The Sommerfeld-Dirac Paradox Reexamined

In 1916 Arnold Sommerfeld derived a relativistic version of the Bohr hydrogen atom: both being Newtonian particle models. It correctly accounted for the observed "fine splitting" in the hydrogen spectrum. A decade later, Dirac derived the first relativistic wavefunction model of the hydrogen atom by incorporating Pauli spin matrices. This created a paradox: How could Sommerfeld possibly have obtained the correct spectrum without including (the then inconceivable notion of) electron spin?

Surprisingly, opinions about resolving this paradox have been varied and contentious during the intervening decades, with most physicists who are aware of the paradox assuming Sommerfeld just made a lucky mistake. Other luminaries, like Heisenberg and Weinberg, have suggested that the resolution might reside in the underlying dynamical symmetries. We offer a new perspective by introducing a discriminator, comprised of the appropriate quantum numbers, that leads to what we call the Über-Relativistic Approximation (URA). Viewed in this setting, it becomes clear that Sommerfeld did not make a mistake. Rather, the URA reveals that what appears to be the same energy eigenvalue formula mathematically for both Sommerfeld and Dirac models, is actually two different approximations that share some numerical values in common. This framework also enables us to explain why prior opinions have been so varied and incomplete.

References

A. Sommerfeld, Atombau und Spektrallinien, Friedrich Vieweg und Sohn, Braunschweig, 1919

M. Born, Mechanics of the Atom, Ungar, 1960 (Orig. German 1924)

P. G. Bergmann, Introduction to the Theory of Relativity, Dover 1976 (Orig. pub. 1942)

H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Springer-Verlag, 1957

W. Heisenberg, "Ausstrahlung von Sommerfelds Werk in der Gegenwart," Physikalische Blätter, (24) 530--537, 1968

S. Weinberg, Gravitation and Cosmology, Wiley, 1972

N. J. Gunther, “Dynamical Symmetry Breaking in The Lewis-Riesenfeld Oscillator”, Journal of Mathematical Physics, vol. 21, no. 7, 286, 1980

L. C. Biedenharn, “The ‘Sommerfeld Puzzle’ Revisted and Resolved,” Foundations of Physics, vol. 13, no. 1, 1983

Y. Granovskii, “Sommerfeld Formula and Dirac’s Theory,” Uspekhi Fiz Nauk, 47(5) 523-524, 2004