In December 1924 Wolfgang Pauli proposed the idea of an inner degree of freedom of the electron, which he insisted should be thought of as genuinely quantum mechanical in nature. Shortly thereafter Ralph Kronig and a little later Samuel Goudsmit and George Uhlenbeck took up a less radical stance by suggesting that this degree of freedom somehow corresponded to an inner rotational motion, though it was unclear from the very beginning how literal one was actually supposed to take this picture, since it was immediately recognised (already by Goudsmit and Uhlenbeck) that it would very likely lead to serious problems with Special Relativity if the model were to reproduce the electron’s values for mass, charge, angular momentum, and magnetic moment. However, probably due to the then overwhelming impression that classical concepts were generally insufficient for the proper description of microscopic phenomena, a more detailed reasoning was never given. In this contribution I shall investigate in some detail what the restrictions on the physical quantities just mentioned are, if they are to be reproduced by rather simple classical models of the electron within the framework of Special Relativity. It turns out that surface stresses play a decisive role and that the question of whether a classical model for the electron does indeed contradict Special Relativity can only be answered on the basis of an exact solution, which has hitherto not been given.
In December 1924 Wolfgang Pauli proposed the idea of an inner degree of freedom of the electron, which he insisted should be thought of as genuinely quantum mechanical in nature. Shortly thereafter Ralph Kronig and a little later Samuel Goudsmit and George Uhlenbeck took up a less radical stance by suggesting that this degree of freedom somehow corresponded to an inner rotational motion, though it was unclear from the very beginning how literal one was actually supposed to take this picture, since it was immediately recognised (already by Goudsmit and Uhlenbeck) that it would very likely lead to serious problems with Special Relativity if the model were to reproduce the electron’s values for mass, charge, angular momentum, and magnetic moment. However, probably due to the then overwhelming impression that classical concepts were generally insufficient for the proper description of microscopic phenomena, a more detailed reasoning was never given. In this contribution I shall investigate in some detail what the restrictions on the physical quantities just mentioned are, if they are to be reproduced by rather simple classical models of the electron within the framework of Special Relativity. It turns out that surface stresses play a decisive role and that the question of whether a classical model for the electron does indeed contradict Special Relativity can only be answered on the basis of an exact solution, which has hitherto not been given.
In December 1924 Wolfgang Pauli proposed the idea of an inner degree of freedom of the electron, which he insisted should be thought of as genuinely quantum mechanical in nature. Shortly thereafter Ralph Kronig and, independently, Samuel Goudsmit and George Uhlenbeck took up a less radical stance by suggesting that this degree of freedom somehow corresponded to an inner rotational motion, though it was unclear from the very beginning how literal one was actually supposed to take this picture, since it was immediately recognised (already by Goudsmit and Uhlenbeck) that it would very likely lead to serious problems with Special Relativity if the model were to reproduce the electron's values for mass, charge, angular momentum, and magnetic moment. However, probably due to the then overwhelming impression that classical concepts were generally insufficient for the proper description of microscopic phenomena, a more detailed reasoning was never given. In this contribution I shall investigate in some detail what the restrictions on the physical quantities just mentioned are, if they are to be reproduced by rather simple classical models of the electron within the framework of Special Relativity. It turns out that surface stresses play a decisive role and that the question of whether a classical model for the electron does indeed contradict Special Relativity can only be answered on the basis of an exact solution, which has hitherto not been given.
Abstract We have seen in the preceding episode that early in 1925 Pauli attributed to the electron a ‘classically not describable two-valuedness’. Within a few months this property of the electron was given a physical description which, it is true, is not satisfactory from the viewpoint of classical physics. This description is regarded as the discovery of the intrinsic angular momentum or spin of the electron. In fact the description was given twice. Already in January Kronig discussed it with Pauli who was completely opposed to the idea so that Kronig did not publish it. The concept of the electron spin was independently proposed and published [1] by Uhlenbeck and Goudsmit who thus became the discoverers of spin.
The origin of the spin concept was probably the most complicated of all the quantum physics. Indeed, after Bohr's triumph in 1913, Bohr and Sommerfeld's "old quantum theory'', based on the idea of quantum restrictions applied to classical magnitudes, gave a unified explanation of the spectroscopic data. Atomic physics shows that it is not possible to account for the observed effects if we assume that the electron, a point particle, and the three degrees of freedom of translation in space that we have considered up to now. In quantum relativistic mechanics, the structure of the Lorentz group reveals the kinetic momentum to any particle as an intrinsic attribute, which is defined in the same way as its electric charge and its mass. There are several possible representations of the states, accompanied by corresponding representations of the observables, whose use can be more or less convenient according to the problem under consideration.
