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

Vertex and integrated test

Smoothing, vertex, and exact rows › Vertex and integrated test · lines 490–556

Em02-D@4#eq:target, Em02-D@4#eq:Jdelta, Em02-D@4#eq:M, Em02-D@4#eq:multiplier

The trace-adjusted vector for the specified crossed tensor is

Z^{ρ}_{δ} = - \frac{1}{2} A_{δ} (x) \partial_{ρ}^{E} A_{δ} (S - x) .

The Euclidean coordinate gradient occurs because the trace-adjusted $00$ coefficient matrix is one half of the Euclidean identity.

The family is

t_{δ}^{ζ} (Φ) = - i I, J \sum ⟨ K_{ret}^{I, J}, g (y) [Z^{ρ}_{δ} \partial_{y^{ρ}} (ζ Φ_{J}) + \frac{1}{2} D_{δ} ζ Φ_{J}] ⟩ .

The internal distributions are tensored with coordinate volume in $y$ .

Define

ϕ_{j} (a, S, r, x) = (ζ Φ_{j}) (a, a + S, a + r, a + x),

and, for a frozen weight $w$ ,

J_{δ} (S; w) = \int A_{δ} (x) A_{δ} (S - x) w (x) d^{4} x .

The integrated vertex is

M_{δ, j} (a, S, r) = - \frac{1}{2} ρ = 0 \sum^{3} \partial_{S^{ρ}} J_{δ} (S; \partial_{x^{ρ}} ϕ_{j} (a, S, r, \cdot))_{S = S} + \frac{1}{2} J_{δ} (S; ϕ_{j} (a, S, r, \cdot)) .

Propagation derivatives in $S$ freeze the weight. Total derivatives in $S$ also differentiate its parameter dependence.

Integration by parts in $y$ alternatively gives the multiplier

w_{δ} (S, x) = \frac{1}{2} A_{δ} (x) A_{δ} (S - x) + \frac{1}{2} \nabla A_{δ} (x) \cdot \nabla A_{δ} (S - x) - \frac{1}{2} A_{δ} (x) Δ_{E} A_{δ} (S - x) .

Displayed formulas in this passage

Em02-D@4#eq:target · line 504
$t_{δ}^{ζ} (Φ) = - i I, J \sum ⟨ K_{ret}^{I, J}, g (y) [Z^{ρ}_{δ} \partial_{y^{ρ}} (ζ Φ_{J}) + \frac{1}{2} D_{δ} ζ Φ_{J}] ⟩ .$
Em02-D@4#eq:Jdelta · line 525
$J_{δ} (S; w) = \int A_{δ} (x) A_{δ} (S - x) w (x) d^{4} x .$
Em02-D@4#eq:M · line 532
$M_{δ, j} (a, S, r) = - \frac{1}{2} ρ = 0 \sum^{3} \partial_{S^{ρ}} J_{δ} (S; \partial_{x^{ρ}} ϕ_{j} (a, S, r, \cdot))_{S = S} + \frac{1}{2} J_{δ} (S; ϕ_{j} (a, S, r, \cdot)) .$
Em02-D@4#eq:multiplier · line 549
$w_{δ} (S, x) = \frac{1}{2} A_{δ} (x) A_{δ} (S - x) + \frac{1}{2} \nabla A_{δ} (x) \cdot \nabla A_{δ} (S - x) - \frac{1}{2} A_{δ} (x) Δ_{E} A_{δ} (S - x) .$
unlabeled · line 493
$Z^{ρ}_{δ} = - \frac{1}{2} A_{δ} (x) \partial_{ρ}^{E} A_{δ} (S - x) .$
unlabeled · line 519
$ϕ_{j} (a, S, r, x) = (ζ Φ_{j}) (a, a + S, a + r, a + x),$