# Fine-structure constant

In physics, the **fine-structure constant**, also known as **Sommerfeld's constant**, commonly denoted by *α* (the Greek letter *alpha*), is a fundamental physical constant characterizing the strength of the electromagnetic interaction between elementary charged particles. It is a dimensionless quantity related to the elementary charge *e*, which characterizes the strength of the coupling of an elementary charged particle with the electromagnetic field, by the formula 4π*ε*_{0}*ħcα* = *e*^{2}. As a dimensionless quantity, its numerical value, approximately 1/137, is independent of the system of units used.^{[1]}

While there are multiple physical interpretations for *α*, it received its name from Arnold Sommerfeld, who introduced it in 1916, when extending the Bohr model of the atom. *α* quantifies the gap in the fine structure of the spectral lines of the hydrogen atom, which had been measured precisely by Michelson and Morley in 1887.^{[2]}

## Definition

Some equivalent definitions of *α* in terms of other fundamental physical constants are:

where:

*e*is the elementary charge (:= 1.602176634×10^{−19}C);- π is the mathematical constant pi;
*h*is the Planck constant (:= 6.62607015×10^{−34}J⋅s);*ħ*=*h*/2π is the reduced Planck constant (:= 6.62607015×10^{−34}J⋅s/2π);*c*is the speed of light in vacuum (= 299792458 m/s);*ε*_{0}is the electric constant or permittivity in vacuum (or free space);*µ*_{0}is the magnetic constant or permeability in vacuum (or free space)*k*_{e}is the Coulomb constant;*R*_{K}is the von Klitzing constant;*Z*_{0}is the vacuum impedance or impedance in free space.

When the other constants (*c*, *h* and *e*) have defined values, the definition reflects the relationship between *α* and the permeability of free space *µ*_{0}, which equals *µ*_{0} = 2*hα*/*ce*^{2}.
In the 2019 redefinition of SI base units, 4π × 1.00000000082(20)×10^{−7} H⋅m^{−1} is the value for *µ*_{0} based upon more accurate measurements of the fine structure constant.^{[3]}^{[4]}^{[5]}

### In non-SI units

In electrostatic cgs units, the unit of electric charge, the statcoulomb, is defined so that the Coulomb constant, *k*_{e}, or the permittivity factor, 4π*ε*_{0}, is 1 and dimensionless. Then the expression of the fine-structure constant, as commonly found in older physics literature, becomes

In natural units, commonly used in high energy physics, where *ε*_{0} = *c* = *ħ* = 1, the value of the fine-structure constant is^{[6]}

As such, the fine-structure constant is just another, albeit dimensionless, quantity determining (or determined by) the elementary charge: *e* = √4π*α* ≈ 0.30282212 in terms of such a natural unit of charge.

In Hartree atomic units (*e* = *m*_{e} = *ħ* = 1 and *ε*_{0} = 1/4π), the fine structure constant is

## Measurement

The 2018 CODATA recommended value of *α* is^{[7]}

*α*=*e*^{2}/4π*ε*_{0}*ħc*= 0.0072973525693(11).

This has a relative standard uncertainty of 0.15 parts per billion.^{[7]}

This value for *α* gives *µ*_{0} = 4π × 1.00000000054(15)×10^{−7} H⋅m^{−1}, 3.6 standard deviations away from its old defined value, but with the mean differing from the old value by only 0.54 parts per billion.

For reasons of convenience, historically the value of the reciprocal of the fine-structure constant is often specified. The 2018 CODATA recommended value is given by^{[1]}

*α*^{−1}= 137.035999084(21).

While the value of *α* can be *estimated* from the values of the constants appearing in any of its definitions, the theory of quantum electrodynamics (QED) provides a way to measure *α* directly using the quantum Hall effect or the anomalous magnetic moment of the electron. Other methods include the AC Josephson effect and photon recoil in atom interferometry.^{[8]} There is general agreement for the value of *α*, as measured by these different methods. The preferred methods in 2019 are measurements of electron anomalous magnetic moments and of photon recoil in atom interferometry.^{[8]} The theory of QED predicts a relationship between the dimensionless magnetic moment of the electron and the fine-structure constant *α* (the magnetic moment of the electron is also referred to as "Landé *g*-factor" and symbolized as *g*). The most precise value of *α* obtained experimentally (as of 2012) is based on a measurement of *g* using a one-electron so-called "quantum cyclotron" apparatus, together with a calculation via the theory of QED that involved 12672 tenth-order Feynman diagrams:^{[9]}

*α*^{−1}= 137.035999174(35).

