Manuscript XIV-E@1 · XIV-E.tex · proposition · e89fd2a5

Proposition: Exact B 9 root formula

E1: exact calculation of the complete precomposition sum › The complete scalar-response root · lines 739–763 · proof included or linked

XIV-E@1#prop:root, XIV-E@1#eq:B9root

The last term of (eq:fullregroup) is

M_{9} [G; h, ψ] = \int g (a_{1}^{μν} [r_{h}] \partial_{μ} F \partial_{ν} ψ - m^{2} b_{1} [r_{h}] F ψ) d^{4} x .

Both the derivative and mass terms are retained.

Proof. Transposition of the inherited zero-data Green operator gives

\int f p [r_{h}; ψ] = - \int F (\partial_{μ} (g a_{1}^{μν} [r_{h}] \partial_{ν} ψ) + m^{2} g b_{1} [r_{h}] ψ) d^{4} x .

Compactness of $g$ permits integration by parts in the divergence. This gives (eq:B9root). No equation for $ψ$ is assumed.

Displayed formulas in this passage

XIV-E@1#eq:B9root · line 742
$M_{9} [G; h, ψ] = \int g (a_{1}^{μν} [r_{h}] \partial_{μ} F \partial_{ν} ψ - m^{2} b_{1} [r_{h}] F ψ) d^{4} x .$
unlabeled · line 755
$\int f p [r_{h}; ψ] = - \int F (\partial_{μ} (g a_{1}^{μν} [r_{h}] \partial_{ν} ψ) + m^{2} g b_{1} [r_{h}] ψ) d^{4} x .$