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

Energy and positivity are additional questions

What this completion does and does not provide › Energy and positivity are additional questions · lines 1244–1271

I-B-VP-1@1#eq:energy

For a static observer on $M$ , the measured energy density is $- a / r^{4}$ . Its cutoff transverse integral per unit longitudinal length is

\int_{ε < r < R} - \frac{a}{r ^{4}} α r d r d φ = - \frac{σ _{α} a}{2} (ε^{- 2} - R^{- 2}) .

It generally diverges. Likewise,

\int_{ε < r < R} F^{\circ} α r d r d φ = σ_{α} s lo g (R / ε) .

A finite distributional action on smooth compact tests is not a proof of a finite unrenormalized energy per unit length.

If $s > 0$ , no positive distribution on the completion can restrict to $s / r^{2}$ with this density: a positive distribution is a locally finite measure, whereas every neighbourhood of the apex has infinite regular $s / r^{2}$ mass. The constructed finite part is therefore not a positive measure. This does not contradict positivity of the ground-state two-point function on regular tests.

Displayed formulas in this passage

I-B-VP-1@1#eq:energy · line 1249
$\int_{ε < r < R} - \frac{a}{r ^{4}} α r d r d φ = - \frac{σ _{α} a}{2} (ε^{- 2} - R^{- 2}) .$
unlabeled · line 1257
$\int_{ε < r < R} F^{\circ} α r d r d φ = σ_{α} s lo g (R / ε) .$