The number 137.035...
You might also wish to view 137.035... in context on my numbers page, where I discuss the less controversial properties of the number.
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137.035999177(21)...
137.035999177(21)... is a recent (2022) value for the fine-structure constant (or more precisely, its reciprocal), originally called the Sommerfeld fine-structure constant and often referred to by the Greek letter α (alpha). It is a dimensionless constant in physics. The "(21)" at the end of the value "137.035999177(21)" represents the relative uncertainty in the value, which can be thought of as an "error range". The actual, unknown precise value is "probably" within the range 137.035999156 to 137.035999198.
Generally the best known value is found at NIST CODATA, the authoritative source on such things. CODATA, the Committee on Data for Science and Technology, performs calculations to reconcile differences between measurements in different experiments which correlate (determine and/or depend on each other) to varying degrees, producing "recommended" values that are usually better than those from any single experiment up to that point in time [34]. However these calculations and recommendations are only done every 5 years or so, and in the intervening period new experimental work can produce better values for certain constants.
Such was the case with α for a few years from 2009 to 2011, when CODATA still had the July 2006 figure, 1/137.035999679(93) but the Gabrielse et al. results ([20], [21], [23]) were more recent and more accurate. Gabrielse et al. [20] describes the error.
The CODATA 2010 value was published in June 2011, but was based on work on or before 2010 Dec 31 (see [29]). As such, it does not include more recent results such as the 2010 work by Bouchendira et al. [30] using new direct experimental measurements (87Rb rubidium recoil velocity), which was published in Feb 2011. To address this, the following comments were published on 2011 June 28th:
Fine-structure constant α. An improved measurement of the electron magnetic moment anomaly ae, the discovery and correction of an error in its theoretical expression, and an improved measurement of the quotient h/m(87Rb) have led to a new 2010 value of α with a relative standard uncertainty ur = 3.2×10−10 compared to ur = 6.8×10−10 of the 2006 value. Perhaps more significant, because of the correction of the error in the theory, the 2010 value of α shifted significantly and now exceeds the 2006 value by 6.4 times the ur of that value. This change has rather profound consequences, because many constants depend on α.
Here is a summary of α values from original research (e.g. [17], [23], [30]) along with CODATA least-squares reduction over the past 50 years:
Table of Fine Structure Constant Values
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notes
1: gives date of publication; actual date of work is usually
earlier, most notably with CODATA items
2: this CODATA result was soon shown to be in error, see
[20]
The actual α constant is the reciprocal (0.00729735...) of the more familiar "137" value, because the former is more relevant to its use. Expressing this constant as 137.03... goes back to Eddington (as discussed below). Like its actual "exact" value and its integer approximation 137, each of these approximations has a small fan base (also discussed below).
History
This looks 'shopped. I can tell from dimensional analysis
and from having seen lots of electrons in my day.
In the 10-15 year period following Einstein's development of general relativity, much work was done to try to unify the theories of electromagnetism, quantum mechanics and relativity. The fine-structure constant showed up in many formulas modeling electromagnetic phenomena, and it always showed up as a unitless expression involving several other already-accepted physical constants. For example, it was shown to be the ratio between the speed of the electron in its orbit in a classical Bohr atom and the speed of light. It is called the "fine-structure constant" because applying general relativity to the Bohr atom model explains the "fine structure" of the lines in hydrogen's spectrum, and the precise value of the electron's speed determines the width of the bands in the spectrum and other more easily measurable phenomena.
Using the methods of Hughes and Kinoshita the value has been computed quite accurately; the process includes experimental measurement combined with numerical integration of a large number of functions describing many different virtual particle interactions (each with a distinct Feynman diagram). The part involving Feynman diagrams can be approximated with first-order, second-order, third-order, etc. approximations, according to how many virtual event pairs take place. The 1st-order approximation involves just one Feynman diagram, with a single event pair (in which the electron emits and reabsorbs a photon). Its calculation can be done on one page in about 1/2 hour. For a 2nd order approximation there are 7 diagrams; for 3rd order there are 72. For many years the best approximation was 4th order involving 891 Feynman diagrams, and took "many supercomputers over more than 10 years"2 to evaluate. Each "order" reduces the error in the calculation by a factor of about 137, so a 4th-order estimate is off by about 1 part in 1374. The numbers {1, 7, 72, 891, 12672, ...} are Sloane's sequence A005413. It and other related sequences (with lots of example Feynman diagrams) are discussed in the 1978 paper by Cvitanovic, et. al.4.
