Manuscript XIV-C@1 · XIV-C.tex · section · a34d13df

Order and normalization

J0: the pair and extension contract › Order and normalization · lines 499–540

XIV-C@1#eq:R1, XIV-C@1#eq:formal

The retarded convention is

R_{1} (V_{g}; O) = \frac{i}{ℏ} (T_{2}^{ren} (V_{g}, O) - V_{g} ⋆_{H} O) .

The requested order is

R_{λ V_{g}} O = O + λ (R_{1}^{(0)} + ℏ R_{1}^{(1)}) + O (λ^{2}, λ ℏ^{2}) .

Here $λ$ counts one additional vertex insertion. It is not used to reweight the already filled slots.

The formal two-cross-line expression is

\frac{i}{2} ⟨ V_{g}^{''} \otimes O^{''}, H_{T}^{\otimes 2} - H^{\otimes 2} ⟩ .

It remains formal until the complete graph compositions are defined. The difference is kept grouped; no undefined mixed intermediate product is declared to exist.

The inherited ordered values are

supp V_{g} strictly later than supp O ⟹ R_{1} = 0,

supp O strictly later than supp V_{g} ⟹ R_{1} = \frac{i}{ℏ} [O, V_{g}]_{⋆_{H}} .

These refer to full functional supports.

Displayed formulas in this passage

XIV-C@1#eq:R1 · line 502
$R_{1} (V_{g}; O) = \frac{i}{ℏ} (T_{2}^{ren} (V_{g}, O) - V_{g} ⋆_{H} O) .$
XIV-C@1#eq:formal · line 519
$\frac{i}{2} ⟨ V_{g}^{''} \otimes O^{''}, H_{T}^{\otimes 2} - H^{\otimes 2} ⟩ .$
unlabeled · line 510
$R_{λ V_{g}} O = O + λ (R_{1}^{(0)} + ℏ R_{1}^{(1)}) + O (λ^{2}, λ ℏ^{2}) .$
unlabeled · line 530
$supp V_{g} strictly later than supp O ⟹ R_{1} = 0,$
unlabeled · line 534
$supp O strictly later than supp V_{g} ⟹ R_{1} = \frac{i}{ℏ} [O, V_{g}]_{⋆_{H}} .$