Manuscript XIV-D@1 · XIV-D.tex · section · 1a7f173d

A polarization of the entire observable

D1: the complete coefficient before joint composition › A polarization of the entire observable · lines 587–617

XIV-D@1#eq:polarO

Let $C (γ_{1}, γ_{2}; ψ)$ be the symmetric polarization in the metric arguments of $A_{2}^{sc} (γ, 0; ψ)$ :

C (γ_{1}, γ_{2}; ψ) = \frac{1}{2} [A_{2}^{sc} (γ_{1} + γ_{2}, 0; ψ) - A_{2}^{sc} (γ_{1}, 0; ψ) - A_{2}^{sc} (γ_{2}, 0; ψ)] .

Define symmetric bilinear metric-valued maps

D (γ_{1}, γ_{2}) = \frac{1}{2} [r_{D} [γ_{1} + γ_{2}] - r_{D} [γ_{1}] - r_{D} [γ_{2}]],

G (ψ_{1}, ψ_{2}) = - \frac{1}{2} P E_{0}^{R} (g T (ψ_{1}, ψ_{2})) .

Then the complete observable is

O = C (γ, γ; ϕ) + A_{1} (D (γ, γ); ϕ) + A_{1} (G (ϕ, ϕ); ϕ) .

This formula includes both legs of each filled slot.

Displayed formulas in this passage

XIV-D@1#eq:polarO · line 609
$O = C (γ, γ; ϕ) + A_{1} (D (γ, γ); ϕ) + A_{1} (G (ϕ, ϕ); ϕ) .$
unlabeled · line 591
$C (γ_{1}, γ_{2}; ψ) = \frac{1}{2} [A_{2}^{sc} (γ_{1} + γ_{2}, 0; ψ) - A_{2}^{sc} (γ_{1}, 0; ψ) - A_{2}^{sc} (γ_{2}, 0; ψ)] .$
unlabeled · line 598
$D (γ_{1}, γ_{2}) = \frac{1}{2} [r_{D} [γ_{1} + γ_{2}] - r_{D} [γ_{1}] - r_{D} [γ_{2}]],$
unlabeled · line 604
$G (ψ_{1}, ψ_{2}) = - \frac{1}{2} P E_{0}^{R} (g T (ψ_{1}, ψ_{2})) .$