This measurement of *α* has a relative standard uncertainty of 2.5×10^{−10}. This value and uncertainty are about the same as the latest experimental results.^{[10]}

## Physical interpretations

The fine-structure constant, *α*, has several physical interpretations. *α* is:

- The ratio of two energies: (i) the energy needed to overcome the electrostatic repulsion between two electrons a distance of
*d*apart, and (ii) the energy of a single photon of wavelength*λ*= 2π*d*(or of angular wavelength*d*; see Planck relation):

- The ratio of the velocity of the electron in the first circular orbit of the Bohr model of the atom, which is 1/4π
*ε*_{0}*e*^{2}/*ħ*, to the speed of light in vacuum,*c*.^{[11]}This is Sommerfeld's original physical interpretation. Then the square of*α*is the ratio between the Hartree energy (27.2 eV = twice the Rydberg energy = approximately twice its ionization energy) and the electron rest energy (511 keV). - The two ratios of three characteristic lengths: the classical electron radius
*r*_{e}, the Compton wavelength of the electron*λ*_{e}, and the Bohr radius*a*_{0}:

- In quantum electrodynamics,
*α*is directly related to the coupling constant determining the strength of the interaction between electrons and photons.^{[12]}The theory does not predict its value. Therefore,*α*must be determined experimentally. In fact,*α*is one of the empirical parameters in the Standard Model of particle physics, whose value is not determined within the Standard Model. - In the electroweak theory unifying the weak interaction with electromagnetism,
*α*is absorbed into two other coupling constants associated with the electroweak gauge fields. In this theory, the electromagnetic interaction is treated as a mixture of interactions associated with the electroweak fields. The strength of the electromagnetic interaction varies with the strength of the energy field. - In the fields of electrical engineering and solid-state physics, the fine-structure constant is one fourth the product of the characteristic impedance of free space,
*Z*_{0}=*μ*_{0}*c*, and the conductance quantum,*G*_{0}= 2*e*^{2}/*h*:

- .
- The optical conductivity of graphene for visible frequencies is theoretically given by π
*G*_{0}/4, and as a result its light absorption and transmission properties can be expressed in terms of the fine structure constant alone.^{[13]}The absorption value for normal-incident light on graphene in vacuum would then be given by*πα*/(1 +*πα*/2)^{2}or 2.24%, and the transmission by 1/(1 +*πα*/2)^{2}or 97.75% (experimentally observed to be between 97.6% and 97.8%).

- The fine-structure constant gives the maximum positive charge of an atomic nucleus that will allow a stable electron-orbit around it within the Bohr model (element feynmanium).
^{[14]}For an electron orbiting an atomic nucleus with atomic number*Z*,*mv*^{2}/*r*= 1/4*πε*_{0}*Ze*^{2}/*r*^{2}. The Heisenberg uncertainty principle momentum/position uncertainty relationship of such an electron is just*mvr*=*ħ*. The relativistic limiting value for*v*is*c*, and so the limiting value for*Z*is the reciprocal of the fine-structure constant, 137.^{[15]} - The magnetic moment of the electron indicates that the charge is circulating at a radius
*r*with the velocity of light._{Q}^{[16]}It generates the radiation energy*m*_{e}*c*^{2}and has an angular momentum*L*= 1*ħ*=*r*_{Q}m_{e}*c*. The field energy of the stationary Coulomb field is*m*_{e}*c*^{2}=*e*^{2}/4π*ε*_{0}*r*_{e}and defines the classical electron radius*r*_{e}. These values inserted into the definition of alpha yields*α*=*r*_{e}/*r*. It compares the dynamic structure of the electron with the classical static assumption._{Q} - Alpha is related to the probability that an electron will emit or absorb a photon.
^{[17]} - Given two hypothetical point particles each of Planck mass and elementary charge, separated by any distance,
*α*is the ratio of their electrostatic repulsive force to their gravitational attractive force. - The square of the ratio of the elementary charge to the Planck charge

When perturbation theory is applied to quantum electrodynamics, the resulting perturbative expansions for physical results are expressed as sets of power series in *α*. Because *α* is much less than one, higher powers of *α* are soon unimportant, making the perturbation theory practical in this case. On the other hand, the large value of the corresponding factors in quantum chromodynamics makes calculations involving the strong nuclear force extremely difficult.

## Variation with energy scale

In quantum electrodynamics, the more thorough quantum field theory underlying the electromagnetic coupling, the renormalization group dictates how the strength of the electromagnetic interaction grows logarithmically as the relevant energy scale increases. The value of the fine-structure constant *α* is linked to the observed value of this coupling associated with the energy scale of the electron mass: the electron is a lower bound for this energy scale, because it (and the positron) is the lightest charged object whose quantum loops can contribute to the running. Therefore, 1/137.036 is the asymptotic value of the fine-structure constant at zero energy.
At higher energies, such as the scale of the Z boson, about 90 GeV, one measures an *effective* *α* ≈ 1/127, instead.

