Two elementary electric charges \(e\) interact. From the Coulomb force \(F(r)= e^2/ 4 \pi \epsilon_0 r^2\) between them, which decreases as \(1/r^2\), the fine structure constant is defined as:

\(\alpha= \frac{F(r) \ r^2 }{\hbar c}\). This gives \(\alpha = \frac{e^2}{ 4 \pi \epsilon_0 \hbar c}\) , which is Sommerfeld's formula in SI units. The value of \(\alpha\) is measured to be about 1/137.04 and describes the strength of the Coulomb force, or, equivalently, the strength of electromagnetism.

Here, \(c\) is the speed of light, and \(\hbar\) is Planck's constant, and \(r\) is the distance between the charges. Now, a researcher proposes the following argument in two steps:

1. The Coulomb force \(F(r) \) between two elementary charges, i.e. the force due to exchanging *virtual* photons, is surely larger than than the force between two *neutral* (microscopic) *black holes* that appears when they exchange *thermal* photons.

Both forces are electromagnetic in origin, and we neglect gravity effects completely, by assuming extremely small bodies throughout, in the microscopic regime, with negligible masses and/or negligible gravitational interaction. We also assume that the charges are of the same sign, and that the forces are thus repulsive. In particular, the thermal photons from one hot particle push against the other particle and thus induce a force.

2. This last force can be calculated. It is given as

\(F_{BHTP}= \frac{P_{BH}}{c} \frac{A_{BH}/ 2}{ 4 \pi \ r^2 }\)

where \(P_{BH}\) is the power of the black hole Bekenstein-Hawking radiation, \(A_{BH}\) the surface of the black hole, and \(r\) the distance between the two black holes (which is assumed much larger than the black holes, so that space can be approximated as flat).

If we insert the formulae for \(P_{BH}\) and \(A_{BH}\) from the wikipedia (article on Hawking radiation), we get

\(\alpha > \frac{F_{BHTP} \ r^2}{ \hbar c} = \frac{1}{ 7680 \pi} = \frac{1}{ 24127,...}\)

Since alpha needs to be smaller than 1 for many reasons (a simple one: probabilities are always smaller than one), we have deduced:

\(1 > \alpha > \frac{1}{ 24127,...}\)

which is not a great result, but still a result that is better than nothing.

Is this argument correct? Or is there a mistake? In particular, is it correct to assume that

F_(Coulomb) > F _ (neutral black holes)

in the case of negligible gravitation? The algebra of the calculations is correct, but is the argument correct? I have never seen a limit of alpha, ever, and this one, even though it is a rough one, is the first I have ever seen deduced from physical formulae. The issue is really whether the force comparison is valid also for microscopic hot bodies and microscopic black holes. What do you think? The force sequence indirectly assumes that microscopic particles might be "hot"- so it is questionable. But still intriguing.

All this is not my own idea, nor of a friend of mine: I found it via the wikipedia entry on the fine structure constant in this paper by John P. Lestone: https://arxiv.org/abs/physics/0703151 I simplified the argument in the paper, took away all misleading issues, and corrected the mistakes (there are two factors of 4 pi lost in the paper). The idea is already ten years old but I only found it a few days ago.

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A few comments:

A. The expression for \(F_{BHTP}\) might not be an equality, because the effective surface might be different and not exactly given by \(A_{BH}/2\).

B. This reasoning does for electrodynamics the same that Verlinde did for gravity: both assume that a macroscopic force is in fact due to many microscopic degrees of freedom.

C. Also the strength of gravity is conventionally defined using the above expression \(\alpha_G= \frac{F(r) \ r^2 }{\hbar c}\)