Why 137 is the Most Magical Number in Science?

This article is based on the 2009 book "137: Jung, Pauli and the Pursuit of a Scientific Obsession," written by Arthur I. Miller, an emeritus professor of history and philosophy of science at University College London. In his book, he has explained why Carl Jung (world-renowned psychiatrist) and Wolfgang Pauli (theoretical physicist and one of the pioneers of quantum physics) had an obsession with the number 137.

The Fine-structure Constant

To physicists, 137 is the approximate denominator of the fine-structure constant (1/137.03599913), the measure of the strength of the electromagnetic force that controls how charged elementary particles such as the electron and muon interact with photons of light, according to the National Institute of Standards and Technology. The fine-structure constant is one of the key physical constants of the universe. "This immutable number determines how stars burn, how chemistry happens, and even whether atoms exist at all," as Michael Brooks explained in a recent New Scientist article.

The fine-structure constant "characterizes the strength with which matter couples to light, e.g. the probability that an excited atom will decay in a certain time," Paul Davies, Regents Professor of Physics at Arizona State University and a best-selling author of 30 books on science, explains If the constant was bigger, "atoms would decay faster. It is significant too because it is a pure number – a ratio of quantities with equal units. Unlike, say, the speed of light, which is either 186,000 miles per second or 300,000 kilometers per second, depending on which units you prefer." (Davies wrote this 2016 article on the fine-structure constant for Cosmos.)

In this video, British physicist Laurence Eaves explains that if the fine structure constant was a different value, "physics, chemistry, biochemistry would be totally different – and we might not be around to talk about it."

But practically from the time of its discovery in 1915 by German physicist Arnold Sommerfeld, who originally rendered it as 0.00729, the fine-structure constant seemed to signify some larger metaphysical truth as well. The fine-structure constant "determines the distance between an atom's spectral lines, which are the atom's DNA," Miller explains. "And so it is one of those numbers that are at the root of the universe. If it were any other value then the structure of matter would be very different, and so us too. People began referring to it as a mystical number."

Miller continues: "The language of the spectra – the spectral lines where Sommerfeld found it – is a true music of the spheres within the atom," he wrote. "People asked why it has this particular value. Physicists could only conclude that it cannot have this value by accident. It is 'out there,' independent of the structure of our minds."

But in 1929, English astrophysicist Arthur Eddington – who played a key role in establishing the validity of Albert Einstein's General Theory of Relativity and was an early advocate of the Big Bang theory, among other things – began expressing it as 1/137. He also saw it as having larger, spiritual implications. "Arthur Eddington sought a new mysticism which would emerge from the natural sciences," Miller says. "Perhaps, he thought, the clue lay in numbers, particularly the number 137. Eddington's reputation as one of the great astrophysicists of his day put a great deal of weight on this approach."

The fine-structure constant of physics continues to convince esotericists that the universe has numerological fine-tuning: for example, the age of the universe could be considered as roughly 13.7 times 1 billion years, though newer estimates put this value at 13.8.

Relativity and Quantum Theory

As Miler's book details, Austrian-born quantum physics pioneer Wolfgang Pauli became fascinated with the number as well since it figured in the mysterious intersection of relativity and quantum theory that he explored with the help of his friend, psychoanalyst Carl Jung. The scientific fixation on the fine-structure constant was such that in 1936, Nature published an article titled "The Mysterious Number 137."

But as Pauli learned in the 1950s from a religious scholar, 137 had another significance. It was the number associated with the Kabbalah, an esoteric form of Jewish mysticism – what Miller calls "an extraordinary link between mysticism and physics."

In "Deciphering the Cosmic Number" Miller explored how Carl Jung analysed the dream imagery of one of his most famous patients, the groundbreaking physicist Wolfgang Pauli. Pauli’s unconventional and wildlife brought him to the brink of a mental breakdown. He obsessed over how he had made his greatest discovery, feeling that he had tapped into something beyond physics. Jung’s and Pauli’s was a truly unique meeting of the minds. It was, as Jung wrote, to lead both of them into “the no man’s land between Physics and the Psychology of the Unconscious…the most fascinating yet the darkest hunting ground of our times.” Another interesting coincidence is that the hospital room in which Pauli died was also numbered 137.

Physicist Leon M. Lederman numbered his home near Fermilab '137' based on the significance of the number to those in his profession. Lederman expounded on the significance of the number in his book The God Particle: If the Universe Is the Answer, What Is the Question? noting that not only was it the inverse of the fine-structure constant, but was also related to the probability that an electron will emit or absorb a photon—i.e., Feynman's conjecture. He added that it also "contains the crux of electromagnetism (the electron), relativity (the velocity of light), and quantum theory (Planck's constant). It would be less unsettling if the relationship between all these important concepts turned out to be one or three or maybe a multiple of pi. But 137?" The number 137, according to Lederman, "shows up naked all over the place", meaning that scientists on any planet in the universe using whatever units they have for charge or speed, and whatever their version of Planck's constant may be, will all come up with 137 because it is a pure number. Lederman recalled that Richard Feynman had even suggested that all physicists put a sign in their offices with the number 137 to remind them of just how much they do not know.

In the Bohr model, the innermost electron of an atom with Z = 137 would be orbiting just below the speed of light, and the next element (Z = 138) would be "impossible". Since the Bohr model does not include either quantum mechanics or special relativity, the fact that it breaks down in this regime is not unexpected. However, such large atoms (if their nuclei were stable) could be expected to behave rather differently from a naive extrapolation of trends in the periodic table. For a finite nuclear volume, the limit is raised to about Z = 172.

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