Some have suggested that the constant may vary over time as the universe evolves. This can be tested by close measurement of the relative isotope abundances in a natural nuclear fission reactor, such as the one in Oklo, Gabon, Africa. This was attempted by two Los Alamos scientists5. They initially (in 2004) concluded that the "constant" may have decreased by 45 parts per billion over the last 2 billion years. Later they refined their model and the threat to the constancy of α was gone. In 2010 the idea came back in a substantially stronger form: evidence that α varies throughout space as well as time, differing by as much as one part on 105 between opposite ends of the visible universe in the distant past (see [26] and [27]).
The Anthropic Principle Approach
Even if α is found to be constant throughout the entire past, present and future history of the observable universe, there is still the possibility of more space and time beyond that which we observe. The anthropic principle addresses the possibility of many regions of space-time (possibly disconnected from each other, or perhaps merely separated by enough distance as to be beyond each other's event horizons). In each of these regions, there is some combination of fundamental constants resulting in different physical laws and properties of matter. The anthropic principle states that, since life can only arise in an area where the physical constants are conducive to the existence of life, the sole explanation for the values we measure is that we are here to measure them.
Roger Penrose summarized it thus [12]:
"The argument can be used to explain why the conditions happen to be just right for the existence of (intelligent) life on the earth at the present time. For if they were not just right, then we should not have found ourselves to be here now, but somewhere else, at some other appropriate time. This principle was used very effectively [to explain] various striking numerical relations that are observed to hold between the physical constants (the gravitational constant, the mass of the proton, the age of the universe, etc.). [...] [In] any other epoch, so the argument ran, there would be no intelligent life around in order to measure the physical constants in question — so the coincidence had to hold, simply because there would be intelligent life around only at the particular time that the coincidence did hold!"
This approach has the overwhelming advantage of satisfying Occam's razor: the principle that, given a choice of explanations, the simplest explanation is usually the proper one. As stated by Isaac Newton:
We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.
Nevertheless, natural human curiosity motivates physicists to find explanations for everything, and non-physicists, lacking an understanding of the field, simply attack the numbers themselves. So we end up with attempted explanations of the fine structure constant and other such numbers, even if the result is ridiculously unwieldy and not falsifiable.
The Cult of 137
The fine-structure constant holds a special place among cult numbers: unlike its more mundane cousins 17 and 666, the fine-structure constant seduces otherwise sedate people (usually untrained in the field) into seeking mystical truths and developing uncollaborated theories3. The fact that it is unitless, like π, seems to make people think that it should have some precise mathematical value. And, rather than embrace the idea that it is independent of the other unitless constants, they feel they need to express α in terms of some kind of formula. In 1979, physicist Richard Feynman gave a series of lectures in Auckland New Zealand [10] on quantum electrodynamics. In the 4th lecture, at about 19:30 in the video, he begins talking about the Cult of 137 phenomenon:
His later book QED: The Strange Theory of Light and Matter, summarizes it thus:
[...] It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it. Immediately you would like to know where this number for a coupling comes from: is it related to pi 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 by man. [...]
The "Cult of 137" began with scientists who already had quite a reputation, including Wolfgang Pauli and Karl Jung [24], Werner Heisenberg, and most notably Arthur Stanley Eddington. For a while Eddington worked on proving that 1/α was exactly 136 (see the entry on Eddington number); when measurements became more accurate he was compelled (by his own desire, not by others) to redo this work to "prove" it was 137. This type of thing has affected all members of the "Cult of 137". Any attempt to explain the "magic number" in concrete terms is a hypothesis which is falsifiable: it can be tested by more accurate experimental measurements and calculations, or related research; and as it so happens, all the 137-cultists' predictions have proven wrong.
In order to do any real work on explaining these constants, one needs to be able to evaluate the proposed explanation against modern models and theories of physics. To do that one would need to be familiar with:
- Lie groups and Lie algebras
- the Lorentz transformation as used in special relativity
- the Poincaré group of Minkowski space
- Particle physics and representation theory
- Quantum field theory
However, Eddington to some extent, and his successors in later years to a much greater extent, have not bothered with that and instead approach the dimensionless physical constants as mathematical truths in and of themselves, like pi and the Feigenbaum constant, that have a more universal meaning outside of any physical manifestation.