As the energy scale increases, the strength of the electromagnetic interaction in the Standard Model approaches that of the other two fundamental interactions, a feature important for grand unification theories. If quantum electrodynamics were an exact theory, the fine-structure constant would actually diverge at an energy known as the Landau pole—this fact undermines the consistency of quantum electrodynamics beyond perturbative expansions.

## History

Based on the precise measurement of the hydrogen atom spectrum by Michelson and Morley in 1887,^{[18]} Arnold Sommerfeld extended the Bohr model to include elliptical orbits and relativistic dependence of mass on velocity. He introduced a term for the fine-structure constant in 1916.^{[19]} The first physical interpretation of the fine-structure constant *α* was as the ratio of the velocity of the electron in the first circular orbit of the relativistic Bohr atom to the speed of light in the vacuum.^{[20]} Equivalently, it was the quotient between the minimum angular momentum allowed by relativity for a closed orbit, and the minimum angular momentum allowed for it by quantum mechanics. It appears naturally in Sommerfeld's analysis, and determines the size of the splitting or fine-structure of the hydrogenic spectral lines. This constant was not seen as significant until Paul Dirac's linear relativistic wave equation in 1928, which gave the exact fine structure formula.^{[21]}^{:407}

With the development of quantum electrodynamics (QED) the significance of α has broadened from a spectroscopic phenomenon to a general coupling constant for the electromagnetic field, determining the strength of the interaction between electrons and photons. The term *α*/2π is engraved on the tombstone of one of the pioneers of QED, Julian Schwinger, referring to his calculation of the anomalous magnetic dipole moment.

## Is the fine-structure constant actually constant?

Physicists have pondered whether the fine-structure constant is in fact constant, or whether its value differs by location and over time. A varying *α* has been proposed as a way of solving problems in cosmology and astrophysics.^{[22]}^{[23]}^{[24]}^{[25]} String theory and other proposals for going beyond the Standard Model of particle physics have led to theoretical interest in whether the accepted physical constants (not just *α*) actually vary.

In the experiments below, Δ*α* represents the change in *α* over time, which can be computed by *α*_{prev} − *α*_{now}. If the fine-structure constant really is a constant, then any experiment should show that

or as close to zero as experiment can measure. Any value far away from zero would indicate that *α* does change over time. So far, most experimental data is consistent with *α* being constant.

### Past rate of change

The first experimenters to test whether the fine-structure constant might actually vary examined the spectral lines of distant astronomical objects and the products of radioactive decay in the Oklo natural nuclear fission reactor. Their findings were consistent with no variation in the fine-structure constant between these two vastly separated locations and times.^{[26]}^{[27]}^{[28]}^{[29]}^{[30]}^{[31]}

Improved technology at the dawn of the 21st century made it possible to probe the value of *α* at much larger distances and to a much greater accuracy. In 1999, a team led by John K. Webb of the University of New South Wales claimed the first detection of a variation in *α*.^{[32]}^{[33]}^{[34]}^{[35]} Using the Keck telescopes and a data set of 128 quasars at redshifts 0.5 < *z* < 3, Webb *et al.* found that their spectra were consistent with a slight increase in *α* over the last 10–12 billion years. Specifically, they found that

In other words, they measured the value to be somewhere between −0.0000047 and −0.0000067. This is a very small value, but the error bars do not actually include zero. This result either indicates that *α* is not constant or that there is experimental error unaccounted for.

In 2004, a smaller study of 23 absorption systems by Chand *et al.*, using the Very Large Telescope, found no measurable variation:^{[36]}^{[37]}

However, in 2007 simple flaws were identified in the analysis method of Chand *et al.*, discrediting those results.^{[38]}^{[39]}

King *et al.* have used Markov chain Monte Carlo methods to investigate the algorithm used by the UNSW group to determine Δ*α*/*α* from the quasar spectra, and have found that the algorithm appears to produce correct uncertainties and maximum likelihood estimates for Δ*α*/*α* for particular models.^{[40]} This suggests that the statistical uncertainties and best estimate for Δ*α*/*α* stated by Webb *et al.* and Murphy *et al.* are robust.

Lamoreaux and Torgerson analyzed data from the Oklo natural nuclear fission reactor in 2004, and concluded that *α* has changed in the past 2 billion years by 45 parts per billion. They claimed that this finding was "probably accurate to within 20%". Accuracy is dependent on estimates of impurities and temperature in the natural reactor. These conclusions have to be verified.^{[41]}^{[42]}^{[43]}^{[44]}

In 2007, Khatri and Wandelt of the University of Illinois at Urbana-Champaign realized that the 21 cm hyperfine transition in neutral hydrogen of the early universe leaves a unique absorption line imprint in the cosmic microwave background radiation.^{[45]} They proposed using this effect to measure the value of *α* during the epoch before the formation of the first stars. In principle, this technique provides enough information to measure a variation of 1 part in 10^{9} (4 orders of magnitude better than the current quasar constraints). However, the constraint which can be placed on *α* is strongly dependent upon effective integration time, going as *t*^{−1⁄2}. The European LOFAR radio telescope would only be able to constrain Δ*α*/*α* to about 0.3%.^{[45]} The collecting area required to constrain Δ*α*/*α* to the current level of quasar constraints is on the order of 100 square kilometers, which is economically impracticable at the present time.

