The number 137.035...
(You might also wish to view 137.035... in context on my numbers page.)
137.035999074(44)...
137.035999074(44)... is a recent (2010) 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 "(44)" at the end of the value "137.035999074(44)" 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 137.035999030 to 137.035999118. This is from CODATA 2010[30].
Generally the best known value would be found at 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. However these calculations and recommendations are only done from time to time, 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 ([23], [24], [25]) were more recent and more accurate. Gabrielse et al. [23] describes the error.
The present CODATA 2010 value was published in June 2011, but was based on work on or before 2010 Dec 31 (see [31]). As such, it does not include more recent results such as the 2010 work by Bouchendira et al. [32] using new direct experimental measurements (87Rb rubidium recoil velocity), which was published in Feb 2011.
Here is a summary of α values from original research (e.g. [21], [25], [32]) and CODATA least-squares reduction over the past 50 years:
Table of Fine Structure Constant Values
|
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
[23]
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
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. The current best approximation (which is 4th order) took "many supercomputers over more than 10 years"2 to evaluate 891 diagrams. Each "order" reduces the error in the calculation by a factor of about 137, i.e. the current best 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. However, more recently the idea has come back in a substantially stronger form: there now appears to be evidence that α varies throughout space as well as time, differeing by as much as one part on 105 between opposite ends of the visible universe in the distant past (see [28] and [29]).
Cult Appeal
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 engineers and scientists into seeking mystical truths and developing uncollaborated theories11. 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. A few years before his death Richard Feynman had the following to say:
[...] 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 [26], 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 comprises a prediction, and, more accurate experimental measurements or other research eventually prove it 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. Mr. Michael Wales in the UK proposed that it is precisely the cube root of 2573380, or 137.0359896... based on the best available estimates at the time (which were third-order). A more elaborate and mystical approach is taken by D. G. Leahy (see the entry for 82944) who communicated to me directly; I was also notified of his work by reader Jerry Iuliano, who was a collecter of fine structure "formulas" for some time. Later, at Dr. James Gilson's web site www.fine-structure-constant.org proposed this value for α:
29 cos(π/137) tan(π/(137×29)) / π
which gives 1/α = 137.0359997866991075..., which was within the 2002 CODATA error range[19] but is now known to be well off the mark. Gilson's theory relates α to the length of the perimeter of a 137×29-sided regular polygon inscribed in a circle of unit diameter.
Reader Richard Blaber, inspired by the movie Pi, wrote to suggest the formula:
α ≅ e/(12 π3 me) = 0.0072972677066...
which gives 1/α ≅ 137.03759272566(52); the special value me is the ratio of the magnetic moment of the electron to the Bohr magneton, another unitless value given by CODATA as -1.00115965218076(27). He also noted that 6π5=1836.1181... is close to the proton-to-electron mass ratio 1836.1526...., and that eπ2/2α = 1838.2259... is close to the neuton-to-electronmassratio 1838.6836... These don't fall within experimental error: for example, the α is off by 0.000007, but less than 0.0000001 of this is accountable to the known errors in me and the real α.
Jeff Caveney suggests e-K2, where K is given by
K = π × Gamma(2/3) / (3 × ln(Ei(1))) = 2.2181622332459491..
which gives α = 0.00729735231642144.... Here "Gamma" is the Gamma function and "Ei" is the exponential integral function. Like the previous example, this one falls far outside the known error range of current experiment.
The "simplest" formula I have seen so far was given by Steve Krakowski:
(37 / 0.27) - (1 / 964) = 137.0359996926386967880743...
which approximates the 2006 CODATA figure[22], and can also be stated as 137 + 1/27 - 1/964. Krakowski also gives the sequence:
(10(((666-72)/666)+3)/666)2 / (1+(10-3/(666))) = 137.036000...
(10(((666-72)/666)+3)/666)2 / (1+(10-3/(666-1))) = 137.035999692...
(10(((666-72)/666)+3)/666)2 / (1+(10-3/(666-3))) = 137.035999071...
which gives several old approximations of 1/α over the years, expressed mainly in terms of that other cult classic number 666, and the number 72 which appears in the sequence A005413 mentioned above. Krakowski continues to derive formulas to keep up with newer experimental values, more recently he has begun using the Riemann zeta function.
In 2007 Raphie Frank8 proposed a set of expressions that use the July 2006 CODATA figure along with some constants like e and phi, a few trigonometric functions, and the integers 1, 3 and 7 (considered significant because they are the digits of 137) to compute a value that is close to the 2006 value from Gabrielse et al. (More recently Mr. Frank has been exploring ways to express it in terms of 1/(eΦ2) = 0.0729461... and 1+√3 = 2.732050...10)
For several years beginning around 1999, Mark Thomas (see also 3.377×1038) investigated a range of theories and formulas related to physics. Recently (late 2009) he proposed a formula for α as well. His theory used 702 and the Monster group size, among other numbers, and remains consistent with CODATA 2010.
In 2009 another reader (Dan Graham)7 suggested the formula π×e×C×13 / B, where C = √5-1 = 1.236067977..., B = (1+A)(1/(1-A)), and A = 137/105. Graham's formula gives 137.03599906605045... so is within the current best known experimental error bounds. He expresses 13 as (52+1)/2, which gives the formula more symmetry by its similarity to C.
In 2004 Hans de Vries1 described a convergent iterative calculation. It is another candidate still in the running, and is particularly attractive because it constructs an infinite series involving Schwinger's constant α/2π, defining α in terms of itself through this series. To compute α you start with any approximation and then successively evaluate the series multiple times, like using Newton's method to find a root of an equation. It is detailed below with an example implementation in Hypercalc.
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...
At present (mid-2011) the simplest rational fraction approximation is:
α ≅ 48029/6581702,
1/α ≅ 137 + 1729/48029
= 137.0359990838868183805617...
Many other fine structure constant theories are listed by Ivan Gorelik 9.
Appendix A: de Vries Method
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.
3 : Gabrielse (2008): see [24].
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).
7 : Dan Graham, personal correspondence, 2009.
8 : http://www.physforum.com/index.php?showtopic=19245 Raphie Frank, The Fine Structure Constant As Fractal Construct?, A "Physio-theoretical" Exploration, forum discussion, 2007 Nov 22.
9 : 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)
10 : Raphie Frank, personal correspondence, 2010.
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.
References
[12] 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.
[14] 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)
[15] 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
[16] 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.
[17] 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
[18] 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.
[19] http://physics.nist.gov/cuu/Archive/2002RMP.pdf Peter J. Mohr and Barry N. Taylor. CODATA recommended values of the fundamental physical constants: 2002.
[20] 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
[21] http://arxiv.org/abs/hep-ph/0507249v2 Toichiro Kinoshita and M. Nio. Improved α4 Term of the Electron Anomalous Magnetic Moment. On arXiv.org.
[22] 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
[24] 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.
[25] 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).
[27] http://www.meeus-d.be/physique/40years.pdf Jean Pestieau. Variations on the Gauge Sector of the Electroweak Model, 2009 Aug 11.
[29] 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.
[30] http://physics.nist.gov/cuu/Constants/Table/allascii.txt Fundamental Physical Constants — Complete Listing (ASCII text file). (Late in 2011 or early 2012, a detailed paper about the 2010 adjustment should become available here.)
[31] 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.
[32] http://arxiv.org/abs/1012.3627v1 Rym Bouchendira et al.. New determination of the fine structure constant and test of the quantum electrodynamics. Submitted 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.
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