Manuscript Em01-E@2 · Em01-E-v2.tex · section · 54f8f8c2

Source incidence and retained rows

Retained internal formulas and local existence › Source incidence and retained rows · lines 2641–2720

Em01-E@2#eq:K47, Em01-E@2#eq:K5plus, Em01-E@2#eq:K5minus, Em01-E@2#eq:K6, Em01-E@2#eq:K8, Em01-E@2#eq:K9der, Em01-E@2#eq:K9mass

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Set

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

Q^{L, I, J} (v; a, z) = α, β \sum q_{α β}^{L, I, J} (v) \partial_{v}^{α} δ_{D}^{(4)} (v - z) \partial_{v}^{β} δ_{D}^{(4)} (v - a),

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

The seven retained rows are

K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{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) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(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) .

Define

K_{ret} = K_{47} + K_{5, +} + K_{5, -} + K_{6} + K_{8} + K_{9, der} + K_{9, mass} .

These definitions reproduce the retained convention attributed to Em01-B and Em01-C in the supplied materials Em01B,Em01C. No physical plateau or full source theorem is inferred merely from the printed formulas.

Displayed formulas in this passage

Em01-E@2#eq:K47 · line 2664
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-E@2#eq:K5plus · line 2664
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-E@2#eq:K5minus · line 2664
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-E@2#eq:K6 · line 2664
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-E@2#eq:K8 · line 2664
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-E@2#eq:K9der · line 2664
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-E@2#eq:K9mass · line 2664
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
unlabeled · line 2644
$L^{μ, I} (s, a) = A^{λ μ, I} \partial_{a^{λ}} R (s - a),$
unlabeled · line 2649
$Q^{L, I, J} (v; a, z) = α, β \sum q_{α β}^{L, I, J} (v) \partial_{v}^{α} δ_{D}^{(4)} (v - z) \partial_{v}^{β} δ_{D}^{(4)} (v - a),$
unlabeled · line 2657
$T^{L, I, J} (b; a, z) = P^{L}_{M} \int R (b - v) Q^{M, I, J} (v; a, z) d^{4} v .$
unlabeled · line 2711
$K_{ret} = K_{47} + K_{5, +} + K_{5, -} + K_{6} + K_{8} + K_{9, der} + K_{9, mass} .$

Source incidence and retained rows

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