The Fine-Structure Constant

 Yesterday I came across Quanta Magazine article titled:

Physicists Nail Down the ‘Magic Number’ That Shapes the Universe

The article continued:

... the fine-structure constant, denoted by the Greek letter α (alpha), comes very close to the ratio 1/137 ... The constant is everywhere because it characterises the strength of the electromagnetic force affecting charged particles such as electrons and protons. “In our everyday world, everything is either gravity or electromagnetism. And that’s why alpha is so important,” said Holger Müller, a physicist at the University of California, Berkeley. Because 1/137 is small, electromagnetism is weak; as a consequence, charged particles form airy atoms whose electrons orbit at a distance and easily hop away, enabling chemical bonds. On the other hand, the constant is also just big enough: Physicists have argued that if it were something like 1/138, stars would not be able to create carbon, and life as we know it wouldn’t exist.

Today, in a new paper in the journal Nature, a team of four physicists led by Saïda Guellati-Khélifa at the Kastler Brossel Laboratory in Paris reported the most precise measurement yet of the fine-structure constant. The team measured the constant’s value to the 11th decimal place, reporting that \( \alpha \) = 1/137.03599920611. (The last two digits are uncertain.) With a margin of error of just 81 parts per trillion, the new measurement is nearly three times more precise than the previous best measurement in 2018 by Müller’s group at Berkeley, the main competition. (Guellati-Khélifa made the most precise measurement before Müller’s in 2011.) Müller said of his rival’s new measurement of alpha, “A factor of three is a big deal. Let’s not be shy about calling this a big accomplishment.”

As can be seen, even with the latest refinements to the value of \( \alpha \), the value is still very close to 1/137 and so the attention tends to focus on the number 137 itself rather than its reciprocal. The question is asked: what makes 137 so special? The physicists Richard Feynmann and Wolfgang Pauli were intrigued by this the fine-structure constant. Here is a short but informative video about the constant:

We learn from the video that the constant is dimensionless and that it can be expressed in terms of three other fundamental constants. We can write:$$ \alpha=\frac{e^2}{\hbar \,c} \approx \frac{1}{137.03599920611}$$where \(e\) is the charge on the electron, \( \hbar\) is Plank's constant divided by 2\(\pi\) and \(c\) is the speed of light. It's important to note the \(e\) in this formula is not the famous mathematical constant that is sometimes called Euler's number. Another way to write the previous result is as:$$\alpha^{-1} =\frac{\hbar \,c}{e^2}\approx 137.03599920611$$There is an interesting concluding quote from this source:

In his essay, “Pauli (Wolfgang) 1900-1958,” Charles Enz (Pauli’s last assistant) writes; “This number 137 symbolised for Pauli the link with the magic world of the alchemists which has so much fascinated him.”. Recalling his preoccupation with the fine-structure constant and research on synchronicity with Carl Jung, Pauli was moved upon finding his room number was 137 at the Red Cross hospital during his last days when pineal activation can happen in transition.

 The abstract for this source states that:

Wolfgang Pauli was influenced by Carl Jung and the Platonism of Arnold Sommerfeld, who introduced the fine-structure constant. Pauli’s vision of a World Clock is related to the symbolic form of the Emerald Tablet of Hermes and Plato’s geometric allegory otherwise known as the Cosmological Circle attributed to ancient tradition. With this vision Pauli revealed geometric clues to the mystery of the fine-structure constant that determines the strength of the electromagnetic interaction. A Platonic interpretation of the World Clock and the Cosmological Circle provides an explanation that includes the geometric structure of the pineal gland described by the golden ratio. In his experience of archetypal images Pauli encounters the synchronicity of events that contribute to his quest for physical symmetry relevant to the development of quantum electrodynamics.

The article is largely incomprehensible to me but it at least shows that this constant touches on many other areas outside of particle physics. Arthur I Miller has written a book titled 137 in the paperback. On the author's website, he writes:

In Deciphering the Cosmic Number I explore how Carl Jung analysed the dream imagery of one of his most famous patients, the ground-breaking physicist Wolfgang Pauli. Pauli’s unconventional and wild life 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.

It’s the story of two mavericks – Pauli, a scientist who – unlike his peers – was fascinated by the inner reaches of his own psyche and not afraid to dabble in the occult; and Jung, the famous psychologist who nevertheless was sure that science held answers to some of the questions that tormented him. Both made enormous and lasting contributions to their fields. But in their many conversations over dinner and wine at Jung’s Gothic mansion on the shores of Lake Zurich, they went much further, striking sparks off each other as they explored the middle ground between their two subjects.

They deliberated at great length over whether there was a number that everything in the universe hinged on, that explained everything – a primal number that provided insight into the equations of the soul. Might it be three as in the Trinity? Or four as argued in alchemical texts? Could it be the weird number 137, which on the one hand described the DNA of light and on the other is the sum of the Hebrew letters of the word “Kabbalah”?

Deciphering the Cosmic Number is a tale of an extraordinary friendship between two equally brilliant yet very different men. 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.”

There is a radio interview, first broadcast on 2nd May, 2009, in which Arthur I Miller is talking to Gene Heinemeyer about Deciphering the Cosmic Number: link.

I found a link to an article that provides a more mathematical treatment of the constant. The author cites various quartic equations that have a close approximation of the constant as a root. Two of these are:$$x^4-136x^3-136x^2-818x+1=0 \text{ with }x=137.03599916816339$$ $$x^4-137x^3-10x^2+697x-365=0 \text{ with }x=137.03599916836927$$Here is the abstract for this article:

The fine-structure constant, which determines the strength of the electromagnetic interaction, is briefly reviewed beginning with its introduction by Arnold Sommerfeld and also includes the interest of Wolfgang Pauli, Paul Dirac, Richard Feynman and others. Sommerfeld was very much a Pythagorean and sometimes compared to Johannes Kepler. The archetypal Pythagorean triangle has long been known as a hiding place for the golden ratio. More recently, the quartic polynomial has also been found as a hiding place for the golden ratio. The Kepler triangle, with its golden ratio proportions, is also a Pythagorean triangle. Combining classical harmonic proportions derived from Kepler’s triangle with quartic equations determine an approximate value for the fine-structure constant that is the same as that found in our previous work with the golden ratio geometry of the hydrogen atom. These results make further progress toward an understanding of the golden ratio as the basis for the fine-structure constant.

I have little idea what all that means but it shows the interconnectedness of this constant. 

UPDATE: February 4th 2022

Here is another informative YouTube video about topic: