The Fine Structure Constant \(\alpha\) is a fundamental dimensionless physical constant that quantifies the strength of the electromagnetic interaction between elementary charged particles, such as electrons and protons. It has an approximate value of \(\frac{1}{137}\) or 0.00729735256933.

This constant appears in the equations governing fine structure splitting, which refers to the small energy level separations observed in atomic spectra. These splittings arise due to the interaction between an electron’s intrinsic spin and its orbital motion around the nucleus, an effect modulated by the strength of the electromagnetic interaction.

In 1916, Arnold Sommerfeld introduced the fine structure constant as part of his extension of the Bohr model, incorporating relativistic corrections. He defined \(\alpha\) as a parameter governing the strength of electromagnetic interactions in atomic spectra, particularly in explaining the fine structure of hydrogen. While Sommerfeld formalized its role, the numerical presence of \(\alpha\) had already been implicit in prior atomic spectroscopy research. It was demonstrated that the fine structure constant can be expressed as the ratio of the velocity of the ground-state electron in hydrogen \((v_e)\) to the speed of light \((c)\): $$ \alpha = \frac{v_e}{c} = 7.297\ 352\ 5693\ × 10^{-3}$$​

The prominence of \(\alpha\) increased with the advent of quantum electrodynamics (QED) and its formulation in the Dirac equation. With the contributions of Feynman, Schwinger and Tomonaga to QED \(\alpha\) was recognized as the fundamental coupling constant of electromagnetism, determining the interaction strength of charged particles with the electromagnetic field.

Richard Feynman famously referred to the fine structure constant \(\alpha\) as: "It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it... It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man.". Wolfgang Pauli, one of the pioneers of quantum mechanics, was also fascinated by the fine structure constant who reportedly said: "When I die, my first question to the Devil will be: What is the meaning of the fine structure constant?" .Now, more than a century after its discovery, this work presents the solution.

Sommerfeld Extension

Arnold Sommerfeld presented an extension to Bohr’s atomic model to explain the fine structure of atomic energy levels—small shifts in spectral lines that Bohr’s model could not explain. Sommerfeld introduced elliptical orbits and used corrections finding that an electron in an atom moves with a velocity \(v \approx c\). The correction modifies the hydrogen energy levels, leading to small shifts proportional to \(\alpha^2\) given by; \[E_{n,j} = - \frac{m_e c^2}{2} \left( \frac{Z\alpha}{n} \right)^2 \left[ 1 + \frac{\alpha^2 Z^2}{n} \left( \frac{1}{j + \frac{1}{2}} - \frac{3}{4n} \right) \right]\] The Fine structure splitting levels being: \[\Delta E_{\text{fine}} = \frac{m_e c^2}{2} \left( \frac{Z\alpha}{n} \right)^4 \left( \frac{1}{j + \frac{1}{2}} - \frac{3}{4n} \right)\]

Where:
\(m_e\) = electron mass.
\(c\) = speed of light.
\(Z\) = atomic numbe.r
\(\alpha\) = Fine struture constant.
\(n\) = principal quantum number.
\(j\) = total angular momentum quantum number.

Einstein Lorentz

Applying a Lorentz transformation displays the basic principle and validates Sommerfeld's extension. Albert Einstein in his 1905 paper, "The Electrodynamics of Moving Bodies", discussed the concept of \(\beta\) as the ratio of the velocity of an object \(v\) to the speed of light \(c\). $$\beta = \frac{v}{c}$$ From the postulates in the paper, he examined how the coordinates of an event in one inertial frame relate to those in another moving at a constant velocity \(v\). By demanding that the speed of light remains the same in both frames—specifically, that a light pulse expanding as a sphere in one frame remains a sphere in any other—Einstein derived the Lorentz transformation equations. In these transformations, the relative velocity \(v\) appears normalized by \(c\) which is why he introduced \(\beta=\frac{v}{c}\) a dimensionless parameter.

This normalization simplifies the transformation equations and leads directly to the definition of the Lorentz factor: $$\gamma = \frac{1}{\sqrt{1 - \beta^2}} = \frac{1}{\sqrt{1 - \left( \frac{v}{c} \right)^2}}$$ Einstein's paper assumed that light propagates as a sphere. However, due to the motion of the reference frame along the x-axis, light does not actually maintain a perfect spherical expansion but instead extends in the x-direction. The fundamental question is how the situation would unfold, as implied by Einstein, if the Lorentz transformation were applied strictly within a classical framework. The physics community is in agreement with Einstein and Lorentz's basic premise that If the Lorentz factor \(\gamma\) when \(v\) is small compared to \(c\) (i.e., \(\beta \ll 1\)), \(\gamma \approx 1\) classical mechanics is recovered. In Quantum mechanics, which is inherently classical, \(\alpha\) is the fine structure constant and it is generally agreed that the values of \(\alpha\) and \(\beta\) can be calculated from the following: $$\beta = \frac{v}{c} \quad \text{and} \quad \alpha = \frac{v}{c}$$

It is evident that both are identical, which is no coincidence. The distinction, however, lies in the fact that if the electron is regarded as wave-like, the wave front would indeed be spherical and align perfectly with Lorentz's transformation. In atomic physics (as in the Bohr model of hydrogen), it follows that the electron’s speed is: $$ v \approx \alpha c \quad \text{with} \quad \alpha \approx 7.297\ 352\ 5693 × 10^{-3}$$ This gives; $$ \alpha = \frac{v}{c} \approx 7.297\ 352\ 5693 × 10^{-3} $$ and then $$ \gamma = \frac{1}{\sqrt{1 - \beta^2}} \approx \frac{1}{\sqrt{1 - (0.0072973525693)^2}} \approx 1.0000266 $$ Even though \(\gamma\) is not exactly 1, it is very close to 1 which is precisely what Einstein meant by “recovering classical mechanics”. In classical mechanics, it is assumed that the value of \(\gamma \approx 1\) exactly, however that is an idealization, valid when \(\frac{v}{c} \rightarrow 0\). The slight deviation (e.g. \(\gamma \approx 1.0000266\) for an electron in hydrogen) is a minor correction that, for most practical purposes, can be ignored.

Conclusion

The value of the fine-structure constant is mainly determined through experimental measurements. High-precision experiments are essential for accurately determining its value, using techniques such as measuring the electron's magnetic moment or the fine-structure splitting in hydrogen. This reliance on experimental data, rather than theoretical prediction, can lead to misunderstandings. Alternatively, it is suggested that the fine-structure constant is not an arbitrary fixed scalar value but simply represents the proportionality between the orbital velocity of the ground state electron and the speed of light, exactly as suggested by Arnold Sommerfeld. Overall, the fine structure constant is a central parameter in quantum mechanics and quantum electrodynamics, governing the behavior of charged particles and the electromagnetic force. Its ubiquitous presence in these theories highlights its fundamental importance in understanding the interactions that shape the physical universe.