The Secret of Room 137: Unlocking the Fine Structure Constant

In this post we explore the enigmatic journey of Swiss mathematician Armand Wyler, who attempted to unravel the fine structure constant, alpha, related to the number 137. It delves into historical attempts by physicists like Sir Arthur Eddington and Wolfgang Pauli to understand this dimensionless constant, connecting quantum theory, relativity, and electromagnetism. The post also highlights Wyler's geometric interpretation and the profound yet tragic impact of his work, which led him to a psychiatric clinic.

The Mysterious Journey of Armand Wyler

Wolfgang Pauli - Room 137

From the prestigious research institute to a psychiatric clinic—this was the price Swiss mathematician Armand Wyler paid for his attempt to unravel the geometric meaning of the fine structure constant, alpha. This mysterious dimensionless constant, whose reciprocal is close to 137, has long been the philosopher's stone for physicists. Wyler's formula was simple enough for a schoolboy to understand:

Wyler's Formula

Any student (not to mention engineers) with a calculator can easily calculate:

1/α=137.0360824482...

An Astonishing Agreement

Physicists' measurements, based on the theory of light and matter interaction, give the value of:

$1\mathrm{/}\mathrm{\alpha }= 137.03599976(50)$

The agreement is astonishing, especially since the value measured by physicists depends on the energy of the scattered electrons, so no one really knows what "exactly" it is.

Fundamental Constants of Nature

Let's start from the beginning. Physicists long ago discovered that certain values are constants. These constants, classified as fundamental, include the "Elementary charge e," "Planck's constant ℏ," and the "Speed of light c." In the commonly used system of units today, these constants have the following values:

$\mathrm{e }= 1.602176462(63)\times 1{\mathrm{0^(}}^{-\mathrm{19) }}C$

$$ℏ = 1.054571596(82) × 10^(-34) J s

$\mathrm{c }= 299,792,458\text{ m/s}$

The Magic of Alpha

There is also the "Vacuum Impedance ε₀":

ε₀ = 8.854187817...×10^(−12) F

Using the formula:

$e²/(2hε₀c)$

we get a dimensionless value, equal to $7.297352533(27)\times 1{\mathrm{0^(}}^{-\mathrm{3).}}$ This is α, the fine-structure constant. In the days of the CGS system, the expression was simpler:

$\alpha =\frac{\mathrm{e²/\hslash }c}{}$

Alpha connects quantum theory (constant h), relativity (constant c), and electromagnetism (constant e). If this constant could be calculated from purely geometric premises, it could revolutionize physics, possibly earning a Nobel Prize.

α is truly exceptional among physical constants. It is a pure number - it carries no meters or seconds or kilograms. And because it is a pure number, there is an overwhelming feeling that it must come out of pure math! Possibly of some pure geometry, like the number π.

The Eddington Affair

Astrophysicist and mystic Sir Arthur Eddington, in 1929, published speculative arguments suggesting $= 136$. In those days, measurements suggested this value. However, other physicists mocked Eddington, notably in a 1931 paper by Beck, Bethe, and Riezler, which "derived" $1\mathrm{/}\alpha$ from absolute zero temperature—a dubious joke for serious scientists.

Pauli's Obsession

Wolfgang Pauli, known for the Pauli Principle, also delved into this constant. Though he concluded Eddington's ideas were nonsense, Pauli himself was intrigued. When he was very ill, he requested room number 137 in a Zurich clinic, where he eventually died.

W. Pauli - Room 137

Quoting from Arthur I. Miller, "137: Jung, Pauli, and the Pursuit of a Scientific Obsession", W. W. Norton & Company, 2009:

"On Friday, December 5, 1958, as he was teaching his afternoon class, Pauli suddenly began to suffer excruciating stomach pains. Up until then he had been fine. The next day he was rushed to the Red Cross Hospital in Zürich. Charles Enz visited him the day after. Pauli was visibly agitated. Had Enz noticed the number of the room, he asked him?

“No,” replied Enz.

“It’s 137!” Pauli groaned. “I’m never getting out of here alive.”

