View source for Fine-structure constant

{{Short description|Dimensionless number that quantifies the strength of the electromagnetic interaction}}
{{Use dmy dates|date=May 2022}}
{| class="wikitable floatright"
|-
! Value of {{math|''α''}} 
|-
| {{physconst|alpha|ref=no}}
|-
! Value of {{math|''α''{{sup|−1}}}}
|-
| {{physconst|alphainv|ref=no}}
|}

{{Quantum field theory}}
In [[physics]], the '''fine-structure constant''', also known as the '''Sommerfeld constant''', commonly denoted by {{mvar|&alpha;}} (the [[Alpha|Greek letter ''alpha'']]), is a [[Dimensionless physical constant|fundamental physical constant]] that quantifies the strength of the [[electromagnetic interaction]] between elementary charged particles.

It is a [[dimensionless quantity]] ([[dimensionless physical constant]]), independent of the [[system of units]] used, which is related to the strength of the coupling of an [[elementary charge]] {{mvar|e}} with the [[electromagnetic field]], by the formula <math display=inline>\alpha = { e^2 }\left( 4 \pi \varepsilon_0 \hbar c \right)^{-1}</math>. Its [[numerical value]] is approximately {{nobr|{{physconst|alpha|round=13|ref=no}} ≈ {{physconst|alphainv|round=9|ref=no}}<sup>−1</sup>}}, with a relative uncertainty of {{physconst|alpha|after=.|runc=yes}}

The constant was named by [[Arnold Sommerfeld]], who introduced it in 1916<ref name=Sommerfeld-1916>{{cite journal |author=Sommerfeld, Arnold |author-link=Arnold Sommerfeld |year=1916 |title=Zur Quantentheorie der Spektrallinien |journal=[[Annalen der Physik]] |volume=4 |issue=51 |pages=51–52 |url=https://babel.hathitrust.org/cgi/pt?id=nyp.33433090771183&view=1up&seq=65 |access-date=2020-12-06 }} Equation 12a, ''"rund 7·{{10^|-3}}" (about ...)''</ref> when extending the [[Bohr model]] of the atom. <math>\alpha</math> quantified the gap in the [[fine structure]] of the [[spectral lines]] of the hydrogen atom, which had been measured precisely by the [[Michelson–Morley experiment]] in 1887.{{efn|In [[quantum electrodynamics]], <math>\alpha</math> is proportional to the square of the [[coupling constant]] for a charged particle to the electromagnetic field. There are analogous coupling constants that give the interaction strength of the [[nuclear strong force]] and the [[nuclear weak force]].}}

Why the constant should have this value is not understood,<ref name=Feynman1985/> but there are a number of ways to [[Precision tests of QED|measure its value]].

== Definition ==
In terms of other [[physical constant]]s, <math>\alpha</math> may be defined as:{{cn|date=April 2026}}
<math display="block">\begin{align}
\alpha &= \frac{e^2}{2 \varepsilon_0 h c} = \frac{ e^2 }{ 4 \pi \varepsilon_0 \hbar c }\\
&= \frac{e^2\mu_0 c}{2 h} = \frac{ e^2 \mu_0 c}{ 4 \pi \hbar}\\
\end{align}
</math>
where
: {{mvar|e}} is the [[elementary charge]] ({{physconst|e}});
: {{mvar|h}} is the [[Planck constant]] ({{physconst|h}});
: {{mvar|&hbar;}} is the [[reduced Planck constant]], {{sfrac|{{mvar|h}}|2{{mvar|&pi;}}}} ({{physconst|hbar}});
: {{mvar|c}} is the [[speed of light]] ({{physconst|c}});
: {{mvar|&epsilon;}}<sub>0</sub> is the [[vacuum permittivity|electrical permittivity of space]] ({{physconst|eps0}});
:{{mvar|&mu;}}<sub>0</sub> is the [[vacuum permeability|magnetic permeability of space]] ({{physconst|mu0}}).

Since the [[2019 revision of the SI]], the only quantities on this list that do not have exact values in [[SI units]] are the [[vacuum permittivity|electrical permittivity of space]] and [[vacuum permeability|magnetic permeability of space]]; the others are all defining constants.

