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In physics, the fine-structure constant (usually denoted α, the Greek letter alpha) is a fundamental physical constant, namely the coupling constant characterizing the strength of the electromagnetic interaction. The numerical value of α is the same in all systems of units, because α is a dimensionless quantity.



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

\alpha =\ \frac{e^2}{(4 \pi \varepsilon_0)\hbar c}\ =\ \frac{e^2 c \mu_0}{2 h} = \frac{k_\mathrm{e} e^2}{\hbar c},


In electrostatic cgs units, the unit of electric charge, the statcoulomb, is defined so that the Coulomb constant, ke, or the permittivity factor, 4πε0, is 1 and dimensionless. Then the expression of the fine-structure constant becomes the abbreviated

\alpha = \frac{e^2}{\hbar c}

an expression commonly appearing in physics literature.


According to 2006 CODATA, the defining expression and recommended value for α are:[1]

 \alpha = \frac{e^2}{\hbar c \ 4 \pi \varepsilon_0} = 7.297\,352\,5376(50) \times 10^{-3} = \frac{1}{137.035\,999\,679(94)}.
Two example eighth-order Feynman diagrams that contribute to the electron self-interaction. The horizontal line with an arrow represents the electron while the wavy-lines are virtual photons, and the circles represent virtual electron-positron pairs.

However, after the 2006 CODATA adjustment was completed, an error was discovered in one of the main data inputs.[2] Nevertheless, the 2006 CODATA recommended value was republished in 2008.[3] A revised standard value, taking recent research and adjustments to SI units into account, is expected to be published in 2010 or early in 2011.

The definition of α can be estimated from the values of the constants appearing in any of its definitions. However, quantum electrodynamics (QED) provides a way to measure α directly using the quantum Hall effect or the anomalous magnetic moment of the electron.

QED predicts a relationship between the dimensionless magnetic moment of the electron (or the Lande g-factor, g) and the fine structure constant α. The most precise value of α obtained to date is based on a new measurement of g using a one-electron quantum cyclotron, together with a QED calculation involving 891 four-loop Feynman diagrams:[4]

\alpha^{-1} = 137.035\,999\,084(51).

This measurement has a precision of 0.37 ppb. This uncertainty is 20 times smaller than those of the nearest rival methods that include atom-recoil measurements. Comparisons of the measured and calculated values of g test QED very stringently, and limit the possible internal structure of the electron.

Physical interpretations

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

\alpha = \left( \frac{e}{q_\mathrm{P}} \right)^2.
  • The ratio of two energies: (i) the energy needed to overcome the electrostatic repulsion between two electrons when the distance between them is reduced from infinity to some finite d, and (ii) the energy of a single photon of wavelength λ = 2πd (see Planck relation):
\alpha = \frac{e^2}{4 \pi \varepsilon_0 d} \times \frac{\lambda}{h c} = \frac{e^2}{4 \pi \varepsilon_0 d} \times {\frac{2 \pi d}{h c}} = \frac{e^2}{4 \pi \varepsilon_0 \hbar c}.

When perturbation theory is applied to quantum electrodynamics, the resulting perturbative expansions for physical results are expressed as power series in α. Because α is clearly less than 1, higher powers of α are soon unimportant, making perturbation theory practical. In contrast, the large value of the corresponding factors in quantum chromodynamics makes calculations involving the strong force extremely difficult.

According to the theory of the renormalization group, the value of the fine structure constant (the strength of the electromagnetic interaction) grows logarithmically as the energy scale is increased. The observed value of α is 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 value of the fine structure constant at zero energy. Moreover, as the energy scale increases, the strength of the electromagnetic interaction approaches that of the other two fundamental interactions, a fact 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 makes quantum electrodynamics inconsistent beyond the perturbative expansions.


Arnold Sommerfeld introduced the fine-structure constant in 1916, as part of his theory of the relativistic deviations of atomic spectral lines from the predictions of the Bohr model. 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.[5] Equivalently, it was the quotient between the maximum 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.