### Present rate of change

In 2008, Rosenband *et al.*^{[46]} used the frequency ratio of ^{}_{}Al^{+}_{} and ^{}_{}Hg^{+}_{} in single-ion optical atomic clocks to place a very stringent constraint on the present-time temporal variation of *α*, namely *α̇*/*α* = (−1.6±2.3)×10^{−17} per year. Note that any present day null constraint on the time variation of alpha does not necessarily rule out time variation in the past. Indeed, some theories^{[47]} that predict a variable fine-structure constant also predict that the value of the fine-structure constant should become practically fixed in its value once the universe enters its current dark energy-dominated epoch.

### Spatial variation – Australian dipole

In September 2010 researchers from Australia said they had identified a dipole-like structure in the variation of the fine-structure constant across the observable universe. They used data on quasars obtained by the Very Large Telescope, combined with the previous data obtained by Webb at the Keck telescopes. The fine-structure constant appears to have been larger by one part in 100,000 in the direction of the southern hemisphere constellation Ara, 10 billion years ago. Similarly, the constant appeared to have been smaller by a similar fraction in the northern direction, 10 billion years ago.^{[48]}^{[49]}^{[50]}

In September and October 2010, after Webb's released research, physicists Chad Orzel and Sean M. Carroll suggested various approaches of how Webb's observations may be wrong. Orzel argues^{[51]} that the study may contain wrong data due to subtle differences in the two telescopes, in which one of the telescopes the data set was slightly high and on the other slightly low, so that they cancel each other out when they overlapped. He finds it suspicious that the sources showing the greatest changes are all observed by one telescope, with the region observed by both telescopes aligning so well with the sources where no effect is observed. Carroll suggested^{[52]} a totally different approach; he looks at the fine-structure constant as a scalar field and claims that if the telescopes are correct and the fine-structure constant varies smoothly over the universe, then the scalar field must have a very small mass. However, previous research has shown that the mass is not likely to be extremely small. Both of these scientists' early criticisms point to the fact that different techniques are needed to confirm or contradict the results, as Webb, et al., also concluded in their study.

In October 2011, Webb *et al.* reported^{[53]}
a variation in *α* dependent on both redshift and spatial direction. They report "the combined data set fits a spatial dipole" with an increase in *α* with redshift in one direction and a decrease in the other. "Independent VLT and Keck samples give consistent dipole directions and amplitudes...."^{[clarification needed]}

In 2020, the team verified their previous results, finding a dipole structure in the strength of the electromagnetic force using the most distant quasar measurements. Observations of the quasar of the universe at only 0.8 billion years old with AI analysis method employed on the Very Large Telescope (VLT) found a spatial variation preferred over a no-variation model at the level.^{[54]}

## Anthropic explanation

The anthropic principle is a controversial argument of why the fine-structure constant has the value it does: stable matter, and therefore life and intelligent beings, could not exist if its value were much different. For instance, were *α* to change by 4%, stellar fusion would not produce carbon, so that carbon-based life would be impossible. If *α* were greater than 0.1, stellar fusion would be impossible, and no place in the universe would be warm enough for life as we know it.^{[55]}

## Numerological explanations and multiverse theory

As a dimensionless constant which does not seem to be directly related to any mathematical constant, the fine-structure constant has long fascinated physicists.

Arthur Eddington argued that the value could be "obtained by pure deduction" and he related it to the Eddington number, his estimate of the number of protons in the universe.^{[56]} This led him in 1929 to conjecture that the reciprocal of the fine-structure constant was not approximately the integer 137, but precisely the integer 137.^{[57]} Other physicists neither adopted this conjecture nor accepted his arguments but by the 1940s experimental values for 1/*α* deviated sufficiently from 137 to refute Eddington's argument.^{[21]}

The fine-structure constant so intrigued physicist Wolfgang Pauli that he collaborated with psychoanalyst Carl Jung in a quest to understand its significance.^{[58]} Similarly, Max Born believed that would the value of alpha differ, the universe would degenerate. Thus, he asserted that 1/137 is a law of nature.^{[59]}

Richard Feynman, one of the originators and early developers of the theory of quantum electrodynamics (QED), referred to the fine-structure constant in these terms:

There is a most profound and beautiful question associated with the observed coupling constant,

e– the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won't recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. 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. You might say the "hand of God" wrote that number, and "we don't know how He pushed his pencil." We know what kind of a dance to do experimentally to measure this number very accurately, but we don't know what kind of dance to do on the computer to make this number come out, without putting it in secretly!

