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The Fine Structure Constant

June 8, 2020

Memes have existed for a long time prior to the Internet. One of these ancient memes is alpha and omega, in which alpha (α), the first letter of the Greek alphabet, and omega (ω), the last letter of the Greek alphabet, are combined the indicate an entirety of things from the first to the last. The alpha and omega was identified as a Christian symbol in the Bible.

Alpha has been used as the name for two fundamental things in physics. The alpha particle, which is a bare helium nucleus emitted during radioactive decay in which an element transforms into another element by shedding nuclear matter. The fine structure constant, also symbolized by alpha, is a dimensionless physical constant that specifies the strength of the interaction of charged particles. Why alpha has its particular value, and not some other, is one of the mysteries of physics.

Enrico Fermi at a blackboard

Nobel Physics Laureate, Enrico Fermi (1901-1954), at a blackboard after writing the wrong expression for the fine structure constant, alpha.

smile might indicate that this was a joke, or a test question for his audience.

(Image from the Smithsonian Institution, via Wikimedia Commons)

The fine structure constant agrees with theory to eleven decimal places. The fine structure constant is related to some rather fundamental things; namely, the elementary charge e, Planck's constant h, the speed of light c and the mathematical constant π,
α = (2 π e2)/(h c)
So, asking why α has its particular value is the same as asking why these other physical constants have their particular values.

What's just as mysterious is that the reciprocal of α is a small number quite close to 137. Its CODATA 2014 value is 137.035 999 139,[1] but it was indistinguishable from the exact value of 136 early in the 20th century. As I wrote in an earlier article (Numerology, March 2, 2015), physicist, Arthur Eddington (1882-1944), who rose to fame by his observational verification of Einstein's theory of general relativity by observing the bending of starlight around the Sun during solar eclipse of May 29, 1919, proposed an a priori reason why it should be exactly 136.

The number of protons in the universe, now called the Eddington number, is estimated to be about 1080, and Eddington conjectured that this number was actually 136 x 2256; or. later, 137 x 2256. Using 137 and a little calculation allowed Eddington to make the statement in 1939 that "...there are 15 747 724 136 275 002 577 605 653 961 181 555 468 044 717 914 527 116 709 366 231 425 076 185 631 031 296 protons in the universe and the same number of electrons."

Sir Arthur Stanley Eddington (1882-1944)

Physicist, Sir Arthur Stanley Eddington (1882-1944).

Astrometry was primitive in 1919, so Eddington's evidence in confirmation of relativity was tenuous, but sufficient to convince physicists of his era. A 1979 computer-assisted re-analysis of his original eclipse plates confirmed his conclusions.[2]

(United States Library of Congress, Prints and Photographs Division, image ggbain.38064, via Wikimedia Commons, modified for artistic effect.)

Most physicists rejected this strange numerological and teleological approach to physics that was more like the Philosophy of Aristotle (384-322 BC) and less like the observational approach of Galileo (1564-1642). A parody of this approach was published by Guido Beck, Hans Bethe, and Wolfgang Riezler in the respected physics journal, Die Naturwissenschaften.[3] Their short paper, which somehow got through peer review, related absolute zero temperature, T0 = -273 °C, to the fine structure constant,

Beck, Bethe, Riezler parody equation

Almost as interesting is the fact that the journal wants $39.95 for a copy of this 1931 paper.

If you throw sufficient mathematics into an equation, you'll get a close value for alpha, as the following "calculation" of the fine structure constant shows.[4]

Gibson's equation for the fine structure constant

in which

Gibson's equation for the fine structure constant


Gibson's equation for the fine structure constant

This is reminiscent of a 1951 paper about the ratio of proton to electron mass that now has a value of 1836.152673.[5] In this one sentence paper, Friedrich Lenz of Düsseldorf, Germany, showed that the accepted value of this constant at that time, 1836.12, was very nearly 5 (1836.11810845).[5] Just as interesting is the fact that a copy of this one sentence paper can be purchased for $35.00.[5]

The age of the universe is 13.8 billion years with an uncertainty of just 20 million years. This is one of the impressive results of the Lambda cold dark matter model. Has the fine structure constant remained constant in all that time? if α has changed, then either e, h or c would have changed, or quantum electrodynmaics is wrong.