The fine structure constant so intrigued the physicist Wolfgang Pauli that he even collaborated with the psychologist Carl Jung in an extraordinary quest to understand its significance.[6]

Is the fine structure constant truly constant?

Physicists have pondered for many years whether the fine structure constant is in fact constant, i.e., whether or not its value differs by location and over time. Specifically, a varying α has been proposed as a way of solving problems in cosmology and astrophysics.[7][8][9][10] More recently, theoretical interest in varying constants (not just α) has been motivated by string theory and other such proposals for going beyond the Standard Model of particle physics. The first experimental tests of this question examined the spectral lines of distant astronomical objects, and the products of radioactive decay in the Oklo natural nuclear fission reactor. The findings were consistent with no change.[11][12][13][14][15][16]

More recently, improved technology has 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.[17][18][19][20] 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

\frac{\Delta \alpha}{\alpha} \ \stackrel{\mathrm{def}}{=}\ \frac{\alpha _\mathrm{prev}-\alpha _\mathrm{now}}{\alpha_\mathrm{now}} = \left( -0.57\pm 0.10 \right) \times 10^{-5}.

In 2004, a smaller study of 23 absorption systems by Chand et al., using the Very Large Telescope, found no measureable variation:[21][22]

 \frac{\Delta \alpha}{\alpha_\mathrm{em}}= \left(-0.6\pm 0.6\right) \times 10^{-6}.

However, in 2007 simple flaws were identified in the analysis method of Chand et al., discrediting those results.[23][24] Nevertheless, systematic uncertainties are difficult to quantify and so the Webb et al.. results still need to be checked by independent analysis, using quasar spectra from different telescopes.

King et al. have used Markov Chain Monte Carlo methods to investigate the algorithm used by the UNSW group to determine Δα / α from the quasar spectra, and have found that the algorithm appears to produce correct uncertainties and maximum likelihood estimates for Δα / α for particular models[25]. This suggests that the statistical uncertainties and best estimate for Δα / 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 4.5 parts in 108. 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 but provide an interesting direction of study.[26][27][28][29]

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.[30] 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 109 (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 LOFAR telescope would only be able to constrain Δα/α to about 0.3%.[30] The collecting area required to constrain Δα/α to the current level of quasar constraints is on the order of 100 km2, which is impracticable at present.

In 2008, Rosenband et al. [31] used the frequency ratio of Al+ and Hg+ in single-ion optical atomic clocks to place a very stringent constraint on the present time variation of α, namely Δα̇/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 that predict a variable fine structure constant predict that the value of the fine structure constant should become frozen once the universe entered the current dark energy dominated epoch.

Anthropic explanation

The anthropic principle is a controversial explanation of why the fine-structure constant takes on 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 > 0.1, stellar fusion would be impossible and no place in the universe would be warm enough for life.[32]

The fine structure constant plays a central role in John Barrow and Frank Tipler's broad-ranging discussion of astrophysics, cosmology, quantum physics, teleology, and the anthropic principle.[33]

Numerological explanations

As a dimensionless constant which does not seem to be directly related to any mathematical constant, the fine-structure constant has long fascinated physicists. Richard Feynman, one of the founders of quantum electrodynamics, referred to it 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!

Richard P. Feynman (1985), QED: The Strange Theory of Light and Matter, Princeton University Press, p. 129, ISBN 0691083886 

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[34]. This led him in 1929 to conjecture that its reciprocal was precisely the integer 137. 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.[35] Attempts to find a mathematical basis for this dimensionless constant have continued up to the present time. For example, the mathematician James Gilson suggested (earliest entry dated December 2006 [1]), that the fine-structure constant has the value:

 \alpha = \frac{\cos \left(\pi/137 \right)}{137} \ \frac{\tan \left(\pi/(137 \cdot 29) \right)}{\pi/(137 \cdot 29)} \approx \frac{1}{137.0359997867} ,

29 and 137 being the 10th and 33rd prime numbers. The difference between the 2007 CODATA value for α and this theoretical value is about 3 x 10-11, about 6 times the standard error for the measured value.


