Manuscript XVIII-A@2 · XVIII-A-v2.tex · proposition · 0f4336ec

Proposition: Complete formal extraction at order κ 2

The selected quantum coefficient and its inventory · lines 1000–1030 · proof included or linked

XVIII-A@2#prop:inventory, XVIII-A@2#eq:completecoefficient

Under ass:retarded,

[κ^{2} ℏ] R_{V (κ)} (O_{ε} (κ)) = [ℏ] (O_{2, ε} + R_{1} (V_{1}; O_{1, ε}) + R_{1} (V_{2}; O_{0, ε}) + \frac{1}{2} R_{2} (V_{1}, V_{1}; O_{0, ε})) .

Proof. Multilinearity gives

[κ^{2}] R_{1} (V; O) = R_{1} (V_{1}; O_{1, ε}) + R_{1} (V_{2}; O_{0, ε}) .

Two interaction arguments contribute $\frac{1}{2} R_{2} (V_{1}, V_{1}; O_{0, ε})$ . Three or more begin at $κ^{3}$ . The direct insertion contributes $O_{2, ε}$ . Extracting $[ℏ]$ proves the formula.

Displayed formulas in this passage

XVIII-A@2#eq:completecoefficient · line 1003
$[κ^{2} ℏ] R_{V (κ)} (O_{ε} (κ)) = [ℏ] (O_{2, ε} + R_{1} (V_{1}; O_{1, ε}) + R_{1} (V_{2}; O_{0, ε}) + \frac{1}{2} R_{2} (V_{1}, V_{1}; O_{0, ε})) .$
unlabeled · line 1020
$[κ^{2}] R_{1} (V; O) = R_{1} (V_{1}; O_{1, ε}) + R_{1} (V_{2}; O_{0, ε}) .$

Proposition: Complete formal extraction at order κ 2

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