Why Do 137 Electron-Positron Pairs In Their Compton Wavelength Have Electrostatic Potential About Equal The Mass Energy of One Pair?

Question

Why does taking the ratio of the mass energy of an electron positron pair to their electrostatic potential energy at their Compton wavelength yield a number very close to $\alpha^{-1} \approx 137$:

$$137 \approx \frac{2 m_e c^2}{\frac{k_e e^2 2 m_e c}{\hbar}}=\frac{k_ee^2}{\hbar c}=\frac{e^2}{4\pi\epsilon_0 \hbar c}=\alpha^{-1}$$

One interpretation is that 137 pairs may be packed into that small of a space without pair production draining potential energy, but 138 would be an unstable count. While I suspect this has something to do with Freeman Dyson's argument in "Divergence of Perturbation Theory in Quantum Electrodynamics", there the count is of the number of terms of summation (perhaps related to Borel resummation) -- not energy density or pair production therefrom.

Answer 1

This should not come as a surprise since the fine-structure constant is defined by $$\alpha=\frac{k_ee^2}{\hbar c}$$ where $k_e=\frac{1}{4\pi\epsilon_0}$ is Coulomb's constant.