The mystery about α is actually a double mystery. The first mystery — the origin of its numerical value α ≈ 1/137 has been recognized and discussed for decades. The second mystery — the range of its domain — is generally unrecognized.
Malcolm H. Mac Gregor, Malcolm H. Mac Gregor (2007), The Power of Alpha, World Scientific, p. 69, ISBN 9789812569615 
If alpha [the fine structure constant] 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.
Max Born, Arthur I. Miller (2009), Deciphering the Cosmic Number: The Strange Friendship of Wolfgang Pauli and Carl Jung, W.W. Norton & Co., p. 253, ISBN 9780393065329 

See also


  1. ^ NIST. "NIST reference on constants, units, and uncertainty". Retrieved 2009-04-11. 
  2. ^ G. Gabrielse; D. Hanneke; T. Kinoshita; M. Nio; B. Odom (2007). "Erratum: New Determination of the Fine Structure Constant from the Electron g Value and QED [Phys. Rev. Lett. 97, 030802 (2006)]". Phys. Rev. Lett. 99: 039902. doi:10.1103/PhysRevLett.99.039902. 
  3. ^ Mohr, P.J.; B.N. Taylor and D.B. Newell (2008). "CODATA Recommended Values of the Fundamental Physical Constants: 2006" (pdf). Reviews of Modern Physics 80: 633. doi:10.1103/RevModPhys.80.633. 
  4. ^ D. Hanneke; S. Fogwell; G. Gabrielse (2008). "New Measurement of the Electron Magnetic Moment and the Fine Structure Constant". Phys. Rev. Lett. 100: 120801. doi:10.1103/PhysRevLett.100.120801. 
  5. ^ "Introduction to the constants for nonexperts; Current advances: The fine-structure constant and quantum Hall effect". NIST. Retrieved 2009-04-11. 
  6. ^ P. Varlaki, L. Nadai, J. Bokor (2009). "Number Archetypes and Background Control Theory Concerning the Fine Structure Constant". Acta Polytechnica Hungarica 5 (2). 
  7. ^ E.A. Milne (1935). Relativity, Gravitation and World Structure. Clarendon Press. 
  8. ^ P.A.M. Dirac (1937). Nature 139: 323. 
  9. ^ G. Gamow (1967). Physical Review Letters 19: 757. 
  10. ^ G. Gamow (1967). Physical Review Letters 19: 913. 
  11. ^ J.-P. Uzan (2003). "The fundamental constants and their variation: observational status and theoretical motivations". Reviews of Modern Physics (American Physical Society) 75: 403–455. doi:10.1103/RevModPhys.75.403. arΧiv:hep-ph/0205340. 
  12. ^ Jean-Philippe Uzan (2004). "Variation of the constants in the late and early universe". arΧiv:astro-ph/0409424 [astro-ph]. 
  13. ^ K. Olive, Y.-Z. Qian (2003). "Were Fundamental Constants Different in the Past?". Physics Today 57 (10): 40–5. doi:10.1063/1.1825267. 
  14. ^ J.D. Barrow (2002). The Constants of Nature: From Alpha to Omega--the Numbers That Encode the Deepest Secrets of the Universe. Vintage. ISBN 0-09-928647-5. 
  15. ^ J.-P. Uzan, B. Leclercq (2008). The natural laws of the universe - Understanding fundamentral constants. Springer Praxis. ISBN 978-0-387-73454-5. 
  16. ^ F. Yasunori (2004). "Oklo Constraint on the Time-Variability of the Fine-Structure Constant". Astrophysics, Clocks and Fundamental Constants. Lecture Notes in Physics. Springer Berlin. pp. 167–185. ISBN 978-3-540-21967-5. 
  17. ^ J.K. Webb et al. (1999). "Search for Time Variation of the Fine Structure Constant". Physical Review Letters 82 (5): 884–887. doi:10.1103/PhysRevLett.82.884. arΧiv:astro-ph/9803165. 
