Manuscript VI@1 · VI.tex · section · b5b60afb

Coupling order is not loop order

Second-order retarded bookkeeping and the missing quantum step › Coupling order is not loop order · lines 1974–2004

VI@1#eq:two-point-expansion

A declaration of “order $κ^{2}$ ” does not specify a loop order. Classical source responses, tree contributions, and radiative contributions can occur at that coupling order. A quantum statement must also specify its $ℏ$ -degree or graph-loop convention.

For example, suppose fields

A_{f} = A_{f}^{(0)} + κ A_{f}^{(1)} + κ^{2} A_{f}^{(2)}

have already been constructed in an algebra with a fixed state $ω$ . Then the coefficient of their connected two-point function is

[ω (A_{f} A_{ℓ})_{c}]_{κ^{2}} = ω (A_{f}^{(2)} A_{ℓ}^{(0)})_{c} + ω (A_{f}^{(0)} A_{ℓ}^{(2)})_{c} + ω (A_{f}^{(1)} A_{ℓ}^{(1)})_{c} .

If the state itself is varied perturbatively, its expansion contributes additional terms. Equation (eq:four-terms) for one insertion therefore does not by itself specify a two-point response vector.

Figure fig:VI-four-term-prerequisites separates the conditional coefficient identity from the product and state data required for a quantum response.

Displayed formulas in this passage

VI@1#eq:two-point-expansion · line 1988
$[ω (A_{f} A_{ℓ})_{c}]_{κ^{2}} = ω (A_{f}^{(2)} A_{ℓ}^{(0)})_{c} + ω (A_{f}^{(0)} A_{ℓ}^{(2)})_{c} + ω (A_{f}^{(1)} A_{ℓ}^{(1)})_{c} .$
unlabeled · line 1982
$A_{f} = A_{f}^{(0)} + κ A_{f}^{(1)} + κ^{2} A_{f}^{(2)}$

Coupling order is not loop order

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