=== Alternative systems of units ===
The electrostatic [[CGS]] system implicitly sets {{tmath|1= 4\pi\varepsilon_0 = 1 }}, as commonly found in older physics literature, where the expression of the fine-structure constant becomes
<math display="block"> \alpha = \frac{e^2}{\ \hbar c\ } ~.</math>

A normalised system of units [[natural units|commonly used in high energy physics]] selects artificial units for mass, distance, time, and [[electrical charge]] which cause <math> \varepsilon_0 = c = \hbar = 1</math>; in such a system of "[[natural units]]" the expression for the fine-structure constant becomes<ref>
{{cite book |last1=Peskin |first1=M. |last2=Schroeder |first2=D. |year=1995 |title=An Introduction to Quantum Field Theory |publisher=[[Westview Press]] |isbn=978-0-201-50397-5 |page=[https://archive.org/details/introductiontoqu0000pesk/page/125 125] |url=https://archive.org/details/introductiontoqu0000pesk/page/125}}
</ref>
<math display="block"> \alpha = \frac{ e^2 }{\ 4 \pi\ } ~.</math>
As such, the fine-structure constant is chiefly a quantity determining (or determined by) the [[elementary charge]]: {{nobr|<math display="inline">\ e = \sqrt{ 4\pi\alpha\; } \approx</math> {{val|0.30282212}}&nbsp;}} in terms of such a natural unit of charge.

In the system of [[atomic units]], which sets {{tmath|1= e = \hbar = 4\pi\varepsilon_0 = 1 }}, the expression for the fine-structure constant becomes
<math display="block"> \alpha = \frac{ 1 }{\ c\  } ~.</math>

== Measurement ==
[[File:EighthOrderMagMoment.svg|frame|right|Eighth-[[Perturbation theory|order]] [[Feynman diagrams]] on electron self-interaction. The arrowed horizontal line represents the electron, the wavy lines are virtual photons, and the circles are virtual [[electron]]–[[positron]] pairs.]]
The [[CODATA]] recommended value of {{math|''α''}} is{{physconst|alpha|ref=only}}
{{block indent|{{math|1=''α'' = {{sfrac|''e''{{sup|2}}| 4''πε''{{sub|0}}''ħc''}}}} {{=}} {{physconst|alpha|ref=no}}.}} This has a relative standard uncertainty of {{physconst|alpha|runc=yes|after=.}}

This value for {{math|''α''}} gives the following value for the [[Vacuum permeability|vacuum magnetic permeability]] (magnetic constant): {{nowrap|1={{mvar|µ}}{{sub|0}} = 4''π'' × {{val|0.99999999987|(16)|e=-7|u=H.m-1}}}}, with the mean differing from the old defined value by only 0.13&nbsp;[[parts per billion]], 0.8 times the standard uncertainty (0.16 parts per billion) of its recommended measured value.

Historically, the value of the [[multiplicative inverse|reciprocal]] of the fine-structure constant is often given. The [[CODATA]] recommended value is {{physconst|alphainv|ref=only}}
{{block indent|{{math|{{sfrac|1|''α''}}}} {{=}} {{physconst|alphainv|ref=no}}.}}

While the value of {{mvar|α}} can be determined from estimates of the constants that appear in any of its definitions, the theory of [[quantum electrodynamics]] (QED) provides a way to measure {{mvar|α}} directly using the [[quantum Hall effect]] or the [[anomalous magnetic moment]] of the [[electron]].<ref name=Fan-Myers-etal-2023>
{{cite journal |last1=Fan |first1=X. |last2=Myers |first2=T.G. |last3=Sukra |first3=B.A.D. |last4=Gabrielse |first4=G. |date=2023-02-13 |title=Measurement of the Electron Magnetic Moment |url=https://link.aps.org/doi/10.1103/PhysRevLett.130.071801 |journal=Physical Review Letters |volume=130 |issue=7 |article-number=071801 |doi=10.1103/PhysRevLett.130.071801 |pmid=36867820 |arxiv=2209.13084 |bibcode=2023PhRvL.130g1801F }}</ref> Other methods include the A.C.&nbsp;Josephson effect and photon recoil in atom interferometry.<ref name=Yu2019>
{{cite journal
 |last1=Yu     |first1=C.         |last2=Zhong  |first2=W.
 |last3=Estey  |first3=B.         |last4=Kwan   |first4=J.
 |last5=Parker |first5=R.H.       |last6=Müller |first6=H.
 |year=2019
 |title=Atom-interferometry measurement of the fine structure constant
 |journal=Annalen der Physik
 |volume=531  |issue=5  |article-number=1800346
 |doi=10.1002/andp.201800346 |doi-access=free 
 |bibcode=2019AnP...53100346Y
}}
</ref>
There is general agreement for the value of {{mvar|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.<ref name=Yu2019/> The theory of QED predicts a relationship between the [[g-factor (physics)|dimensionless magnetic moment]] of the [[electron]] and the fine-structure constant {{mvar|α}} (the magnetic moment of the electron is also referred to as the [[g-factor (physics)|electron {{mvar|g}}-factor]] {{math|''g''<sub>e</sub>}}). One of the most precise values of {{mvar|α}} obtained experimentally (as of 2023) is based on a measurement of {{math|''g''<sub>e</sub>}} using a one-electron so-called "quantum cyclotron" apparatus,<ref name=Fan-Myers-etal-2023/> together with a calculation via the theory of QED that involved {{val|12672}} tenth-order [[Feynman diagrams]]:<ref name=Aoyama12>
{{cite journal
 |last1=Aoyama    |first1=T.     |last2=Hayakawa |first2=M.
 |last3=Kinoshita |first3=T.     |last4=Nio      |first4=M.
 |year=2012
 |title=Tenth-order QED contribution to the electron {{nowrap|''g'' − 2}} and an improved value of the fine structure constant
 |journal=[[Physical Review Letters]]
 |volume=109 |issue=11 |article-number=111807
 |arxiv=1205.5368 |bibcode=2012PhRvL.109k1807A
 |doi=10.1103/PhysRevLett.109.111807 |pmid=23005618
 |s2cid=14712017
}}
</ref>
: {{nobr|<math> \frac{ 1 }{ \alpha } =</math> {{val|137.035999166|(15)}} .}}

