δ}] .

Adding the $3 δ_{D} (d) Γ_{2}$ term leaves

G_{2}^{r} [U_{δ}] - G_{1}^{r} [D U_{δ}] + G_{0}^{r} [D^{2} U_{δ}] .

The $- 2 D_{p}^{2}$ term similarly leaves

- 2 G_{1}^{r} [\partial_{r^{0}} U_{δ}] + 2 G_{0}^{r} [\partial_{r^{0}} D U_{δ}] .

The remaining boundary term is $G_{0}^{r} [\partial_{r^{0}}^{2} U_{δ}]$ . The undifferentiated Heaviside terms give (eq:5z-interior).

Displayed formulas in this passage

Em02-D@4#eq:5z-boundary · line 1592
$t_{δ, 5 z}^{bdry} = i \int d^{4} a d^{3} s {G_{0}^{r} [D^{2} U_{δ}] - G_{1}^{r} [D U_{δ}] + 2 G_{0}^{r} [\partial_{r^{0}} D U_{δ}] + G_{2}^{r} [U_{δ}] - 2 G_{1}^{r} [\partial_{r^{0}} U_{δ}] + G_{0}^{r} [\partial_{r^{0}}^{2} U_{δ}]}_{S = s},$
Em02-D@4#eq:5z-interior · line 1607
$t_{δ, 5 z}^{int} = i \int_{d (S) > 0} d^{4} a d^{4} S {Γ_{3}^{r} (S; U_{δ}) - 2 Γ_{2}^{r} (S; \partial_{r^{0}} U_{δ}) + Γ_{1}^{r} (S; \partial_{r^{0}}^{2} U_{δ})} .$
unlabeled · line 1628
$D_{p}^{3} (Θ (d) Γ) D_{p}^{2} (Θ (d) Γ) D_{p} (Θ (d) Γ) = Θ (d) Γ_{3} + 3 δ_{D} (d) Γ_{2} + 3 δ_{D}^{'} (d) Γ_{1} + δ_{D}^{''} (d) Γ_{0}, = Θ (d) Γ_{2} + 2 δ_{D} (d) Γ_{1} + δ_{D}^{'} (d) Γ_{0}, = Θ (d) Γ_{1} + δ_{D} (d) Γ_{0} .$
unlabeled · line 1585
$t_{δ, 5 z}^{ζ} = t_{δ, 5 z}^{bdry} + t_{δ, 5 z}^{int},$
unlabeled · line 1641
$\int δ_{D}^{''} (d) Γ_{0} (S; U_{δ} (S)) d S^{0} = G_{2}^{r} [U_{δ}] + 2 G_{1}^{r} [D U_{δ}] + G_{0}^{r} [D^{2} U_{δ}],$
unlabeled · line 1649
$3 \int δ_{D}^{'} (d) Γ_{1} (S; U_{δ} (S)) d S^{0} = - 3 G_{2}^{r} [U_{δ}] - 3 G_{1}^{r} [D U_{δ}] .$
unlabeled · line 1655
$G_{2}^{r} [U_{δ}] - G_{1}^{r} [D U_{δ}] + G_{0}^{r} [D^{2} U_{δ}] .$
unlabeled · line 1660
$- 2 G_{1}^{r} [\partial_{r^{0}} U_{δ}] + 2 G_{0}^{r} [\partial_{r^{0}} D U_{δ}] .$

Proposition: Exact 5z decomposition

Displayed formulas in this passage

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