Conversely, statistician I. J. Good argued that a numerological explanation would only be acceptable if it could be based on a good theory that is not yet known but "exists" in the sense of a Platonic Ideal.^{[60]}

Attempts to find a mathematical basis for this dimensionless constant have continued up to the present time. However, no numerological explanation has ever been accepted by the physics community.

In the early 21st century, multiple physicists, including Stephen Hawking in his book *A Brief History of Time*, began exploring the idea of a multiverse, and the fine-structure constant was one of several universal constants that suggested the idea of a fine-tuned universe.^{[61]}

## Quotes

The mystery about

αis actually a double mystery. The first mystery – the origin of its numerical valueα≈ 1/137 – has been recognized and discussed for decades. The second mystery – the range of its domain – is generally unrecognized.

## See also

- Dimensionless physical constant
- Electric constant
- Hyperfine structure
- Planck constant
- Speed of light

## References

- ^
^{a}^{b}Mohr, P. J.; Taylor, B. N.; Newell, D. B. (2019). "Inverse fine structure constant".*CODATA Internationally recommended 2018 values of the fundamental physical constants*. National Institute of Standards and Technology. Retrieved 20 May 2019. **^***α*is proportional to the square of the coupling constant for a charged particle to the electromagnetic field. There are analogous constants that parameterize the interaction strength of the nuclear strong force, known as*α*_{s}(≈1), and the nuclear weak force, known as*α*_{w}(≈10^{−6}to 10^{−7}). "Coupling Constants for the Fundamental Forces".*HyperPhysics*. Georgia State University. Retrieved 12 May 2020.**^**"Convocationde la Conférence générale des poids et mesures (26e réunion)" (PDF).**^**Parker, Richard H.; Yu, Chenghui; Zhong, Weicheng; Estey, Brian; Müller, Holger (13 April 2018). "Measurement of the fine-structure constant as a test of the Standard Model".*Science*.**360**(6385): 191–195. arXiv:1812.04130. Bibcode:2018Sci...360..191P. doi:10.1126/science.aap7706. ISSN 0036-8075. PMID 29650669.**^**Davis, Richard S. (2017). "Determining the value of the fine-structure constant from a current balance: Getting acquainted with some upcoming changes to the SI".*American Journal of Physics*.**85**(5): 364–368. arXiv:1610.02910. Bibcode:2017AmJPh..85..364D. doi:10.1119/1.4976701. ISSN 0002-9505.**^**Peskin, M.; Schroeder, D. (1995).*An Introduction to Quantum Field Theory*. Westview Press. p. 125. ISBN 978-0-201-50397-5.- ^
^{a}^{b}Mohr, P. J.; Taylor, B. N.; Newell, D. B. (2019). "Fine structure constant".*CODATA Internationally recommended 2018 values of the fundamental physical constants*. National Institute of Standards and Technology. - ^
^{a}^{b}Yu, C.; Zhong, W.; Estey, B.; Kwan, J.; Parker, R. H.; Müller, H. (2019). "Atom‐Interferometry Measurement of the Fine Structure Constant".*Annalen der Physik*.**531**: 1800346. doi:10.1002/andp.201800346. **^**Aoyama, T.; Hayakawa, M.; Kinoshita, T.; Nio, M. (2012). "Tenth-order QED contribution to the electron*g*−2 and an improved value of the fine structure constant".*Physical Review Letters*.**109**(11): 111807. arXiv:1205.5368. Bibcode:2012PhRvL.109k1807A. doi:10.1103/PhysRevLett.109.111807. PMID 23005618.**^**Bouchendira, Rym; Cladé, Pierre; Guellati-Khélifa, Saïda; Nez, François; Biraben, François (2011). "New determination of the fine-structure constant and test of the quantum electrodynamics" (PDF).*Physical Review Letters*(Submitted manuscript).**106**(8): 080801. arXiv:1012.3627. Bibcode:2011PhRvL.106h0801B. doi:10.1103/PhysRevLett.106.080801. PMID 21405559.**^**NIST Reference on Constants, Units, and Uncertainty Current advances: The fine-structure constant and quantum Hall effect**^**Riazuddin, Fayyazuddin (2012).*A Modern Introduction to Particle Physics*(Third ed.). World Scientific. p. 4. ISBN 9789814338837. Retrieved 20 April 2017.**^**Nair, R. R.; Blake, P.; Grigorenko, A. N.; Novoselov, K. S.; Booth, T. J.; Stauber, T.; Peres, N. M. R.; Geim, A. K. (2008). "Fine Structure Constant Defines Visual Transparency of Graphene".*Science*.**320**(5881): 1308. arXiv:0803.3718. Bibcode:2008Sci...320.1308N. doi:10.1126/science.1156965. PMID 18388259.**^**Subrahmanyan Chandrasekhar (April 1984). "On Stars, Their Evolution and Their Stability".*Reviews of Modern Physics*. Retrieved 11 May 2020.**^**Bedford, D.; Krumm, P. (2004). "Heisenberg indeterminacy and the fine structure constant".*American Journal of Physics*.**72**(7): 969. Bibcode:2004AmJPh..72..969B. doi:10.1119/1.1646135.**^**Poelz, G. (5 October 2016). "An Electron Model with Synchrotron Radiation". arXiv:1206.0620v24 [physics.class-ph].**^**Lederman, Leon,*The God Particle: If the Universe is the Answer, What is the Question?*(1993), Houghton Mifflin Harcourt, pp. 28–29.**^**Michelson, Albert A.; Morley, Edward W. (1887). "Method of making the wave-length of sodium light the actual and practical standard of length".*The American Journal of Science*. 3rd series.**34**(204): 427–430.- Reprinted in: Michelson, Albert A.; Morley, Edward W. (1887). "Method of making the wave-length of sodium light the actual and practical standard of length".
*The Philosophical Magazine*. 5th series.**24**(151): 463–466.