This measurement of {{mvar|α}} has a relative standard uncertainty of {{val|1.1|e=-10}}. This value and uncertainty are about the same as the latest experimental results.<ref>
{{cite journal
 |last1=Bouchendira |first1=Rym          |last2=Cladé |first2=Pierre
 |last3=Guellati-Khélifa |first3=Saïda   |last4=Nez   |first4=François
 |last5=Biraben     |first5=François
 |year=2011
 |title=New determination of the fine-structure constant and test of the quantum electrodynamics
 |journal=[[Physical Review Letters]]
 |volume=106  |issue=8  |article-number=080801
 |arxiv=1012.3627  |bibcode=2011PhRvL.106h0801B
 |doi=10.1103/PhysRevLett.106.080801 |pmid=21405559
 |s2cid=47470092
 |type=Submitted manuscript
 |url=https://hal.archives-ouvertes.fr/hal-00547525/file/MesureAlpha2010.pdf |archive-url=https://web.archive.org/web/20181104125931/https://hal.archives-ouvertes.fr/hal-00547525/file/MesureAlpha2010.pdf |archive-date=2018-11-04 |url-status=live
}}</ref>

Further refinement of the experimental value was published by the end of 2020, giving the value
{{block indent|{{math|{{sfrac|1|''α''}}}} {{=}} {{val|137.035999206|(11)}},}}
with a relative accuracy of {{val|8.1|e=-11}}, which has a significant discrepancy from the previous experimental value.<ref name=morel2020>
{{cite journal
 |last1=Morel |first1=Léo     |last2=Yao |first2=Zhibin 
 |last3=Cladé |first3=Pierre  |last4=Guellati-Khélifa |first4=Saïda 
 |year=2020
 |title=Determination of the fine-structure constant with an accuracy of 81&nbsp;parts per trillion
 |journal=[[Nature (journal)|Nature]]
 |volume=588 |pages=61–65
 |issue=7836 
 |doi=10.1038/s41586-020-2964-7
 |pmid=33268866 
 |bibcode=2020Natur.588...61M 
 |s2cid=227259475 
 |url=https://hal.archives-ouvertes.fr/hal-03107990/file/main.pdf 
}}</ref>

== Physical interpretations ==
The fine-structure constant, {{mvar|α}}, has several physical interpretations. {{mvar|α}} is:{{unordered list
| The ratio of two energies:{{ordered list |type=lower-roman
  | the energy needed to overcome the [[electrostatic repulsion]] between two electrons a distance of {{mvar|d}} apart, and
  | the energy of a single [[photon]] of wavelength {{math|''λ'' {{=}} 2''πd''}} (or of [[wavelength#Angular wavelength|angular wavelength]] {{mvar|d}}; see ''[[Planck relation]]''): <math display="block">\alpha = \left. { \left( \frac{e^2}{4\pi \varepsilon_0 d} \right) }\right/ { \left( \frac{hc}{\lambda} \right) } = \frac{e^2}{4\pi\varepsilon_0 d } \times {\frac{ 2 \pi d }{hc}} = \frac{e^2}{ 4 \pi \varepsilon_0 d } \times {\frac{d}{ \hbar c }} = \frac{e^2}{ 4 \pi \varepsilon_0 \hbar c } .</math>}}