When the doctors operated, they found a massive pancreatic carcinoma. Pauli died in Room 137 on December 15. His last request had been to speak to Carl Jung."

Einstein and Feynman's Insights

Einstein, in 1950, expressed his belief that there should be no dimensionless constants in the laws of nature. Richard Feynman famously said every theoretical physicist should have a sign on their door saying, "137 - how little we know."

Wyler's Geometric Interpretation

Armand Wyler

In 1968-1969, Zurich mathematician Armand Wyler published papers suggesting alpha has a geometric interpretation, related to higher-dimensional geometry. Physicists couldn't understand his mathematics but took it seriously. Freeman Dyson invited Wyler to IAS, Princeton to explain his ideas, but reportedly, Dyson understood nothing. After three months, Wyler returned home and ended up in a psychiatric clinic, publishing nothing more.

IAS Princeton

An Unexpected Encounter

I inquired AI (Perplexity, having access to online resources) about Wyler's work and his fate  after his return from IAS. Here is what I got:

#### **Focus on Pure Mathematics**

- **Topological Singularities**: Wyler's dissertation at ETH Zürich focused on topological singularities, employing rigorous mathematical tools from algebraic topology, differential topology, and singularity theory. This work was more aligned with pure mathematics and the theoretical aspects of topology.

- **Mathematical Foundations**: His work in Switzerland was heavily grounded in the mathematical foundations laid by his advisor Heinz Hopf and other prominent mathematicians. This period was characterized by a deep dive into the mathematical structures and their properties.

#### **Initial Forays into Theoretical Physics**

- **Proton/Electron Mass Ratio**: Wyler applied Clifford algebra for spinors and Green functions for the Shilov Boundaries $ Q^5 $ and $ Q^4 $, calculating the proton/electron mass ratio to be $ 6\pi^5 = 1836.118 $, which is remarkably close to the actual value[1]. This work hinted at his interest in bridging mathematics and physics.

...

Before his time at Princeton, Armand Wyler was in a stable and productive mental state, marked by academic success and innovative research. However, his experience at the Institute for Advanced Study significantly deteriorated his mental health. The hostile and isolating environment, combined with intense scrutiny and lack of recognition, led to a nervous breakdown upon his return to Switzerland. This tragic outcome highlights the importance of a supportive and inclusive academic environment for researchers, especially those working on groundbreaking and unconventional ideas.

Armand Wyler is no longer living. According to the information gathered, Wyler passed away in 1971. This is confirmed by multiple sources, including a blog post that discusses his work and mentions his tragic destiny, stating explicitly "Armand Wyler -1971" and detailing his life and contributions up until his death[1].

Citations:

[1] https://physicalgeometry.wordpress.com/2016/08/30/armand-wyler/

[2] https://www.e-periodica.ch/cntmng?pid=swo-001%3A1940%3A0%3A%3A1080

[3] https://academic.oup.com/book/25794/chapter-abstract/193393218?redirectedFrom=fulltext

[4] https://www.scirp.org/journal/paperinformation?paperid=31409

[5] https://www.math.columbia.edu/~woit/wordpress/?p=272

[6] https://en.geneanet.org/fonds/bibliotheque/?country=CHE&go=1&nom=WYLER&page=1&size=50

[7] https://cosmosmagazine.com/science/mathematics/the-number-that-fascinates-physicists-above-all-others/

[8] https://www.physicsforums.com/threads/the-7-strangest-coincidences-in-the-laws-of-nature-s-hossenfelder.1062281/

In 2007, a journalist from Zurich approached me for material on Wyler. The last message I received was that Wyler had been seen on the street. Thus we have a real mystery.

My colleague at CNRS in Marseille and I have worked on similar issues, citing Wyler's work [2,3]. Years later, Laura and I visited my colleague in Marseille and stayed in Cassis, where we were assigned room number 137. Jung would have laughed at the coincidence.