  18. ^ M.T. Murphy et al. (2001). Monthly Notices of the Royal Astronomical Society 327: 1208. 
  19. ^ J.K. Webb et al. (2001). "Further Evidence for Cosmological Evolution of the Fine Structure Constant". Physical Review Letters 87 (9): 091301. doi:10.1103/PhysRevLett.87.091301. arΧiv:astro-ph/0012539. 
  20. ^ M.T. Murphy, J.K. Webb, V.V. Flambaum (2003). "Further evidence for a variable fine-structure constant from Keck/HIRES QSO absorption spectra". Monthly Notices of the Royal Astronomical Society 345: 609. doi:10.1046/j.1365-8711.2003.06970.x. 
  21. ^ H. Chand et al. (2004). "Probing the cosmological variation of the fine-structure constant: Results based on VLT-UVES sample". Astron. Astrophys. 417: 853. doi:10.1051/0004-6361:20035701. 
  22. ^ 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: 121302. doi:10.1103/PhysRevLett.92.121302. 
  23. ^ 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: 239001. doi:10.1103/PhysRevLett.99.239001. 
  24. ^ 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: 1053. doi:10.1111/j.1365-2966.2007.12695.x. 
  25. ^ King, Julian; Mortlock, Daniel; Webb, John; Murphy, Michael (2009). "Markov Chain Monte Carlo methods applied to measuring the fine structure constant from quasar spectroscopy". arΧiv:0910.2699 [astro-ph]. 
  26. ^ R. Kurzweil (2005). The Singularity Is Near. Viking Penguin. pp. 139–140. ISBN 0-670-03384-7. 
  27. ^ S.K. Lamoreaux, J.R. Torgerson (2004). "Neutron Moderation in the Oklo Natural Reactor and the Time Variation of Alpha". Physical Review D 69. arΧiv:nucl-th/0309048v3. 
  28. ^ E.S. Reich (30 June 2004). "Speed of light may have changed recently". New Scientist. Retrieved 2009-01-30. 
  29. ^ "Scientists Discover One Of The Constants Of The Universe Might Not Be Constant". ScienceDaily. 12 May 2005. Retrieved 2009-01-30. 
  30. ^ 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: 111301. doi:10.1103/PhysRevLett.98.111301. arΧiv:astro-ph/0701752. 
  31. ^ Rosenband, T., Hume, D., Schmidt, P. O., Chou, C. W., Brusch, A., Lorini, L. Oskay, W. H., Drullinger, R. E., Fortier, T. M., Stalnaker, J. E., Diddams, S. A., Swann, W. C., Newbury, N. R., Itano, W. M., Wineland, D. J., Berquist, J. C. (2008). "Frequency Ratio of Al+ and Hg+ Single-Ion Optical Clocks; Metrology at the 17th Decimal Place.". Science 319 (5871): 1808-. doi:10.1126/science.1154622. 
  32. ^ J.D. Barrow (2001). "Cosmology, Life, and the Anthropic Principle". Annals of the New York Academy of Sciences 950 (1): 139–153. doi:10.1111/j.1749-6632.2001.tb02133.x (inactive 2008-06-25). 
  33. ^
    [Edit this reference]
    Barrow, John D.; Tipler, Frank J. (19 May 1988). The Anthropic Cosmological Principle. foreword by John A. Wheeler. Oxford: Oxford University Press. LC 87-28148. ISBN 9780192821478. Retrieved 31 December 2009. 
  34. ^ Eddington, A.S., The Constants of Nature in "The World of Mathematics", Vol. 2 (1956) Ed. Newman, J.R., Simon and Schuster, pp. 1074-1093.
  35. ^ Helge Kragh, "Magic Number: A Partial History of the Fine-Structure Constant", Archive for History of Exact Sciences 57:5:395 (July, 2003) doi:10.1007/s00407-002-0065-7

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