| The ratio of the velocity of the electron in the first circular orbit of the [[Bohr model of the atom]], which is {{math|{{sfrac|1|4π{{var|ε}}{{sub|0}}}}{{sfrac|''e''{{sup|2}}|''ħ''}}}}, to the [[speed of light]] in vacuum, {{mvar|c}}.<ref>
{{cite book
 |last=Sommerfeld |first=A. |author-link=Arnold Sommerfeld
 |title=Atombau und Spektrallinien |lang=de
 |place=Braunschweig, DE
 |publisher=Friedr. Vieweg & Sohn
 |edition=2
 |year=1921
 |pages=241–242, Equation&nbsp;8
 |url=https://archive.org/stream/atombauundspekt00sommgoog?ref=ol#page/n261/mode/2up
 |quote=Das Verhältnis <math>v_{1}/c</math> nennen wir {{mvar|α}}. |trans-quote=The ratio <math>v_{1}/c</math> we call {{mvar|α}}.
}}<br/>
: {{cite book |url=https://babel.hathitrust.org/cgi/pt?id=mdp.39015078632786&view=1up&seq=233 |title=English translation |year=1923 |publisher=Methuen & co. }}
</ref> This is [[Arnold Sommerfeld|Sommerfeld]]'s original physical interpretation.
| <math>\ \alpha^2\ </math> is the ratio of the potential energy of the electron in the first circular orbit of the [[Bohr model of the atom]] and the energy {{math|''m''{{sub|e}}''c''{{sup|2}}}} equivalent to the mass of an electron. Using the [[virial theorem]] in the [[Bohr model of the atom]] {{tmath|1= U_\mathsf{el} = 2 U_\mathsf{kin} }}, which means that {{tmath|1= U_\mathsf{el} = m_\mathsf{e} v_\mathsf{e}^2 = m_\mathsf{e} (\alpha c)^2 = \alpha^2 (m_\mathsf{e} c^2) }}. Essentially this ratio follows from the electron's velocity being {{tmath|1= v_\mathsf{e} = \alpha c }}.

| The two ratios of three characteristic lengths: the [[classical electron radius]] {{math|''r''{{sub|e}}}}, the [[reduced Compton wavelength]] of the electron {{math|''ƛ''{{sub|e}}}}, and the [[Bohr radius]] {{math|''a''{{sub|0}}}}: {{math|1=''r''{{sub|e}} = ''αƛ''{{sub|e}} = ''α''{{sup|2}}''a''{{sub|0}}}}.

| In [[quantum electrodynamics]], {{mvar|α}} is directly related to the [[coupling constant]] determining the strength of the interaction between [[electron]]s and [[photon]]s.<ref>{{cite book |last1=Riazuddin |first1=Fayyazuddin |year=2012 |title=A Modern Introduction to Particle Physics |edition=3rd |publisher=World Scientific |isbn=978-981-4338-83-7 |page=4 |url=https://books.google.com/books?id=dbysnBTHF4QC |access-date=20 April 2017 }}</ref> The theory does not predict its value. Therefore, {{mvar|α}} must be determined experimentally. In fact, {{mvar|α}} is one of the empirical [[Standard Model#Theoretical aspects|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]], {{mvar|α}} is absorbed into two other [[coupling constant]]s associated with the electroweak [[gauge theory|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]], <math>\ Z_0 = \mu_0 c\ ,</math> and the [[conductance quantum]], <math>\ G_0 = 2 e^2 / h\ :</math> <math>\ \alpha = \tfrac{ 1 }{\ 4\ } Z_0\ G_0 ~.</math>  The [[optical conductivity]] of [[graphene]] for visible frequencies is theoretically given by {{math|{{sfrac| {{var|π}} |4}}{{var|G}}{{sub|0}}}}, and as a result its light absorption and transmission properties can be expressed in terms of the fine-structure constant alone.<ref name="NairBlake2008">{{cite journal |last1=Nair |first1=R.R. |last2=Blake |first2=P. |last3=Grigorenko |first3=A.N. |last4=Novoselov |first4=K.S. |last5=Booth |first5=T.J. |last6=Stauber |first6=T. |last7=Peres |first7=N. M.R. |last8=Geim |first8=A.K. |display-authors=6 |year=2008 |title=Fine Structure Constant Defines Visual Transparency of Graphene |journal=[[Science (journal)|Science]] |volume=320 |issue=5881 |page=1308 |bibcode=2008Sci...320.1308N |doi=10.1126/science.1156965 |pmid=18388259 |arxiv=0803.3718 |s2cid=3024573 }}</ref> The absorption value for normal-incident light on graphene in vacuum would then be given by {{math|{{sfrac|π{{var|α}}| (1 + π{{var|α}}/2){{sup|2}}}} }} or 2.24%, and the transmission by {{math|{{sfrac|1|(1 + π{{var|α}}/2){{sup|2}}}}}} or 97.75% (experimentally observed to be between 97.6% and 97.8%). The reflection would then be given by {{math|{{sfrac| π{{sup|2}} {{var|α}}{{sup|2}}| 4 (1 + π{{var|α}}/2){{sup|2}}}}}}.

