Manuscript VII-H@1 · VII-H.tex · theorem · d20d2a9f

Theorem: Explicit invariant quantum response

An exactly evaluated nonzero invariant · lines 1046–1085 · proof included or linked

VII-H@1#thm:main, VII-H@1#eq:main-value

For every admissible prescription in Definition def:prescriptions,

⟨ R^{P}, F_{*} ⟩ = - \frac{ℏ t _{0}}{2 π ^{2}} \neq = 0.

This value is unchanged by every finite normalization change in the complete class, not only by a selected regulator conversion.

Proof. Since $F_{*} (0) = 0$ , Theorem thm:ambiguity removes every allowed difference. The common value can be evaluated with the punctured cone kernel.

For fixed $t > 0$ , integrate spatially in polar coordinates and put $u = t^{2} - r^{2}$ . Near $u = 0$ ,

4 π r^{2} d r = 2 π r ∣ d u ∣.

The $δ^{'}$ pairing therefore gives

\int_{R^{3}} δ^{'} (t^{2} - r^{2}) (t^{2} - r^{2}) b (t) b (r) d^{3} x = - \frac{d}{d u} (2 π t^{2} - u u b (t) b (t^{2} - u))_{u = 0} = - 2 π t b (t)^{2} .

Multiply by $ℏ/ (4 π^{3})$ , integrate in $t$ , and apply (eq:bump-moments).

Displayed formulas in this passage

VII-H@1#eq:main-value · line 1050
$⟨ R^{P}, F_{*} ⟩ = - \frac{ℏ t _{0}}{2 π ^{2}} \neq = 0.$
unlabeled · line 1067
$4 π r^{2} d r = 2 π r ∣ d u ∣.$
unlabeled · line 1071
$\int_{R^{3}} δ^{'} (t^{2} - r^{2}) (t^{2} - r^{2}) b (t) b (r) d^{3} x = - \frac{d}{d u} (2 π t^{2} - u u b (t) b (t^{2} - u))_{u = 0} = - 2 π t b (t)^{2} .$

Theorem: Explicit invariant quantum response

Displayed formulas in this passage

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