- Reprinted in: Michelson, Albert A.; Morley, Edward W. (1887). "Method of making the wave-length of sodium light the actual and practical standard of length".
**^**Sommerfeld, A. (1916). "Zur Quantentheorie der Spektrallinien" [On the quantum theory of spectral lines].*Annalen der Physik*. 4th series (in German).**51**(17): 1–94. From p.91:*"Wir fügen den Bohrschen Gleichungen (46) und (47) die charakteristische Konstante unserer Feinstrukturen (49) α = 2πe*(We add the characteristic constant of our fine structures (49) α = 2πe^{2}/ch hinzu, die zugleich mit der Kenntnis des Wasserstoffdubletts oder des Heliumtripletts in §10 oder irgend einer analogen Struktur bekannt ist."^{2}/ch to Bohr's equations (46) and (47), which is recognized at the same time from knowledge of the hydrogen doublet or the helium triplet in §10 or any analogous structure.)**^**"Introduction to the Constants for Nonexperts – Current Advances: The Fine-Structure Constant and Quantum Hall Effect".*The NIST Reference on Constants, Units, and Uncertainty*. NIST. Retrieved 11 April 2009.- ^
^{a}^{b}H. Kragh (2003). "Magic Number: A Partial History of the Fine-Structure Constant".*Archive for History of Exact Sciences*.**57**(5): 395–431. doi:10.1007/s00407-002-0065-7. JSTOR 41134170. **^**Milne, E. A. (1935).*Relativity, Gravitation and World Structure*. Clarendon Press.**^**Dirac, P. A. M. (1937). "The Cosmological Constants".*Nature*.**139**(3512): 323. Bibcode:1937Natur.139..323D. doi:10.1038/139323a0.**^**Gamow, G. (1967). "Electricity, Gravity, and Cosmology".*Physical Review Letters*.**19**(13): 759–761. Bibcode:1967PhRvL..19..759G. doi:10.1103/PhysRevLett.19.759.**^**Gamow, G. (1967). "Variability of Elementary Charge and Quasistellar Objects".*Physical Review Letters*.**19**(16): 913–914. Bibcode:1967PhRvL..19..913G. doi:10.1103/PhysRevLett.19.913.**^**Uzan, J.-P. (2003). "The Fundamental Constants and Their Variation: Observational Status and Theoretical Motivations".*Reviews of Modern Physics*.**75**(2): 403–455. arXiv:hep-ph/0205340. Bibcode:2003RvMP...75..403U. doi:10.1103/RevModPhys.75.403.**^**Uzan, J.-P. (2004). "Variation of the Constants in the Late and Early Universe".*AIP Conference Proceedings*.**736**: 3–20. arXiv:astro-ph/0409424. Bibcode:2004AIPC..736....3U. doi:10.1063/1.1835171.**^**Olive, K.; Qian, Y.-Z. (2003). "Were Fundamental Constants Different in the Past?".*Physics Today*.**57**(10): 40–45. Bibcode:2004PhT....57j..40O. doi:10.1063/1.1825267.**^**Barrow, J. D. (2002).*The Constants of Nature: From Alpha to Omega—the Numbers That Encode the Deepest Secrets of the Universe*. Vintage. ISBN 978-0-09-928647-9.**^**Uzan, J.-P.; Leclercq, B. (2008).*The Natural Laws of the Universe*.*The Natural Laws of the Universe: Understanding Fundamental Constants*. Springer Praxis. Bibcode:2008nlu..book.....U. doi:10.1007/978-0-387-74081-2. ISBN 978-0-387-73454-5.**^**Yasunori, F. (2004). "Oklo Constraint on the Time-Variability of the Fine-Structure Constant".*Astrophysics, Clocks and Fundamental Constants*. Lecture Notes in Physics.**648**. pp. 167–185. arXiv:astro-ph/0309817. Bibcode:2004LNP...648.....K. doi:10.1007/b13178. ISBN 978-3-540-21967-5.**^**Webb, J. K.; et al. (1999). "Search for Time Variation of the Fine Structure Constant".*Physical Review Letters*(Submitted manuscript).**82**(5): 884–887. arXiv:astro-ph/9803165. Bibcode:1999PhRvL..82..884W. doi:10.1103/PhysRevLett.82.884.**^**Murphy, M. T.; et al. (2001). "Possible evidence for a variable fine-structure constant from QSO absorption lines: motivations, analysis and results".*Monthly Notices of the Royal Astronomical Society*.