| 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]]).<ref>
{{cite journal 
|last=Chandrasekhar |first=S. |author-link=Subrahmanyan Chandrasekhar
 |date=1 April 1984
 |title=On stars, their evolution and their stability
 |journal=[[Reviews of Modern Physics]]
 |volume=56 |issue=2 |pages=137–147
 |doi=10.1103/RevModPhys.56.137
 |bibcode=1984RvMP...56..137C
 |s2cid=2317589
}}
</ref> For an electron orbiting an atomic nucleus with atomic number {{mvar|Z}} the relation is {{math| {{sfrac|{{var|m}}{{var|v}}{{sup|2}}|{{var|r}}}} {{=}} {{sfrac|1| 4π{{var|ε}}{{sub|0}}}} {{sfrac|{{var|Z}}{{var|e}}{{sup|2}}|{{var|r}}{{sup|2}}}} }}. The Heisenberg [[uncertainty principle]] momentum/position uncertainty relationship of such an electron is just {{math|{{var|m}}{{var|v}}{{var|r}} {{=}} {{var|ħ}}}}. The relativistic limiting value for {{mvar|v}} is {{mvar|c}}, and so the limiting value for {{mvar|Z}} is the reciprocal of the fine-structure constant, 137.<ref>
{{cite journal
 |last1=Bedford |first1=D.
 |last2=Krumm   |first2=P.
 |year=2004
 |title=Heisenberg indeterminacy and the fine structure constant
 |journal=[[American Journal of Physics]]
 |volume=72 |issue=7 |page=969
 |doi=10.1119/1.1646135 |bibcode=2004AmJPh..72..969B
}}
</ref>

}}

When [[perturbation theory (quantum mechanics)|perturbation theory]] is applied to [[quantum electrodynamics]], the resulting [[perturbative]] expansions for physical results are expressed as sets of [[power series]] in {{mvar|α}}. Because {{mvar|α}} is much less than one, higher powers of {{mvar|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 [[logarithm]]ically as the relevant [[energy scale]] increases. The value of the fine-structure constant {{mvar|α}} is linked to the observed value of this coupling associated with the energy scale of the [[electron mass]]: the electron's mass gives a lower bound for this energy scale, because it (and the [[positron]]) is the lightest charged object whose [[Quantum electrodynamics#Renormalizability|quantum loops]] can contribute to the running. Therefore, {{sfrac|1| 137.03600 }} 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&nbsp;[[GeV]], one [[Coupling constant#QED and the Landau pole|instead]] measures an ''effective'' {{mvar|α}} ≈ 1/127.<ref>
{{cite journal
 |last=Fritzsch |first=Harald
 |year=2002
 |title=Fundamental constants at high energy
 |journal=Fortschritte der Physik
 |volume=50 |issue=5–7 |pages=518–524
 |doi=10.1002/1521-3978(200205)50:5/7<518::AID-PROP518>3.0.CO;2-F
 |arxiv=hep-ph/0201198 |bibcode=2002ForPh..50..518F |s2cid=18481179
}}</ref>

As the energy scale increases, the strength of the electromagnetic interaction in the [[Standard Model]] approaches that of the other two [[fundamental interaction]]s, a feature important for [[grand unification]] theories. In [[perturbative]] quantum electrodynamics as a stand-alone theory, the coupling diverges at an energy known as the [[Landau pole]] – this fact undermines the consistency of quantum electrodynamics in perturbative expansions. However, non-perturbative numerical experiments suggest that instead the Landau pole becomes inaccessible at such energies and QED becomes trivial.<ref>
{{cite journal
 | last1 = Göckeler | first1 = M.
 | last2 = Horsley | first2 = R.
 | last3 = Linke | first3 = V.
 | last4 = Rakow | first4 = P. E. L.
 | last5 = Schierholz | first5 = G.
 | last6 = Stüben | first6 = H.
 | title = Is there a Landau pole problem in QED?
 | journal = Physical Review Letters
 | volume = 80
 | issue = 19
 | pages = 4119–4122
 | year = 1998
 | doi = 10.1103/PhysRevLett.80.4119
 | eprint = hep-th/9712244
}}</ref><ref>
{{cite journal
 | last1 = Gies | first1 = H.
 | last2 = Jaeckel | first2 = J.
 | title = Renormalization flow of QED
 | journal = Physical Review Letters
 | volume = 93
 | issue = 11
 | article-number = 110405
 | year = 2004
 | doi = 10.1103/PhysRevLett.93.110405
 | arxiv = hep-ph/0405183
}}</ref>

