Manuscript VII-I@1 · VII-I.tex · section · 907e96ab

The full second-order bookkeeping

Interactions, normalization, and a controlled diagnostic › The full second-order bookkeeping · lines 1918–1944

VII-I@1#eq:four-pieces

If the necessary retarded products are constructed, then

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

The bare $O_{f}^{(2)}$ contains no separately solved interacting scalar coefficients. A second metric coefficient contributes through its affine $k$ -slot once.

Local interaction vertices belong to the ordinary polynomial domain, but mixed time-ordered products with a nonlocal prepared or sharp insertion require their own construction. The Bogoliubov formula is applicable only after those time-ordered products exist Duetsch,BDF. Free Wick closure is not that existence theorem.

Functional support reaches into $D_{f}$ , including the initial surface for the sharp object. Causal factorization cannot be based only on $supp f$ .

Displayed formulas in this passage

VII-I@1#eq:four-pieces · line 1921
$[κ^{2}] R_{V} (O_{f}) = O_{f}^{(2)} + R_{1} (V_{1}; O_{f}^{(1)}) + R_{1} (V_{2}; O_{f}^{(0)}) + \frac{1}{2} R_{2} (V_{1}, V_{1}; O_{f}^{(0)}) .$