Manuscript I-B-VP-1@1 · I-B-VP-1-v1.tex · proposition · 6d700be9

Proposition: Exact finite conversions

Simultaneous extensions and the exact Ward repair › The direct cutoff prescription need not obey the Ward equations · lines 754–804 · proof included or linked

I-B-VP-1@1#prop:conversion, I-B-VP-1@1#eq:convertV, I-B-VP-1@1#eq:convertQ, I-B-VP-1@1#eq:convertS, I-B-VP-1@1#eq:rawdefects

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The direct finite parts satisfy

\frac{1}{4} Δ U_{L} - \frac{1}{2} \partial_{i} U_{L} \frac{1}{2} K_{ij} U_{L} = V_{L}^{c} - \frac{σ _{α}}{4} Δ δ_{W}, = Q_{L, i}^{c} + \frac{σ _{α}}{4} \partial_{i} δ_{W}, = S_{L, ij}^{c} + σ_{α} (\frac{7}{16} δ_{ij} Δ - \frac{3}{8} \partial_{i} \partial_{j}) δ_{W} .

In particular,

\partial_{i} S_{L, ij}^{c} = - \frac{σ _{α}}{16} \partial_{j} Δ δ_{W}, S_{L, ii}^{c} = - 2 V_{L}^{c} .

Proof. Profile independence permits the sharp cutoff in this comparison. Twice integrating by parts on $r > ε$ , and expanding the boundary test jets, gives

\partial_{i} U_{L} \partial_{i} \partial_{j} U_{L} = R_{L} [\partial_{i} r^{- 2}] - \frac{σ _{α}}{2} \partial_{i} δ_{W}, = R_{L} [\partial_{i} \partial_{j} r^{- 2}] - \frac{3 σ _{α}}{4} \partial_{i} \partial_{j} δ_{W} - \frac{σ _{α}}{8} δ_{ij} Δ δ_{W} .

The boundary computations are detailed in Appendix app:boundary. Their trace gives (eq:convertV); the first identity gives (eq:convertQ); subtracting the traced second identity gives (eq:convertS). Taking a divergence of (eq:convertS) proves the first identity in (eq:rawdefects). The second follows either by tracing the regular angular kernel before applying the linear finite-part operation, or directly from the moment subtractions.

Displayed formulas in this passage

I-B-VP-1@1#eq:convertV · line 757
$\frac{1}{4} Δ U_{L} - \frac{1}{2} \partial_{i} U_{L} \frac{1}{2} K_{ij} U_{L} = V_{L}^{c} - \frac{σ _{α}}{4} Δ δ_{W}, = Q_{L, i}^{c} + \frac{σ _{α}}{4} \partial_{i} δ_{W}, = S_{L, ij}^{c} + σ_{α} (\frac{7}{16} δ_{ij} Δ - \frac{3}{8} \partial_{i} \partial_{j}) δ_{W} .$
I-B-VP-1@1#eq:convertQ · line 757
$\frac{1}{4} Δ U_{L} - \frac{1}{2} \partial_{i} U_{L} \frac{1}{2} K_{ij} U_{L} = V_{L}^{c} - \frac{σ _{α}}{4} Δ δ_{W}, = Q_{L, i}^{c} + \frac{σ _{α}}{4} \partial_{i} δ_{W}, = S_{L, ij}^{c} + σ_{α} (\frac{7}{16} δ_{ij} Δ - \frac{3}{8} \partial_{i} \partial_{j}) δ_{W} .$
I-B-VP-1@1#eq:convertS · line 757
$\frac{1}{4} Δ U_{L} - \frac{1}{2} \partial_{i} U_{L} \frac{1}{2} K_{ij} U_{L} = V_{L}^{c} - \frac{σ _{α}}{4} Δ δ_{W}, = Q_{L, i}^{c} + \frac{σ _{α}}{4} \partial_{i} δ_{W}, = S_{L, ij}^{c} + σ_{α} (\frac{7}{16} δ_{ij} Δ - \frac{3}{8} \partial_{i} \partial_{j}) δ_{W} .$
I-B-VP-1@1#eq:rawdefects · line 773
$\partial_{i} S_{L, ij}^{c} = - \frac{σ _{α}}{16} \partial_{j} Δ δ_{W}, S_{L, ii}^{c} = - 2 V_{L}^{c} .$
unlabeled · line 786
$\partial_{i} U_{L} \partial_{i} \partial_{j} U_{L} = R_{L} [\partial_{i} r^{- 2}] - \frac{σ _{α}}{2} \partial_{i} δ_{W}, = R_{L} [\partial_{i} \partial_{j} r^{- 2}] - \frac{3 σ _{α}}{4} \partial_{i} \partial_{j} δ_{W} - \frac{σ _{α}}{8} δ_{ij} Δ δ_{W} .$

Proposition: Exact finite conversions

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