== History ==
[[File:Sommerfeld-Muenchen.jpg|thumb|right|upright|Sommerfeld memorial at [[LMU Munich]] ]]
Based on the precise measurement of the hydrogen atom spectrum by [[Albert A. Michelson]] and [[Edward W. Morley]] in 1887,{{efn|
"Among other substances [that were] tried in the preliminary experiments, were thallium, lithium, and hydrogen. ... It may be noted, that in [the] case of the red hydrogen line, the interference phenomena disappeared at about 15,000&nbsp;wave-lengths, and again at about 45,000&nbsp;wave-lengths: So that the red hydrogen line must be a double line with the components about one-sixtieth as distant as the sodium lines."{{refn|
 {{cite journal
  |last1=Michelson |first1=Albert A.  |author1-link=Albert A. Michelson
  |last2=Morley    |first2=Edward W.  |author2-link=Edward W. Morley 
  |year=1887
  |title=Method of making the wave-length of sodium light the actual and practical standard of length
  |journal=[[The American Journal of Science]]
  |volume=34 |issue=204 |pages=427–430
  |url=https://babel.hathitrust.org/cgi/pt?id=coo.31924084352636&view=1up&seq=461
  |series=3rd series
 }} — Article reprinted same year in ''[[the Philosophical Magazine]]''.<ref>
  {{cite journal
   |last1=Michelson |first1=Albert A.  |author1-link=Albert A. Michelson
   |last2=Morley    |first2=Edward W.  |author2-link=Edward W. Morley 
   |year=1887
   |title=Method of making the wave-length of sodium light the actual and practical standard of length
   |journal=[[The Philosophical Magazine]]
   |volume=24 |issue=151 |pages=463–466
   |url=https://babel.hathitrust.org/cgi/pt?id=mdp.39015024088174&view=1up&seq=493
   |series=5th series
   |type=reprint
  }}
  </ref>
}}{{rp|style=ama|p=430}}
}} 
[[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.{{efn|
''"Wir fügen den Bohrschen Gleichungen (46) und (47) die charakteristische Konstante unserer Feinstrukturen'' {{nowrap|1=(49) ''&alpha;'' = {{sfrac|2''&pi;e''<sup>2</sup>|''ch''}}}} ''hinzu, die zugleich mit der Kenntnis des Wasserstoffdubletts oder des Heliumtripletts in §10 oder irgend einer analogen Struktur bekannt ist."''<br/>
{{---}}<br/>
(We add, to Bohr's equations (46) and (47), the characteristic constant of our fine structures {{nowrap|1=(49) ''&alpha;'' = {{sfrac|2''&pi;e''<sup>2</sup>|''ch''}}}} which is known at once from knowledge of the hydrogen doublet or the helium triplet in §10 or any analogous structure.)<ref>
 {{cite journal
  |last=Sommerfeld |first=A. |author-link=Arnold Sommerfeld
  |date=1916
  |title=Zur Quantentheorie der Spektrallinien |lang=de
  |trans-title=On the quantum theory of spectral lines
  |journal=Annalen der Physik
  |volume=51 |issue=17 |pages=1–94
  |series=4th series
  |doi=10.1002/andp.19163561702
  |bibcode=1916AnP...356....1S
  |url=https://zenodo.org/records/1424309/files/article.pdf
 }}
</ref>{{rp|style=ama|p=[https://babel.hathitrust.org/cgi/pt?id=nyp.33433090771183&view=1up&seq=107  91]}}
}}
The first physical interpretation of the fine-structure constant {{mvar|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.<ref>
{{cite web
 |title=Current advances: The fine-structure constant and quantum Hall effect
 |series=Introduction to the constants for nonexperts
 |website=The NIST Reference on Constants, Units, and Uncertainty (physics.nist.gov/cuu)
 |publisher=[[National Institute for Standards and Technology]]
 |url=http://physics.nist.gov/cuu/Constants/alpha.html
 |access-date=11 April 2009
}}</ref>
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|fine-structure]] of the hydrogenic [[Lyman series|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.<ref name=Kragh-2003/>{{rp|407}}

With the development of [[quantum electrodynamics]] (QED) the significance of {{math|''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 {{math|{{sfrac|''α''|2''π''}}}} is engraved on the tombstone of one of the pioneers of QED, [[Julian Schwinger]], referring to his calculation of the [[anomalous magnetic dipole moment]].