In more recent years the tradition has spread to the larger community of science theory hobbyists, who usually construct expressions involving integers and functions that are taught in high school, for example:
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All this might not seem quite so amazing when you realize that there are many nifty formulas for any number. For example, my RIES program finds this in less than 4 seconds:
1/α ≈ ππe/2 + √e^3-1 = 137.035999746803...
(using the command ries -l3 -x 137.035999679). That was within the 2006 error bounds but is now known to be wrong. But you only need to search a little deeper (I used ries -l5 137.035999084 -NSCTA -Ox -x), to get this answer in about a minute:
1/α ≈ 5×(32 - 1/9 - e3/2) = 137.0359990927541203314341...
here is another:
1/α ≈ 7(e-1/(e×81/π)) = 137.0359990651809516838326...
In mid-2011 the simplest rational fraction approximation was:
α ≈ 48029/6581702,
1/α ≈ 137 + 1729/48029
= 137.0359990838868183805617...
Many other fine-structure constant theories are listed by Ivan Gorelik [22].
Appendix A: de Vries Method
One of the α "formulas", by Hans de Vries1, is kind of fun. One begins with any approximation to α, then repeats the following calculations as many times as desired:
G = 1 + α(1 + α/2π(1 + α/(4π2)(1 +
α/(8π3)(1 + ...)))))
α' = G2/(eπ2/2)
then repeating with the new value α'. This converges on 1/α = 137.0359990958297... For automatic calculation one may wish to rewrite the G expression as
G = 1 + α + α2/2π + α3/(2π)3 + α4/(2π)6 + α5/(2π)10 + α6/(2π)15 + ...
where the exponents of 2π are the triangular numbers. Here it is in Hypercalc's BASIC interpreter:
C1 = old fine-structure C1 = list 10 scale = 20; a = 0.007; 20 for x = 1 to 11; 25 t = 0; g = 0; 30 for i = 0 to 5; 40 g = g + a^i/(2*pi)^t; 50 t = t + i; 60 next i; 70 a = g^2/(e^(pi^2/2)); 80 fsc = 1/a 90 next x 99 end C1 = run R1 = fsc: 137.11712418125116745 R2 = fsc: 137.03717654594675111 R3 = fsc: 137.03601619522281466 R4 = fsc: 137.03599934415588632 R5 = fsc: 137.03599909943602158 R6 = fsc: 137.03599909588207334 R7 = fsc: 137.03599909583046107 R8 = fsc: 137.03599909582971153 R9 = fsc: 137.03599909582970065 R10 = fsc: 137.03599909582970049 R11 = fsc: 137.03599909582970048To get more digits, double each of the scale and loop limits (from 20, 11 and 5 to 40, 22 and 10 respectively).
Footnotes
1 : http://www.chip-architect.com/news/2004_10_04_The_Electro_Magnetic_coupling_constant.html Hand de Vries, An exact formula for the Electro Magnetic coupling constant, web page, 2004 Oct 4.
2 : G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio, and B. Odom, New Determination of the Fine Structure Constant from the Electron g Value and QED, Physical Review Letters 97, 030802 (2006), week ending 2007 July 21st.
3 : My earlier wording of this line actually offended one or two of my readers, most of whom are otherwise sedate:
This constant holds a special place among cult numbers: unlike its more mundane cousins 17 and 666, the Fine Structure Constant seduces otherwise sane engineers and scientists into seeking mystical truths and developing farfetched theories.
4 : http://www.nbi.dk/~predrag/papers/PRD18-78.pdf P. Cvitanovic, B. Lautrup and R. B. Pearson, The number and weights of Feynman diagrams, Physical Review D vol. 18, pp. 1939-1949 (1978).
5 : Lamoreaux and Thorgerson, Phys. Rev. D 69, 121701 (2004).
References
[6] http://physics.nist.gov/cuu/Archive/1969RMP.pdf B. N. Taylor, W. H. Parker, and D. N. Langenberg. Determination of e/h, Using Macroscopic Quantum Phase Coherence in Superconductors: Implications for Quantum Electrodynamics and the Fundamental Physical Constants. Reviews of Modern Physics 41(3), July 1969.
[7] Jean Pestieau and Probir Roy. Lepton Symmetry and Self-Mass. Physical Review Letters 23(6) 349-351 (1969).
