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The physical interpretation of the fine-structure constant $\alpha\approx \frac{1}{137.036...}$

by Luca Varani, Federico Intini, Lino Reggiani

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Authors (as registered SciPost users):

Luca Varani

Abstract

The fine-structure constant $\alpha$ is a fundamental, dimensionless constant in physics, introduced by Sommerfeld in 1916, that is still an empirical parameter in the Standard Model for physical particles, the most advanced theory of modern physics. Its numerical value 1/137 is continuosly receiving wide attention in respect of the accuracy of its value, now claimed to stay at 11 digits in fair agreement with the Sommerfeld expression. This number was physically interpreted as a way of quantifying the strength of the electromagnetic interaction. Since then, several physicists (like Dirac, Pauli, Feynman and others) expressed their unsatisfaction of this situation waiting for a more direct interpretation of this number. Here we address this problem by giving a direct microscopic meaning to 137. We show the existence of a one-to-one correspondence between 137 and the variance of the photon average number $\overline N$ inside a black-body of given volume and temperature. The explicit expression writes: $\overline{\delta N^2} = \gamma \overline{N} = \frac{1}{\alpha}$ with the variance and the average determined by the Planck's law applied to a cubic cavity of side $L=3.3$~cm exact and an equilibrium temperature of 0.516~K corresponding to the equivalent energy of the measured fine-structure line of about 45~$\mu$eV, with $\gamma =1.37$ being the Fano factor. As a broader perspective we propose that any atomic spectral line be associated with a structure-constant that provides the statistical information on the photon gas inside the corresponding black-body. From this point of view, the expression of the Sommerfeld constant is wieved as a particular case that coincides exactly with our results when the emission line is of 45~$\mu$eV. Present findings open new experimental possibilities to measure the statistical properties of thermal photons and in particular of the fine-structure constant by using a recently developed counting-statistics technique.

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