One crude indicator of the constancy of α is that too large of a change would have prevented production of carbon by stellar nucleosynthesis, so life would not have existed in the universe. An extremely small change could have happened over 13.8 billion years, so there are frequent attempts to measure the fine structure constant of the very early universe and compare it with today's value. A recent publication in Science Advances by a huge international team of physicists and astronomers examined four direct measurements of the fine-structure constant as it was 13 billion years ago.[6-7] They discovered no evidence for a temporal change in the fine structure constant over our local value.[6-7] They did, however, find a change of α with direction at the 3.9 σ level.[6-7]

This research team was composed of members from the University of New South Wales Sydney (Sydney, Australia), the University of Leicester (Leicester, UK), the University of Cambridge (Cambridge, UK), University College London (London, UK), the Institute of Astronomy (Cambridge, UK), the University of Szczecin (Szczecin, Poland), Lawrence Berkeley National Laboratory (Berkeley, California), the Centre for Astrophysics of the University of Porto (Porto, Portugal), the Instituto de Astrofísica e Ciências do Espaço (Porto, Portugal), the University of Porto (Porto, Portugal), the Universität Hamburg (Hamburg, Germany), the Maritime University of Szczecin (Szczecin, Poland), the European Southern Observatory (Garching bei München, Germany), the Ludwig-Maximilians-Universität (Munich, Germany), and the Astronomical Observatory of Trieste (Trieste, Italy).

The research team measured the fine-structure constant in four high-redshift absorption systems, and these measurements are the highest-redshift direct measurements of alpha to date.[6] This high redshift corresponds to a time when the universe was less than a billion years old.[7] One motivation for the study was not just a look at a possible temporal change of alpha, but to further investigate past hints that alpha might be slightly different in different directions.[7] The current observations were designed to ensure that the spatial variation observed in the past was not just observational error.[7] Their 3.9 sigma validation of this result indicates a high statistical significance.

Figure 3 from Wilczynska et al., 2020

Figure 3 from ref. 6.[6] Shown are previous quasar measurements, with the combined data from the cited study appearing as the point at a redshift (z) of 5.87. No temporal trend is seen. (Licensed under the Creative Commons Attribution NonCommercial License 4.0. Click for larger image.)

Says John K. Webb of the University of New South Wales Sydney and corresponding author of the paper,
“...There could be some direction or preferred direction in the universe where the laws of physics change, but not in the perpendicular direction. In other words, the universe in some sense, has a dipole structure to it... Putting all the data together, electromagnetism seems to gradually increase the further we look, while towards the opposite direction, it gradually decreases. In other directions in the cosmos, the fine structure constant remains just that - constant."[7]
Some recent Xray observations are further indicative of this directionality of the universe.[7]


  1. Peter J. Mohr, David B. Newell, and Barry N. Taylor, "CODATA Recommended Values of the Fundamental Physical Constants:2014," arXiv, July 21, 2015.
  2. Daniel Kennefick, "Not Only Because of Theory: Dyson, Eddington and the Competing Myths of the 1919 Eclipse Expedition," arXiv, September 5, 2007
  3. G. Beck, H. Bethe, and W. Riezler, "Remarks on the quantum theory of the absolute zero of temperature," Die Naturwissenschaften, vol. 2 (1931), p. 39.
  4. Giuseppe Dattoli, "The fine structure constant and numerical alchemy," arXiv (September 9, 2010).
  5. Friedrich Lenz, "The Ratio of Proton and Electron Masses," Phys. Rev., vol. 82, no. 4 (May, 1951), pp. 554–554.
  6. Michael R. Wilczynska, John K. Webb, Matthew Bainbridge, John D. Barrow, Sarah E. I. Bosman, Robert F. Carswell, Mariusz P. Dąbrowski, Vincent Dumon, Chung-Chi Lee, Ana Catarina Leite, Katarzyna Leszczyńska, Jochen Liske, Konrad Marosek, Carlos J. A. P. Martins, Dinko Milaković, Paolo Molaro, and Luca Pasquini, "Four direct measurements of the fine-structure constant 13 billion years ago," Science Advances, vol. 6, no. 17 (April 24, 2020), Article eaay9672, DOI: 10.1126/sciadv.aay9672. This is an open access publication with a PDF file here
  7. Lachlan Gilbert, "New findings suggest laws of nature not as constant as previously thought, University of New South Wales Press Release, April 27, 2020.

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