=== History of measurements ===
{| class="wikitable"
|+ Successive values determined for the fine-structure constant<ref name="The number 137.035... at MROB">
 {{cite web
  |title = The number 137.035...
  |website=MROB
  |url=https://mrob.com/pub/num/n-b137_035.html
 }}
</ref>{{efn|Numbers in parentheses (e.g. the "(11)" appearing at the end of the value "137.035999206(11)") give its [[standard uncertainty]] referred to the least significant preceding digit.}}
! Date
! {{math|''α''}}
! {{math|1/''α''}}
! Sources
|-
| 1969 Jul
| 0.007297351(11)
| 137.03602(21)
| CODATA 1969
|-
| 1973
| 0.0072973461(81)
| 137.03612(15)
| CODATA 1973
|-
| 1987 Jan
| 0.00729735308(33)
| 137.0359895(61)
| CODATA 1986
|-
| 1998
| 0.007297352582(27)
| 137.03599883(51)
| Kinoshita
|-
| 2000 Apr
| 0.007297352533(27)
| 137.03599976(50)
| CODATA 1998
|-
| 2002
| 0.007297352568(24)
| 137.03599911(46)
| CODATA 2002
|-
| 2007 Jul
| 0.0072973525700(52)
| 137.035999070(98)
| Gabrielse (2007)
|-
| 2008 Jun
| 0.0072973525376(50)
| 137.035999679(94)
| CODATA 2006
|-
| 2008 Jul
| 0.0072973525692(27)
| 137.035999084(51)
| Gabrielse (2008), Hanneke (2008)
|-
| 2010 Dec
| 0.0072973525717(48)
| 137.035999037(91)
| Bouchendira (2010)
|-
| 2011 Jun
| 0.0072973525698(24)
| 137.035999074(44)
| CODATA 2010
|-
| 2015 Jun
| 0.0072973525664(17)
| 137.035999139(31)
| CODATA 2014
|-
| 2017 Jul
| 0.0072973525657(18)
| 137.035999150(33)
| Aoyama ''et al''. (2017)<ref name=Aoyama-2018>
 {{cite journal
  |first1=Tatsumi  |last1=Aoyama
  |first2=Toichiro |last2=Kinoshita
  |first3=Makiko   |last3=Nio
  |date=8 February 2018
  |title=Revised and improved value of the QED tenth-order electron anomalous magnetic moment
  |journal=[[Physical Review D]]
  |volume=97 |issue=3 |article-number=036001
  |doi=10.1103/PhysRevD.97.036001 |arxiv=1712.06060
  |bibcode=2018PhRvD..97c6001A |s2cid=118922814
 }}</ref>
|-
| 2018 Dec
| 0.0072973525713(14)
| 137.035999046(27)
| Parker, Yu, ''et al''. (2018)<ref name="Parker">
 {{cite journal
  |first1=Richard H. |last1=Parker       |first2=Chenghui |last2=Yu
  |first3=Weicheng   |last3=Zhong        |first4=Brian    |last4=Estey
  |first5=Holger     |last5=Müller
  |year=2018
  |title=Measurement of the fine-structure constant as a test of the Standard Model
  |journal=[[Science (journal)|Science]]
  |volume=360 |issue=6385 |pages=191–195
  |doi=10.1126/science.aap7706
  |pmid=29650669 |arxiv=1812.04130 |bibcode=2018Sci...360..191P |s2cid=4875011
 }}</ref>
|-
| 2019 May
| 0.0072973525693(11)
| 137.035999084(21)
| CODATA 2018
|-
| 2020 Dec
| 0.0072973525628(6)<!--Reciprocals of published max/avg/min are ...622014/...627871/...633729, rounded to ...622/628/634-->
| 137.035999206(11)
| Morel ''et al''. (2020)<ref name=morel2020/>
|-
| 2022 Dec
| 0.0072973525643(11)
| 137.035999177(21)
| CODATA 2022
|-
| 2023 Feb
| 0.0072973525649(8)
| 137.035999166(15)
| Fan ''et al''. (2023)<ref name=Fan-Myers-etal-2023/>{{efn|This is not an experimentally measured value; instead it is a value determined ''by the current theory'' from an experimentally determined value of the [[electron magnetic moment]].}}
|}

The CODATA values in the above table are computed by averaging other measurements; they are not independent experiments.