Cassis

References

1. Astrophysics, Clocks and Fundamental Constants, S. G. Karshenboim and E. Peik (Eds), Lecture Notes in Physics 648, Springer Heidelberg – New York 2004
2. Born's Reciprocity on a Conformal Domain, A. Jadczyk, in "Spinors, Twistors, Clifford Algebras and Quantum Deformations”, Z. Oziewicz, B. Jancewicz, and A. Borowiec (Eds), Kluwer Academic, Dordrecht, 1993
3. Conformal Theories, Curved Phase Spaces, Relativistic Wavelets and the Geometry of Complex Domains", R. Coquereaux, A. Jadczyk, Rev. Math. Phys. 2, (1990), p. 1-44
4. Wyler, A., On the Conformal Groups in the Theory of Relativity and their Unitary Representations, Arch. Rat. Mech. and Anal., 31:35-50, (1968)
5. Wyler, A., L'espace symetrique du groupe des equations de Maxwell' C. R. Acad. Sc. Paris, 269:743-745 (1969)
6. Wyler, A., 'Les groupes des potentiels de Coulomb et de Yukawa', C. R. Acad. Sc. Paris, 271:186-188

P.S.1. The following excerpt from the freely available and eye opening book "AGAINST THE TIDE  A Critical Review by Scientists of  How Physics and Astronomy Get Done", Martín López Corredoira &  Carlos Castro Perelman (Eds.), 2008.

In the chapter  "Is Ethics in Science an Oxymoron?", by Castro Perelman, we find an additional take on the mystery:

"A particular example of the detrimental psychological effects of the “silence treatment” on a scientist is the story about Armand Wyler's sabbatical leave from the University of Zurich to the Institute for Advanced Study at Princeton University in the early 1970's. He was invited to Princeton by F. Dyson based on his interest on Wyler’s work in bounded complex homogeneous domains and his geometrical derivation of the fine structure constant (1/137) involving the Electro-Magnetic interaction, and whose value is crucial for the emergence of life in our Universe. According to a personal communication with Frank Tony Smith (another blacklistee by the Cornell arXiv.org e-archives) the atmosphere under which Wyler had to live during his sabbatical leave at Princeton was filled with such hostile silence that Wyler felt utterly invisible. His “silence treatment” was so detrimental to Wyler’s mental health that when he returned back to Zurich he basically experienced a nervous breakdown. Even worse, it seems that F. Dyson himself might have felt this hostile silence treatment towards Wyler to the point that he gave R. Gilmore (a Group theory expert) the only copy of Wyler’s work during his Princeton stay. Wyler gave a copy of his recent work at the time (in June 1972) to Dyson. It was related to Operations of Symplectic and Spinor Groups and the Complex Light Cone. At the top of the page one can see Wyler’s warm words (in French) thanking Professor Dyson. However, in the right hand margin one can read Dyson’s words to R. Gilmore saying: “This seems to be the only copy I have. Don’t tell Wyler that I gave it to you, F. D.”. Perhaps F. Dyson did not wish to hurt Wyler’s feelings and this is why he asked R. Gilmore not to tell Wyler that he had dispensed of the only copy he had (containing the warm words of gratitude from Wyler to Dyson). Frank Tony Smith posted in his website the 57 pages of Wyler’s two articles at Princeton. This was possible since R. Gilmore gave the alleged copy to Frank Tony Smith  later  on.  One  of  the  reasons  why  Frank  Tony  Smith  has  been  blacklisted  is  because  he  extended Wyler’s  derivation  of  the  fine  structure  constant  to  the  weak  and  strong  couplings  (in  addition  to  other parameters of the Standard Model). Ideas that have been ridiculed so many times by so many very powerful individuals  (David  Gross,  for  example).   

P.S.2. Even with the help of AI I was unable to find the year of birth of Armand Wyler.

P.S.3. A complementary reading on the subject can be found in the article "The number that fascinates physicists" by Paul Davis.

P.S.4 The paper by Robert Coquereaux and myself  [3], where we, in particular, referred to the works of Armand Wyler, was recently (June 27, 2024) cited by Martin Ammon, Jakob Hollweck, Tobias Hossel, and Katharina Wolfl in "Conformal Blocks in Two and Four Dimensions from Oscillator Representations".