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

Flat and flux limits

Exact Friedrichs observables and analytical checks › Flat and flux limits · lines 1354–1380

At fixed $r > 0$ , for $β = 0$ ,

P_{0} = \frac{μ}{π} + O (μ^{2}), C_{0} = \frac{2 μ}{π} + O (μ^{2}),

hence

\frac{⟨ ϕ ^{2} ⟩ _{ren}}{μ} ⟶ \frac{1}{48 π ^{3} r ^{2}}, \frac{⟨ T ^{t} _{t} ⟩ _{ren}}{μ} ⟶ \frac{1}{720 π ^{3} r ^{4}} (ξ = 1/6) .

These are not uniform apex or wave-front limits.

At fixed nonzero $β$ , $α \to 1$ retains the punctured flat-space Aharonov–Bohm effect:

F ⟶ - \frac{β ( 1 - β )}{8 π ^{2} r ^{2}},

and the complex conformal tensor coefficient tends to $- β^{2} (1 - β)^{2} / (24 π^{2} r^{4})$ . The conformal energy may have either sign as the parameters vary.

Displayed formulas in this passage

unlabeled · line 1357
$P_{0} = \frac{μ}{π} + O (μ^{2}), C_{0} = \frac{2 μ}{π} + O (μ^{2}),$
unlabeled · line 1362
$\frac{⟨ ϕ ^{2} ⟩ _{ren}}{μ} ⟶ \frac{1}{48 π ^{3} r ^{2}}, \frac{⟨ T ^{t} _{t} ⟩ _{ren}}{μ} ⟶ \frac{1}{720 π ^{3} r ^{4}} (ξ = 1/6) .$
unlabeled · line 1373
$F ⟶ - \frac{β ( 1 - β )}{8 π ^{2} r ^{2}},$