Most students of science and engineering know that every elementary particle, such as electrons, neutrons, photons, neutrinos, etc., has a quantum mechanical property called “spin” which can be measured (perhaps not easily, but at least in principle) and has a quantized value, including zero. The vast majority of these students mentally visualize spin as the angular momentum associated with the elementary particle spinning or rotating about its own axis (like a top or a planetary object). This mental picture, although convenient and comforting, is actually somewhat crude and certainly incomplete. Landau and Lifshitz, in their classic textbook on quantum mechanics [1], wrote “[the spin] property of elementary particles is peculiar to quantum theory. [It] has no classical interpretation... It would be wholly meaningless to imagine the ‘intrinsic’ angular momentum of an elementary particle as being the result of its rotation about its own axis.” The simplistic notion of self-rotation about an axis, shown in Fig. 1.1, can- not explain many features of spin, such as why its magnitude cannot assume continuous values and why it is quantized to certain specific values. It also causes serious problems if taken too literally. As we will see later (Problem 1.2), if we think of an electron as a solid sphere of radius equal to the Lorentz radius e2/(4πǫ0m0c 2) (where e is the electron’s charge, m0 is the mass, c is the speed of light in vacuum, and ǫ0 is the dielectric constant of vacuum), then the velocity on the surface of a rotating electron would have to be many times the velocity of light in vacuum if such a rotation were to generate an angular momentum equal to the electron’s spin. Obviously that would not be permitted by the theory of relativity. Indeed, a deep understanding of quantum mechanics is required to understand how the spin property comes about. Its origin is in relativistic quantum mechanics and really was first appreciated by Paul Andrew Maurice Dirac when he derived the Dirac equation which is the cornerstone of relativistic quantum mechanics. Richard Feynman, noted for his distaste for mysticism in physics, wrote about the notion of “spin”: “It appears to be one of the few places in physics where there is a rule which can be stated very simply, but for which no one has found a simple and easy explanation. The explanation is down deep in relativistic quantum mechanics.
In 1925 the concept of electron spin was introduced to explain two experimental observations: the splitting of silver atom beams in an inhomogeneous magnetic field (Stern–Gerlach, 1922), and the fine structure of atomic spectra. The physical picture proposed — an electron spinning on its own axis — was immediately shown by Hendrik Lorentz to require surface velocities exceeding the speed of light. The paper was published anyway. Dirac subsequently rescued the two-valuedness mathematically, whereupon the physical question was declared irrelevant. For one hundred years the field has been unable to state what spin physically is. This paper traces the sequence of observations, inferences, and decisions that produced this situation and identifies two distinct wrong turns. The first wrong turn was publishing a model known to be physically impossible. Lorentz's calculation was correct — at the classical electron radius (2.818 fm), the required surface velocity is 274 times the speed of light. But the classical radius is not the electron's physical rotation radius. The correct rotation radius, derived from the Sagnac closure condition, is 285.5 fm — 101 times larger. At that radius, the rotation velocity is 0.822c, the bare magnetic moment is 0.608 μB, and Lorentz's objection does not arise. Both values are computed explicitly with no free parameters. The second wrong turn was a conflation. The closure geometry of the electron involves two independent numerical facts, both related to the number ½: (1) the least-work oscillation path that closes the Sagnac phase condition is a two-lobe ellipse, giving a winding number of n = ½ in the mass equation; (2) the electron's S³ closure surface presents two poles to the external field, giving a topological factor of 2 in the g-factor. In 1925, both facts were numerically present in the spectroscopic data, but they were collapsed into a single postulate: intrinsic angular momentum ℏ/2. That collapse created a half-integer angular momentum with no classical analogue, forced Dirac to construct the SU(2) spinor formalism to accommodate it, and caused the spinor to appear as a new kind of mathematical object requiring new physics — when it was always a description of closure surface topology. Every experimental result attributed to spin-½ as an intrinsic quantum property — two spots in Stern–Gerlach, the electron magnetic moment, Pauli exclusion, and fine structure — is reproduced by real rotation with two stable orientations, without superluminal motion, without abstract mathematical structure disconnected from physical content, and without the hundred-year admission of ignorance that has accompanied the standard treatment. The mathematical apparatus of quantum mechanics — the Dirac equation, the Pauli matrices, the SU(2) structure — remains valid. What changes is the interpretation. The spinor is not a mysterious new object. It is the natural representation of the two-pole topology of the S³ closure surface. The conflation of 1925 prevented this from being seen. Related work: The Sagnac closure condition and the γcause invariant used in this paper are derived in: D. J. Hallman, "The Sagnac Formula Inverted Reveals Mass, Gravity, and Particle Structure," Zenodo (2026). doi:10.5281/zenodo.20193279