Manuscript Em01-B@3 · Em01-B-v3.tex · section · 69189992

Internal source and branch formulas

Frozen expression and complete contraction › Internal source and branch formulas · lines 456–559

Em01-B@3#eq:Q, Em01-B@3#eq:R, Em01-B@3#eq:Kret, Em01-B@3#eq:K47, Em01-B@3#eq:K5plus, Em01-B@3#eq:K5minus, Em01-B@3#eq:K6, Em01-B@3#eq:K8, Em01-B@3#eq:K9

Back to manuscript

Let $I, J, L, M$ denote component indices in the fixed symmetric-tensor bases. Define constant maps by

a_{1}^{α μ} [r] = A^{α μ, I} r_{I}, b_{1} [r] = b^{I} r_{I},

and

L^{μ, I} (s, a) = σ_{ε} (a) A^{λ μ, I} \partial_{a^{λ}} R (s - a) .

Write

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

The designated local source derivative has coefficients $q_{α β}^{L, I, J} \in C^{\infty}$ with $∣ α ∣ + ∣ β ∣ \leq 2$ , and kernel

Q^{L, I, J} (v; a, z) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q_{α β}^{L, I, J} (v) \partial_{v}^{α} δ^{(4)} (v - z) \partial_{v}^{β} δ^{(4)} (v - a) .

All coefficients of the designated source are retained. The analytic results below require its finite local form, not an unprinted numerical simplification.

The fixed retarded kernels are

R (t, x) = \frac{Θ ( t )}{2 π} δ (t^{2} - ∣ x ∣^{2}), B = R * R = \frac{1}{8 π} Θ (t) Θ (t^{2} - ∣ x ∣^{2}) .

Set

T^{L, I, J} (b; a, z) = P^{L}_{M} \int R (b - v) Q^{M, I, J} (v; a, z) d^{4} v, F = E_{m}^{A} f .

The complete retained sum is

K_{ret} = K_{47} + K_{5, +} + K_{5, -} + K_{6} + K_{8} + K_{9}, K_{47} = K_{4} + K_{7},

with

K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9}^{I, J} = - \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) \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) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)] - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .

These formulas, with the fixed coefficient arrays, make the retained input independently specified as a distributional expression. Their physical identification remains the designated inherited identity.

Displayed formulas in this passage

Em01-B@3#eq:Q · line 479
$Q^{L, I, J} (v; a, z) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q_{α β}^{L, I, J} (v) \partial_{v}^{α} δ^{(4)} (v - z) \partial_{v}^{β} δ^{(4)} (v - a) .$
Em01-B@3#eq:R · line 492
$R (t, x) = \frac{Θ ( t )}{2 π} δ (t^{2} - ∣ x ∣^{2}), B = R * R = \frac{1}{8 π} Θ (t) Θ (t^{2} - ∣ x ∣^{2}) .$
Em01-B@3#eq:Kret · line 508
$K_{ret} = K_{47} + K_{5, +} + K_{5, -} + K_{6} + K_{8} + K_{9}, K_{47} = K_{4} + K_{7},$
Em01-B@3#eq:K47 · line 514
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)] - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-B@3#eq:K5plus · line 514
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)] - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-B@3#eq:K5minus · line 514
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)] - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-B@3#eq:K6 · line 514
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)] - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-B@3#eq:K8 · line 514
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)] - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-B@3#eq:K9 · line 514
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)] - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
unlabeled · line 460
$a_{1}^{α μ} [r] = A^{α μ, I} r_{I}, b_{1} [r] = b^{I} r_{I},$
unlabeled · line 465
$L^{μ, I} (s, a) = σ_{ε} (a) A^{λ μ, I} \partial_{a^{λ}} R (s - a) .$
unlabeled · line 471
$a_{2} ˙^{α μ} [G; h] = D^{α μ, I, J} h_{I} G_{J} .$
unlabeled · line 501
$T^{L, I, J} (b; a, z) = P^{L}_{M} \int R (b - v) Q^{M, I, J} (v; a, z) d^{4} v, F = E_{m}^{A} f .$