[8] http://physics.nist.gov/cuu/Archive/1973JPCRD.pdf E. Richard Cohen and B. N. Taylor. The 1973 Least-Squares Adiustment of the Fundamental Constants. J. Phys. Chem. Ref. Data 2(4) 663-734 (1973)
[9] http://arxiv.org/abs/hep-ph/0512126 R. Decker and Jean Pestieau. Lepton Self-Mass, Higgs Scalar and Heavy Quark Masses. Presented at DESY Workshop, October 22-24, 1979. Available at arxiv.org/abs/hep-ph/0512126
[10] Richard Feynman gave a series of 4 lectures at the University of Auckland, New Zealand, in 1979. In the 4th lecture at about 19:30 into the video, Feynman describes the 137 cult phenomenon, and the folly of trying to explain a number by an arbitrary formula:
[...]
That summarizes all of the problems associated with Quantum Electrodynamics.
The most beautiful one is the coupling constant, 137 point and so
on, and all good theoretical physicists put that up on their wall and worry
about it. There is at the present time no idea of any utility for getting
at that number.
There have been from time to time suggestions, but they
didn't turn out to be useful. They would predict that the number was
exactly 137, when it — well. The first idea was by Eddington, and
experiments were very crude in those days and the number was very
close to 136, so he proved by pure logic that it had to be 136. Then it
turned out that experiment showed that it was a little off, and it was
nearer 137. So he found a slight error the logic and proved with pure logic
that it had to be exactly the integer 137. It's not the integer.
It's 137 point oh three six oh.
Every once in a while someone comes out and they find out that if they
combine pi's and e's and 2's and 5's with the right powers and square roots
that you can make that number.
It seems to be a fact that's not fully appreciated by people that play
with arithmetic — that you'd be surprised how many numbers you can make
by playing with pi's and 2's and 5's and so on — and if you haven't
anything to guide you except the answer, you can always make it come out
even to several decimal places by a suitable jiggling about.
It's surprising how close you can make an arbitrary number by playing
around with nice numbers like pi and e. It's, and therefore throughout
the history of physics, there — paper after paper of people who have
noticed that certain specific combinations give answers which are very close
in several decimal places to experiment disagrees with it — so it
doesn't mean anything.
[11] http://physics.nist.gov/cuu/pdf/codata86.pdf E. Richard Cohen and Barry N. Taylor. The 1986 CODATA Recommended Values of the Fundamental Physical Constants.
[12] Roger Penrose, The Emperor's New Mind: Concerning Computers, Minds and The Laws of Physics, 1989
[13] http://arxiv.org/pdf/hep-ph/9504350 Gabriel Castro and Jean Pestieau. Determination of the Higgs Boson Mass by the Cancellation of Ultraviolet Divergeneces in the SU(2)L×U(1) Theory. Modern Physics Letters A 10(15-16) 1155-1157 (1995). Available at arxiv.org/abs/hep-ph/9504350
[14] http://www.physics.nist.gov/cuu/Archive/1998RMP.pdf Peter J. Mohr and Barry N. Taylor. CODATA recommended values of the fundamental physical constants: 1998.
[15] http://physics.nist.gov/cuu/Archive/2002RMP.pdf Peter J. Mohr and Barry N. Taylor. CODATA recommended values of the fundamental physical constants: 2002.
[16] http://www.lorentz.leidenuniv.nl/history/zeeman/lorentzveltman/Leiden2002lect.pdf M. Veltman, Was lorentz our first particle physicist?, transcript with illustrations, figures and formulas from a talk given at Leiden, 2002 Oct 11. Mentions
[17] http://arxiv.org/abs/hep-ph/0507249v2 Toichiro Kinoshita and M. Nio. Improved α4 Term of the Electron Anomalous Magnetic Moment. On arXiv.org.
[18] http://physics.nist.gov/cuu/Constants/RevModPhys_80_000633acc.pdf Peter J. Mohr, Barry N. Taylor, and David B. Newell, CODATA recommended values of the fundamental physical constants: 2006
[19] G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio, and B. Odom, New Determination of the Fine Structure Constant from the Electron g Value and QED, Physical Review Letters 97, 030802 (2006), week ending 2006 July 21st.
[20] G. Gabrielse, D. Hanneke, T. Kinoshita, M. Nio, and B. Odom. Erratum: New Determination of the Fine Structure Constant from the Electron g Value and QED. Physical Review Letters 99, 039902 (2007), week ending 2007 July 20th.
[21] http://www.phys.uconn.edu/icap2008/invited/icap2008-gabrielse.pdf G. Gabrielse. New Measurement of the Electron Magnetic Moment and the Fine Structure Constant. submitted to the 21st International Conference on Atomic Physics, Storrs, Connecticut, USA; 2008 July 27.
