137 - Arthur I. Miller [124]
The fine structure constant is entirely different. Even though it is made up of these three fundamental constants, it is simply a number, because the dimensions of the charge of the electron, Planck’s constant, and the speed of light cancel out. This means that in any number system it will always be the same, like pi which is always 3.141592…. So why is the fine structure constant 137? Physicists could only conclude that it cannot have this value by accident. It is “out there,” independent of the structure of our minds.
Never before in the history of modern science had a pure number with no dimensions been found to play such a pivotal role. People began referring to it as a “mystical number.” “The language of the spectra”—the spectral lines, where Sommerfeld had found it—“is a true music of the spheres within the atom,” he wrote.
Arthur Eddington and his mania for 137
In 1957, when Pauli was fifty-seven, he wrote to his sister Hertha:
I do not believe in the possible future of mysticism in the old form. However, I do believe that the natural sciences will out of themselves bring forth a counter pole in their adherents, which connects with the old mystic elements.
Perhaps the clue lay in numbers—more specifically, the number 137.
Until 1929, the fine structure constant was always written 0.00729. That year the English astrophysicist Arthur Eddington had a bright idea. He tried dividing 1 by 0.00729. The result was 137.17, to two decimal place accuracy. The actual measurement of the fine structure constant as ascertained in the laboratory, Eddington pointed out, was close to that (it was somewhere between 1/137.1 and 1/137.3). There were two ways in which scientists determined the fine structure constant. When they calculated it from the measured values of the charge of the electron, Planck’s constant, and the velocity of light, the result was 0.00729. Or they could measure the actual fine structure of spectral lines—that is, determine it in the laboratory; the most frequent value that resulted was 0.007295 ± .000005. However, the latter method required input from a particular theory and this presented a problem, for the theory of how electrons interacted with light—quantum electrodynamics—was still in flux, as indicated by the difficulties Heisenberg and Pauli experienced in their work on this very subject. With this in mind, Eddington felt justified in throwing numerical accuracy to the wind and writing the fine structure constant simply as 1/137.
Eddington had a strong mystical streak. To him, mysticism offered an escape from the closed logical system of physics. “It is reasonable to inquire whether in the mystical illusions of man there is not a reflection of an underlying reality,” he mused. Like Pauli, he struggled with the dichotomy between the two worlds, both equally invisible, of science and the spirit. He was sure that mathematics was the key that would open the door between these two worlds and he set about an obsessive quest to derive 137 however he could.
The equation for the fine structure constant is
and the charge of the electron, e, appears in this equation as e × e, or e2. As a result, besides being a measure of the fine structure of spectral lines, the fine structure constant (1/137) also measures how strongly two electrons interact.
Eddington argued that according to relativity theory, particles cannot be considered in isolation but only in relation to each other and therefore any theory of the electron has to deal with at least two electrons. Applying a special mathematics that he had invented, Eddington found that each electron could be described using sixteen E-numbers (E stood for “Eddington”). Multiplying 16 by 16 gave a total of 256 different ways in which electrons could combine with each other. He then showed that, of these 256 ways, only 136 are actually possible; 120 are not. He wrote this mathematically as 256 = 136 + 120. Like pulling a rabbit out of a hat, he thus magically produced the number 136 from purely mathematical