**327**(4): 1208–1222. arXiv:astro-ph/0012419. Bibcode:2001MNRAS.327.1208M. doi:10.1046/j.1365-8711.2001.04840.x.**^**Murphy, M. T.; et al. (2001). "Further Evidence for Cosmological Evolution of the Fine Structure Constant".*Physical Review Letters*.**87**(9): 091301. arXiv:astro-ph/0012539. Bibcode:2001PhRvL..87i1301W. doi:10.1103/PhysRevLett.87.091301. PMID 11531558.**^**Murphy, M. T.; Webb, J .K.; Flambaum, V. V. (2003). "Further Evidence for a Variable Fine-Structure Constant from Keck/HIRES QSO Absorption Spectra".*Monthly Notices of the Royal Astronomical Society*.**345**(2): 609–638. arXiv:astro-ph/0306483. Bibcode:2003MNRAS.345..609M. doi:10.1046/j.1365-8711.2003.06970.x.**^**Chand, H.; et al. (2004). "Probing the Cosmological Variation of the Fine-Structure Constant: Results Based on VLT-UVES Sample".*Astronomy & Astrophysics*.**417**(3): 853–871. arXiv:astro-ph/0401094. Bibcode:2004A&A...417..853C. doi:10.1051/0004-6361:20035701.**^**R. Srianand; et al. (2004). "Limits on the Time Variation of the Electromagnetic Fine-Structure Constant in the Low Energy Limit from Absorption Lines in the Spectra of Distant Quasars".*Physical Review Letters*.**92**(12): 121302. arXiv:astro-ph/0402177. Bibcode:2004PhRvL..92l1302S. doi:10.1103/PhysRevLett.92.121302. PMID 15089663.**^**M. T. Murphy; J.K. Webb; V. V. Flambaum (2007). "Comment on "Limits on the Time Variation of the Electromagnetic Fine-Structure Constant in the Low Energy Limit from Absorption Lines in the Spectra of Distant Quasars"".*Physical Review Letters*.**99**(23): 239001. arXiv:0708.3677. Bibcode:2007PhRvL..99w9001M. doi:10.1103/PhysRevLett.99.239001. PMID 18233422.**^**M. T. Murphy; J.K. Webb; V. V. Flambaum (2008). "Revision of VLT/UVES Constraints on a Varying Fine-Structure Constant".*Monthly Notices of the Royal Astronomical Society*.**384**(3): 1053–1062. arXiv:astro-ph/0612407. Bibcode:2008MNRAS.384.1053M. doi:10.1111/j.1365-2966.2007.12695.x.**^**J. King; D. Mortlock; J. Webb; M. Murphy (2009). "Markov Chain Monte Carlo methods applied to measuring the fine-structure constant from quasar spectroscopy".*Memorie della Societa Astronomica Italiana*.**80**: 864. arXiv:0910.2699. Bibcode:2009MmSAI..80..864K.**^**R. Kurzweil (2005).*The Singularity Is Near*. Viking Penguin. pp. 139–140. ISBN 978-0-670-03384-3.**^**S. K. Lamoreaux; J.R. Torgerson (2004). "Neutron Moderation in the Oklo Natural Reactor and the Time Variation of Alpha".*Physical Review D*.**69**(12): 121701. arXiv:nucl-th/0309048. Bibcode:2004PhRvD..69l1701L. doi:10.1103/PhysRevD.69.121701.**^**E. S. Reich (30 June 2004). "Speed of Light May Have Changed Recently".*New Scientist*. Retrieved 30 January 2009.**^**"Scientists Discover One Of The Constants Of The Universe Might Not Be Constant".*ScienceDaily*. 12 May 2005. Retrieved 30 January 2009.- ^
^{a}^{b}R. Khatri; B. D. Wandelt (2007). "21-cm Radiation: A New Probe of Variation in the Fine-Structure Constant".*Physical Review Letters*.**98**(11): 111301. arXiv:astro-ph/0701752. Bibcode:2007PhRvL..98k1301K. doi:10.1103/PhysRevLett.98.111301. PMID 17501040. **^**T. Rosenband; et al. (2008). "Frequency Ratio of Al^{+}and Hg^{+}Single-Ion Optical Clocks; Metrology at the 17th Decimal Place".*Science*.**319**(5871): 1808–12. Bibcode:2008Sci...319.1808R. doi:10.1126/science.1154622. PMID 18323415.**^**J. D. Barrow; H.B. Sandvik; J. Magueijo (2002). "The Behaviour of Varying-Alpha Cosmologies".*Phys. Rev. D*.**65**(6): 063504. arXiv:astro-ph/0109414. Bibcode:2002PhRvD..65f3504B. doi:10.1103/PhysRevD.65.063504.**^**H. Johnston (2 September 2010). "Changes spotted in fundamental constant".*Physics World*. Retrieved 11 September 2010.