Manuscript X-B@3 · X-B-v3.tex · section · 2d415d58

The explicit truncation

B3: first-order Ward identity and the second-order boundary › The explicit truncation · lines 1226–1259

X-B@3#eq:truncation, X-B@3#eq:classical-A2

The physical first-vertex correction is $κ R_{1}$ , so

O_{κ} + κ R_{1} (V_{1, g}; O_{κ}) = O^{(0)} + κ A_{1} + κ^{2} (O^{(2)} + R_{1} (V_{1, g}; O^{(1)})) (mod κ^{3}) .

The classical coefficient is instead

A_{2}^{cl} = O^{(2)} - \int f X_{1} \cdot \partial ρ_{1, g} d^{4} x + \int f ρ_{2, g} d^{4} x,

with $ρ_{2, g}$ displayed in (eq:rhotwo).

The last term is not supplied by the present map. A full quantum completion would require the second-vertex insertion and the iterated first-vertex insertion. Neither is constructed here. Their classical sum is displayed, not replaced by a contact and not assigned the name of a constructed quantum map.

The bare $O^{(2)}$ is already admitted. Its classical transformation is retained through the copied metric and coordinate laws. That fact does not complete the second-order dynamics.

Displayed formulas in this passage

X-B@3#eq:truncation · line 1229
$O_{κ} + κ R_{1} (V_{1, g}; O_{κ}) = O^{(0)} + κ A_{1} + κ^{2} (O^{(2)} + R_{1} (V_{1, g}; O^{(1)})) (mod κ^{3}) .$
X-B@3#eq:classical-A2 · line 1239
$A_{2}^{cl} = O^{(2)} - \int f X_{1} \cdot \partial ρ_{1, g} d^{4} x + \int f ρ_{2, g} d^{4} x,$

The explicit truncation

Displayed formulas in this passage

Nearby in this section