Manuscript Em01-C@3 · Em01-C-v3.tex · section · f0953702

The complete seven-row sum

Component conventions and the retained internal expression › The complete seven-row sum · lines 587–643

Em01-C@3#eq:Kret, Em01-C@3#eq:K47, Em01-C@3#eq:K5plus, Em01-C@3#eq:K5minus, Em01-C@3#eq:K6, Em01-C@3#eq:K8, Em01-C@3#eq:K9der, Em01-C@3#eq:K9mass

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Define

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

where

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) 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_{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) .

These formulas reproduce the retained architecture Em01B,XIVH,XIVJ. Its physical interpretation is that $K_{47} = K_{4} + K_{7}$ ; both placements of $K_{5}$ and both terms of $K_{9}$ are displayed separately. The seven rows are bookkeeping rows, not seven new interaction branches.

Displayed formulas in this passage

Em01-C@3#eq:Kret · line 590
$K_{ret} = K_{47} + K_{5, +} + K_{5, -} + K_{6} + K_{8} + K_{9, der} + K_{9, mass},$
Em01-C@3#eq:K47 · line 597
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-C@3#eq:K5plus · line 597
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-C@3#eq:K5minus · line 597
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-C@3#eq:K6 · line 597
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-C@3#eq:K8 · line 597
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-C@3#eq:K9der · line 597
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-C@3#eq:K9mass · line 597
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) σ_{ε} (z) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) σ_{ε} (a) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$

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