## Thursday, January 6, 2011

### Fundamental Constant Numerology

My father was obsessed with units of measurement and fundamental constants. He got his engineering degree at the Royal Institute of Technology (Kungliga Tekniska Högskolan) in Stockholm, Sweden. His favorite professor there was Erik Hallén, who was famous for his work on antenna theory and for laying groundwork for the world's most widely used system of measurement, the SI system.

My dad nearly failed Hallén's class, which could be one reason for his lifelong obsession with units. The other reason was that Dad was convinced that he could explain some of physics' deepest questions about the nature of matter by applying the electromagnetic theory he learned in Hallén's class. Dad   explained the structure of the electron by modeling it as a circulating charge wave in a resonant cavity formed by general relativistic warping of space. In his notes, he wrote
To me it looks like all the puzzle is defined and ripe to be put together and the extension to other particles will not be difficult- only time consuming.
Dad used these insights to come up with a relationship between the gravitational constant and the mass and charge of the electron. Here's his equation:
G = 6/π 10-44 Z0 c (e/m)2
where G = the gravitational constant (which defines the force that holds the universe together), Z0 is the impedance of free space, c is the speed of light, and e and m are the charge and mass of the electron.

Here's a prettier version of that equation, made using Roger's Equation Editor from the following TeX code:
G = \frac{6}{\pi} 10^{-44} Z_0 c (\frac{e}{m})^2
(TeX is the most commonly used formatting language for mathematics.)

A derivation of this equation would easily have earned my dad a Nobel Prize, but without a derivation and explanation of the underlying physics, it was just numerology. If you plug in the numbers, Dad's equation is within 0.025% off of the consensus value for the gravitational constant, whose experimental uncertainty is about 0.01%. Dad understood that his equation was worthless without an explanation, so he spent endless hours studying Bessel equations and all sort of obscure mathematics. He was sure that somehow, somewhere, there existed a solution to Maxwell's equations combined with general relativity to explain the 6/π and confirm his resonant cavity. (He said the 10-44 was just using the right units- I never understood that!)

My dad was not alone in physics numerology. The fine structure constant, which is very close to 1/137, has attracted all sorts of numerological explanations. (Dick Lipton calls it a "miracle number") Arthur Eddington, one of the most famous physicists of his time, had an explanation for why the fine structure constant should be exactly 1/137 involving the number of protons in the universe. A more modern numerological result is that of James Gilson, whose suggested value for the fine structure constant is only 30 parts per trillion off.

While fundamental constant numerology has deservedly been on the fringes of science, new internet search technologies may soon change that. Last year, scientific publisher Springer introduced a beta service called LaTeX Search that allows researchers to search for LaTeX formatted equations in all of Springer's journals. (LaTex is a dialect of TeX most widely used for scientific publishing.) That's something you can't do with Google, or any other search engine. The ability to connect obscure mathematical discoveries from disparate fields of science could soon be facilitating new avenues of research, perhaps even new methodologies.

For example, I can search for a fragment of my dad's equation and get at least one result that seems relevant. I don't know of any meaningful discoveries that have been made so far with LaTeX search, but if my dad had been able to search all of the mathematical literature to connect his numerological result with a mathematical solution, perhaps he would have explained the gravitational constant and structure of the electron and would have won his Nobel Prize.

He would have been 83 today. Happy Birthday, Dad! We miss you.