[22] http://www.oocities.com/igorelik/fine.html Ivan Gorelik, "Fine Structure Constant Collection", web page. (Also here, and formerly at http://www.geocities.com/Area51/Nebula/3735/fine.html)
[23] http://hussle.harvard.edu/~gabrielse/gabrielse/papers/2008/HarvardMagneticMoment2008.pdf D. Hanneke, S. Fogwell, and G. Gabrielse. New Measurement of the Electron Magnetic Moment and the Fine Structure Constant. Phys. Rev. Lett. 100, 120801 (2008).
[24] Arthur Miller. Deciphering the cosmic number: the strange friendship of Wolfgang Pauli and Carl Jung. (2009) ISBN 0-393-06532-4. Published in paperback under the title 137: Jung, Pauli, and the pursuit of a scientific obsession. This book has much about the crossover between scientific and mystical culture in the early and middle 20th century.
[25] http://www.meeus-d.be/physique/40years.pdf Jean Pestieau. Variations on the Gauge Sector of the Electroweak Model, 2009 Aug 11.
[26] J. C. Berengut and V. V. Flambaum. Manifestations of a spatial variation of fundamental constants on atomic clocks, Oklo, meteorites, and cosmological phenomena, 2010. Available at arxiv.org/abs/1008.3957.
[27] http://arxiv.org/abs/1008.3907 J. K. Webb, et al. Evidence for spatial variation of the fine structure constant, 2010 Aug 23. Available at arxiv.org/abs/1008.3907.
[28] http://physics.nist.gov/cuu/Constants/Table/allascii.txt Fundamental Physical Constants — Complete Listing (ASCII text file). (This table changes as NIST's published values change. A summary of the 2010 adjustment should be linked from here.)
[29] Regarding the release date of CODATA 2010, the following is from NIST :
The values of the constants provided at this site are recommended for international use by CODATA and are the latest available. Termed the "2010 CODATA recommended values," they are generally recognized worldwide for use in all fields of science and technology. The values became available on 2 June 2011 and replaced the 2006 CODATA set. They are based on all of the data available through 31 December 2010. The 2010 adjustment was carried out under the auspices of the CODATA Task Group on Fundamental Constants. Also available is an Introduction to the constants for nonexperts.
[30] http://arxiv.org/abs/1012.3627v1 Rym Bouchendira et al.. "New determination of the fine structure constant and test of the quantum electrodynamics". Submitted to arXiv 2010 Dec 16, published in Physical Review Letters in 2011 Feb (Phys.Rev.Lett.106:080801,2011). Preprint available at arxiv.org/abs/1012.3627v1.
[31] https://journals.aps.org/prd/abstract/10.1103/PhysRevD.96.019901 Tatsumi Aoyama, Masashi Hayakawa, Toichiro Kinoshita, and Makiko Nio "Erratum: Tenth-order electron anomalous magnetic moment: Contribution of diagrams without closed lepton loops [Phys. Rev. D 91, 033006 (2015)]", Phys. Rev. D 96, 019901 (Published 10 July 2017)
[32] https://arxiv.org/abs/1812.04130 Richard H. Parker, Chenghui Yu, Weicheng Zhong, Brian Estey, Holger Müller "Measurement of the fine-structure constant as a test of the Standard Model." (revision v2 submitted to arXiv on 12 Dec 2018).
[33] https://hal.archives-ouvertes.fr/hal-03107990/file/main.pdf ] Léo Morel, Zhibin Yao, Pierre Cladé and Saïda Guellati-Khélifa "Determination of the fine-structure constant with an accuracy of 81 parts per trillion", Nature 588(7836) pp. 61-65 (2020)
(Same value of the fine-structure constant was also reported in: Morel, L., Yao, Z., Cladé, P. and Guellati-Khélifa, S., 2021. "Accurate Determination of the Fine Structure Constant Using Atom Interferometry". FFK 2021, p.45. (The International Conference on Precision Physics and Fundamental Physical Constants, Slovakia, October 2021))
[34] National Institute of Standards and Technology, Fundamental Physical Constants (website). Gives values for a lot of related physical constants, with means and error ranges that are produced from different types of experiments and correlated with one another via the known relationships between the values.
This page was written in the "embarrassingly readable" markup language RHTF, and was last updated on 2024 Nov 28.
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