**^**J. K. Webb; King, J. A.; Murphy, M. T.; Flambaum, V. V.; Carswell, R. F.; Bainbridge, M. B. (23 August 2010). "Evidence for spatial variation of the fine-structure constant".*Physical Review Letters*.**107**(19): 191101. arXiv:1008.3907. Bibcode:2011PhRvL.107s1101W. doi:10.1103/PhysRevLett.107.191101. PMID 22181590.**^**J. A. King (2010).*Searching for variations in the fine-structure constant and the proton-to-electron mass ratio using quasar absorption lines*(PhD thesis). University of New South Wales.**^**Orzel, Chad (14 October 2010). "Why I'm Skeptical About the Changing Fine-Structure Constant". ScienceBlogs.**^**Carroll, Sean M. (18 October 2010). "The Fine Structure Constant is Probably Constant".**^**J. K. Webb; J. A. King; M. T. Murphy; V. V. Flambaum; R. F. Carswell; M. B. Bainbridge (4 November 2011). "Indications of a Spatial Variation of the Fine Structure Constant" (PDF).*Physical Review Letters*.**107**(19): 191101. arXiv:1008.3907. Bibcode:2011PhRvL.107s1101W. doi:10.1103/PhysRevLett.107.191101. hdl:1959.3/207294. PMID 22181590.**^**Michael R. Wilczynska, John K. Webb; et al. (2020). "Four direct measurements of the fine-structure constant 13 billion years ago".*Science Advances*.**6**(17): 9672. Bibcode:2008Sci...319.1808R. doi:10.1126/sciadv.aay9672.**^**J. D. Barrow (2001). "Cosmology, Life, and the Anthropic Principle".*Annals of the New York Academy of Sciences*.**950**(1): 139–153. Bibcode:2001NYASA.950..139B. doi:10.1111/j.1749-6632.2001.tb02133.x. PMID 11797744.**^**A. S. Eddington (1956). "The Constants of Nature". In J.R. Newman (ed.).*The World of Mathematics*.**2**. Simon & Schuster. pp. 1074–1093.**^**Whittaker, Edmund (1945). "Eddington's Theory of the Constants of Nature".*The Mathematical Gazette*.**29**(286): 137–144. doi:10.2307/3609461. JSTOR 3609461.**^**P. Varlaki; L. Nadai; J. Bokor (2008). "Number Archetypes and Background Control Theory Concerning the Fine Structure Constant" (PDF).*Acta Polytechnica Hungarica*.**5**(2): 71.**^**A. I. Miller (2009).*Deciphering the Cosmic Number: The Strange Friendship of Wolfgang Pauli and Carl Jung*. W.W. Norton & Co. p. 253. ISBN 978-0-393-06532-9.**Max Born**: If alpha were bigger than it really is, we should not be able to distinguish matter from ether [the vacuum, nothingness], and our task to disentangle the natural laws would be hopelessly difficult. The fact however that alpha has just its value 1/137 is certainly no chance but itself a law of nature. It is clear that the explanation of this number must be the central problem of natural philosophy.**^**I. J. Good (1990). "A Quantal Hypothesis for Hadrons and the Judging of Physical Numerology". In G. R. Grimmett; D. J. A. Welsh (eds.).*Disorder in Physical Systems*. Oxford University Press. p. 141. ISBN 978-0-19-853215-6.**I. J. Good**: There have been a few examples of numerology that have led to theories that transformed society: see the mention of Kirchhoff and Balmer in Good (1962, p. 316) … and one can well include Kepler on account of his third law. It would be fair enough to say that numerology was the origin of the theories of electromagnetism, quantum mechanics, gravitation.... So I intend no disparagement when I describe a formula as numerological. When a numerological formula is proposed, then we may ask whether it is correct. … I think an appropriate definition of correctness is that the formula has a good explanation, in a Platonic sense, that is, the explanation could be based on a good theory that is not yet known but ‘exists’ in the universe of possible reasonable ideas.**^**Stephen Hawking (1988).*A Brief History of Time*. Bantam Books. pp. 7, 125. ISBN 978-0-553-05340-1.