Manuscript I-B-VP@3 · I-B-VP-v3.tex · theorem · 93c04eb4

Theorem: Friedrichs induced current

Exact Friedrichs observables and analytical checks · lines 1239–1280 · proof included or linked

I-B-VP@3#thm:current

For one charged complex field,

⟨ j_{φ} ⟩_{ren} = - \frac{e β ( 1 - β ) ( 1 - 2 β )}{12 π ^{2} α ^{2} r ^{2}},

⟨ j_{φ^} ⟩_{ren} = - \frac{e β ( 1 - β ) ( 1 - 2 β )}{12 π ^{2} α ^{3} r ^{3}}, j_{t} = j_{r} = j_{z} = 0.

It is conserved and independent of $ξ$ . With the fractional-part convention it is periodic in flux, odd under $β \mapsto 1 - β$ , and zero at $0, 1/2$ .

Proof. Time split at $Δ φ = 0$ . The direct azimuthal term vanishes. Differentiating (eq:modes),

\partial_{Δ φ} G_{F, E} = \frac{i Σ ( a )}{8 π ^{2} α r ^{2} sinh η},

where

Σ (a) = n \sum q_{n} e^{- ∣ q_{n} ∣ a} = - \frac{d}{d a} \frac{sinh (( 1/2 - β ) a )}{sinh ( a /2 )} = \frac{β ( 1 - β ) ( 1 - 2 β )}{3} a + O (a^{3}) .

Since $\partial_{φ} - \partial_{φ^{'}} = 2 \partial_{Δ φ}$ , (eq:current-prescription) gives the result. The other components vanish by the time and longitudinal evenness and radial endpoint symmetry. Conservation follows from $j^{r} = 0$ and angular independence. The remaining properties follow from the polynomial and mode relabeling.

Displayed formulas in this passage

unlabeled · line 1242
$⟨ j_{φ} ⟩_{ren} = - \frac{e β ( 1 - β ) ( 1 - 2 β )}{12 π ^{2} α ^{2} r ^{2}},$
unlabeled · line 1247
$⟨ j_{φ^} ⟩_{ren} = - \frac{e β ( 1 - β ) ( 1 - 2 β )}{12 π ^{2} α ^{3} r ^{3}}, j_{t} = j_{r} = j_{z} = 0.$
unlabeled · line 1261
$\partial_{Δ φ} G_{F, E} = \frac{i Σ ( a )}{8 π ^{2} α r ^{2} sinh η},$
unlabeled · line 1266
$Σ (a) = n \sum q_{n} e^{- ∣ q_{n} ∣ a} = - \frac{d}{d a} \frac{sinh (( 1/2 - β ) a )}{sinh ( a /2 )} = \frac{β ( 1 - β ) ( 1 - 2 β )}{3} a + O (a^{3}) .$

Theorem: Friedrichs induced current

Displayed formulas in this passage

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