== Potential variation over time ==
{{Further|Time-variation of fundamental constants}}
Physicists have pondered whether the fine-structure constant is in fact constant, or whether its value differs by location and over time. A varying {{mvar|α}} has been proposed as a way of solving problems in [[physical cosmology|cosmology]] and [[astrophysics]].<ref>
{{cite book
 |last=Milne |first=E.A. |author-link=E. A. Milne
 |year=1935
 |title=Relativity, Gravitation, and World Structure
 |publisher=[[Clarendon Press]]
}}
</ref><ref>
{{cite journal
 |last=Dirac |first=Paul A.M. |author-link=Paul Dirac
 |year=1937
 |title=The cosmological constants
 |journal=[[Nature (journal)|Nature]]
 |volume=139 |issue=3512 |page=323
 |bibcode=1937Natur.139..323D
 |doi=10.1038/139323a0 |s2cid=4106534
}}
</ref><ref>
{{cite journal
 |last=Gamow |first=G. |author-link=George Gamow
 |year=1967
 |title=Electricity, gravity, and cosmology
 |journal=[[Physical Review Letters]]
 |volume=19 |issue=13 |pages=759–761
 |bibcode=1967PhRvL..19..759G
 |doi=10.1103/PhysRevLett.19.759
}}
</ref><ref>
{{cite journal
 |last=Gamow |first=G.  |author-link=George Gamow
 |year=1967
 |title=Variability of elementary charge and quasistellar objects
 |journal=[[Physical Review Letters]]
 |volume=19 |issue=16 |pages=913–914
 |bibcode=1967PhRvL..19..913G
 |doi=10.1103/PhysRevLett.19.913
}}
</ref> [[String theory]] and other proposals for going beyond the [[Standard Model]] of particle physics have led to theoretical interest in whether the accepted [[physical constant]]s (not just {{mvar|α}}) actually vary.

In the experiments below, {{math|Δ''α''}} represents the change in {{mvar|α}} over time, which can be computed by  {{mvar|α}}<sub>past</sub> − {{mvar|α}}<sub>now</sub>&nbsp;. If the fine-structure constant really is a constant, then any experiment should show that
<math display="block"> \frac{\ \Delta \alpha\ }{\alpha} ~~ \overset{\underset{\mathsf{~def~}}{}}{=} ~~ \frac{\ \alpha _\mathsf{past} - \alpha _\mathsf{now}\ }{\alpha_\mathsf{now}} ~~=~~ 0 ~,</math>
or as close to zero as experiment can measure. Any value far away from zero would indicate that {{mvar|α}} does change over time. So far, most experimental data is consistent with {{mvar|α}} being constant, up to 10 digits of accuracy.

=== Past rate of change ===
The first experimenters to test whether the fine-structure constant might actually vary examined the [[spectral line]]s 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.<ref>
{{cite journal
 |last=Uzan |first=J.-P.
 |year=2003
 |title=The fundamental constants and their variation: Observational status and theoretical motivations
 |journal=[[Reviews of Modern Physics]]
 |volume=75 |issue=2 |pages=403–455
 |arxiv=hep-ph/0205340 |bibcode=2003RvMP...75..403U
 |doi=10.1103/RevModPhys.75.403 |s2cid=118684485
}}
</ref><ref>
{{cite journal
 |last=Uzan |first=J.-P.
 |year=2004
 |title=Variation of the constants in the late and early universe
 |journal=[[AIP Conference Proceedings]]
 |volume=736  |pages=3–20
 |arxiv=astro-ph/0409424 |bibcode=2004AIPC..736....3U
 |doi=10.1063/1.1835171 |s2cid=15435796
}}
</ref><ref>
{{cite magazine
 |last1=Olive |first1=K.
 |last2=Qian  |first2=Y.-Z.
 |year=2003
 |title=Were fundamental constants different in the past?
 |magazine=[[Physics Today]]
 |volume=57 |issue=10 |pages=40–45
 |bibcode=2004PhT....57j..40O
 |doi=10.1063/1.1825267
}}
</ref><ref>
{{cite book
 |last=Barrow |first=J.D.
 |year=2002
 |title=The Constants of Nature: From Alpha to Omega – the Numbers That Encode the Deepest Secrets of the Universe
 |publisher=[[Random House|Vintage]]
 |isbn=978-0-09-928647-9
}}
</ref><ref>
{{cite book
 |last1=Uzan     |first1=J.-P.
 |last2=Leclercq |first2=B.
 |year=2008
 |title=The Natural Laws of the Universe: Understanding fundamental constants
 |series=Springer-Praxis Books in Popular Astronomy
 |publisher=[[Springer Science+Business Media|Springer Praxis]]
 |isbn=978-0-387-73454-5
 |bibcode=2008nlu..book.....U
}}
</ref><ref>
{{cite book
 |last=Fujii  |first=Yasunori
 |year=2004
 |chapter=Oklo constraint on the time-variability of the fine-structure constant
 |title=Astrophysics, