The Fine Structure Constant - the Curious de Vries formula

The Fine Structure Constant - Adventure Plus

Roughly 20 years ago, on October 4, 2004, Hans de Vries made a subtle yet significant contribution to physics by posting his recursive formula for the fine structure constant, α, on his Chip Architect website. This formula, which can be described as implicit or even recursive, sparked interest and debate among physicists and mathematicians alike.

Here is the formula;

Hans de Vries formula

Who is Hans de Vries?

If you’ve never heard of Hans de Vries, allow me to introduce him. According to his LinkedIn profile, de Vries is a High Tech Consultant and Physics Software Engineer with a solid background in Electrical Engineering, Computer and Telecommunication Technology. He’s also the mind behind the Physics Quest website, where he is currently working on a graduate-level physics book titled “Understanding Relativistic Quantum Field Theory.” Some chapters are available online for the curious reader.

 A Formula in the Spotlight

Fast forward to last year, and de Vries’ formula became a topic of extensive discussion, particularly in German-speaking circles and on the math.stackexchange forum. This formula caught the attention of Gudlaugur Kristinn Ottarsson (GKO), who took a serious approach to calculating α using de Vries’ formula. In July 2018, GKO published a preprint titled “The Fine Structure Constant and Discrete Calculus – GKO2018,” where he rewrote the de Vries formula in a closed form and tested it numerically.

Crunching the Numbers

Out of sheer curiosity, I decided to verify GKO’s results. Ottarsson’s method involves rewriting de Vries’ formula by taking the square roots of both sides and then putting it to the test with a high-precision calculation engine capable of handling 100 decimal digits. 

GKO version of de Vries formula

His results? The fine structure constant to 27 significant digits:

1/α = 137.035999095829799489647400 + ε ;  ε < 10^(-24).

However, GKO also noted that current experimental values are a bit perplexing. It seems we’ll need to wait a while before experiments consistently align with the numbers predicted by de Vries’ formula. For now, we can only be confident up to about 8 digits: 137.035999.

experimental values taken from Wikipedia

 Beyond that, things start to get a bit fuzzy. It’s also important to remember that α is a "running constant," meaning its value depends on the energy levels involved in the measurement, while de Vries’ formula likely provides the low-energy limit for α.

A Personal Dive into the Calculation

Never one to pass up an opportunity for some intellectual exercise, I decided to put Ottarsson’s calculations to the test using Wolfram's Mathematica. My no-brainer code churned out the exact same high-precision number that Ottarsson had obtained. 

But I didn’t stop there—I also estimated the error by limiting the series on the right-hand side to the sixth power of α. Thankfully, the series converges rapidly, much faster than a geometric series, making this task a breeze. However, these are minor details, and I’ll spare you the nitty-gritty.

Reflecting on the Formula

There are a couple of points worth mentioning. First, de Vries’ formula aligns much more closely with the experimental value of α than the mysterious Wyler’s formula, which is discussed in TheSecret of Room 137: Unlocking the Fine Structure Constant. Wyler’s formula results in 1/α=137.03608245, which is further from the mark. However, de Vries’ formula, while more accurate, doesn’t exactly unravel the deep mystery of α. It seems more like a "balance equation" inspired by the perturbative methods of Quantum Electrodynamics rather than a key to new physical insights. Unlike Wyler’s excursions into hyperdimensional physics and conformal symmetry, de Vries’ formula doesn’t open up new adventures in the realms of light and matter.

Second, de Vries’ formula isn’t a one-trick pony - it has more than one solution. By tweaking the last line of my Mathematica code, I found a second possible value of α, roughly 1576.56 times larger than the "correct" one. 

Second solution for α

The significance of this fact remains unclear. Luke Kenneth Casson Leighton noticed the multiple solutions in his paper "An Explanation of the de Vries Formula for the Fine Structure Constant" (January 2017), but he didn’t delve into the potential consequences.

While preparing this post, I kept an old saying in mind: "Follow the clues, stick to the shadows, and most importantly - trust no one." In my quest for knowledge, I checked out other papers by GKO on ResearchGate. 

https://www.researchgate.net/profile/Gudlaugur-Ottarsson

Some of the titles immediately piqued my interest:

• Unified Horizons of Electricity and Gravity

• Matter as Poisson Degeneracy in Natural Space

• The Equations of Maxwell and Gauss in Natural Space

• Operations in d+i Dimensional Natural Space and Einstein's Quest for Asymmetric Tensors

These titles are tantalizing, and I’m planning to dive into them soon. The mystery deepened when I learned that GKO’s education spans Polytechnic Engineering, Mathematical Physics, Computer Science, and Fine Art. To top it all off, he’s also an accomplishedmusician, according to Wikipedia!

Conclusion

The exploration of the fine structure constant, α, through Hans de Vries’ formula is a journey full of numbers, curiosity, and a dash of mystery. While it may not solve the enigma of α, it certainly adds a fascinating chapter to the ongoing quest to understand one of the most fundamental constants in physics. As I continue to follow the clues, I’m reminded that in science, as in life, the answers often lead to more questions—and that’s what makes the journey so exciting.

To be continued...