Manuscript XV-D@1 · XV-D.tex · section · 967136eb

Supplied internal expressions

Frozen branch and placement ledger › Supplied internal expressions · lines 460–515

XV-D@1#eq:K5a, XV-D@1#eq:K5z, XV-D@1#eq:K47, XV-D@1#eq:K6, XV-D@1#eq:pole

The supplied formulas are Hkernel,Iamplitude

K_{5}^{[a], I, J} (a, b, z) K_{5}^{[z], I, J} (a, b, z) K_{47}^{I, J} (a, b, z) K_{6}^{I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)] .

Both $K_{5}$ placements are finite differential expressions in the tensor-product pullbacks of $R (b - a) R (a - z)$ and $R (b - z) R (z - a)$ , which exist in the independent variables $(a, b, z)$ including at $a = z$ Hkernel,XVB. The internal pole chart is

M_{R} = - \frac{s}{2 ρ} (ω - s ρ - i 0)^{- 1} + \frac{s}{2 ρ ( ω + s ρ )}, ρ > 0, ∣ ω - s ρ ∣ < c ρ, 0 < c < 1,

with $(λ - i 0)^{- 1} = PV \frac{1}{λ} + iπ δ (λ)$ Ipole. The delta part is fixed by the retarded line and is not adjusted.

Displayed formulas in this passage

XV-D@1#eq:K5a · line 463
$K_{5}^{[a], I, J} (a, b, z) K_{5}^{[z], I, J} (a, b, z) K_{47}^{I, J} (a, b, z) K_{6}^{I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)] .$
XV-D@1#eq:K5z · line 463
$K_{5}^{[a], I, J} (a, b, z) K_{5}^{[z], I, J} (a, b, z) K_{47}^{I, J} (a, b, z) K_{6}^{I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)] .$
XV-D@1#eq:K47 · line 463
$K_{5}^{[a], I, J} (a, b, z) K_{5}^{[z], I, J} (a, b, z) K_{47}^{I, J} (a, b, z) K_{6}^{I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)] .$
XV-D@1#eq:K6 · line 463
$K_{5}^{[a], I, J} (a, b, z) K_{5}^{[z], I, J} (a, b, z) K_{47}^{I, J} (a, b, z) K_{6}^{I, J} (a, b, z) = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)] .$
XV-D@1#eq:pole · line 503
$M_{R} = - \frac{s}{2 ρ} (ω - s ρ - i 0)^{- 1} + \frac{s}{2 ρ ( ω + s ρ )}, ρ > 0, ∣ ω - s ρ ∣ < c ρ, 0 < c < 1,$