The various formulae look like they were inspired by the hallucinations of a mathematical poet. Among the simplest is a peculiar polynomial in π: 4π3 + π2 + π. Another takes the algebraic form (9/16π4)(π5/5!)1/4, and a third involves the 10th and 33rd prime numbers, 29 and 137, and takes the shape (29/π)cos(π/137)tan(π/29·137). A further formula in the collection is the strikingly simple (π2 + 1372)1/2.
What draws these formulae together is that they are all remarkably close approximations to a number that physicists know very well: the so-called fine structure constant, α = e2/ħc, which plays a fundamental role in the quantum theory of electrodynamics. Arnold Sommerfeld first introduced α into physics in 1916 when proposing relativistic corrections to Bohr's model of the atom — corrections that gave 'fine structure' to its energy levels and to atomic spectra, hence the name. Later, during the development of quantum field theory, α emerged as a far more profound dimensionless number linking quantum theory with electrodynamics and relativity.
Even so, its value still finds no first-principles explanation in any theory of physics. “All good theoretical physicists”, Richard Feynman once remarked, “put this number up on their wall and worry about it. Is it related to π or perhaps to the base of natural logarithms? Nobody knows. It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding.”
Hence the allure of “cabbalistic” (Sommerfeld's word) efforts to find numbers, like those listed above, that come far closer to either α or 1/α than would seem possible by accident. For example, the inverse of (9/16π4)(π5/5!)1/4 is equal to 137.036076, which is very close to 1/α as measured in experiment (roughly 137.0359997). Even so, this isn't quite as close as (π2 + 1372)1/2 = 137.036016. And the trigonometric formula given above is even better than either of these, being equal to the experimental value of α to about 1 part in 1011. The very simple 4π3 + π2 + π doesn't do too badly either for 1/α, coming in at 137.036297.
But if these numbers work, it seems that no one really has a clue as to why. Blended from equal parts low integers, prime numbers and transcendental numbers such as π, they're mathematical concoctions that may possibly point to deep truths we don't yet understand. Or which may instead point only to the ability of the human mind to dupe itself (indeed, “numerical alchemy” is the nice phrase used by the author of a recent exploration of the topic; see Giuseppe Dattoli, arXiv:1009.1711). How seriously should we take these seemingly amazing formulae?
It's one of the greatest damn mysteries of physics.
Some of the derivations behind these numbers involve gravity and hierarchical structures in the Universe, whereas others focus on analyses of quantum waves travelling around atoms. Arthur Eddington got it all going in the 1930s, speculating that the reciprocal of the fine-structure constant should be equal to the expression 1/2n2(n2 − 1) + 1 for the case n = 4, which is exactly 137. He based this on his conviction that α should be directly linked to the total number of particles in the Universe, which he thought he could estimate on other grounds. It's not clear that any of this made much sense, but Eddington's ideas fell out of favour a few years later when experiments showed α to depart from the perfect 1/137.
The first formula I mentioned above, (9/16π4)(π5/5!)1/4, was proposed by Swiss mathematician Armand Wyler in 1969, based on arguments involving the conformal group — the group consisting of ordinary Lorentz transformations, plus others such as spacetime dilations that leave the Maxwell equations invariant. It seems that Freeman Dyson was so impressed by this formula that he sought Wyler out at the time in an attempt to understand his reasoning, but was ultimately unable to understand Wyler's argument.
The result for α appears at the very end of the second paper, reported there as the “coefficient of the Green function of the Dirac equation in momentum space”. In the paper, the formula is linked to the transformation of volume elements in a five-dimensional space, but just how this pertains to α — which plays the role of a coupling constant, and Wyler doesn't even mention interactions between particles — is far from clear to me. Elsewhere, it turns out, Wyler also reported another incredible formula, pointing out that 6π5 = 1836.118 is incredibly close to the ratio of proton and electron masses. This does seem quite remarkable. (One report suggests that, tragically, Wyler went mad soon after completing this work and was institutionalized — perhaps because he could understand something extremely profound yet could not explain it to others.)
But Wyler's formula isn't the most accurate. That title goes to the trigonometric formula listed above, proposed in the 1990s by mathematician James Gilson. Looking through the justification he provides (at http://www.maths.qmul.ac.uk/~jgg/gilg.pdf, for example) I must say I'm even less convinced than by Wyler's arguments. Gilson considers the relativistic motion of an extended linear object around a point. The calculation bears a crude resemblance to early efforts to understand quantum phenomena largely in classical terms. I'm certainly willing to believe that a few rare gems of insight implicit in the older formalism could well have been passed over in the historical rush to change, but Gilson moves to the right number in a way that's never quite clear or convincing. It reminds me of the child's trick for dividing 64 by 16: one merely notes that for 64/16 the 6 in the numerator should cancel the 6 in the denominator, so that 64/16 immediately becomes 4/1 = 4, the correct answer.
But perhaps my suspicion is misplaced. Perhaps this formula and Wyler's formula really are pointing us to something deep and still unknown. The nub of the matter, I suppose, remains our near total ignorance of just how difficult it is to find relatively simple formulae that come close to any number we might choose. Given the number of conceivable rudimentary formulae, algebraic, trigonometric, continued fractions, and so on, might it actually be easy — perhaps almost certain given persistence? Given the amazing power of the human mind to find order in complete randomness, I lean towards the latter possibility, and suspect that much of this work, however alluring, really is equivalent to numerical alchemy. Although Wyler's strange formulae, in particular, do keep me wondering.
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Buchanan, M. Think of a number. Nature Phys 6, 833 (2010). https://doi.org/10.1038/nphys1839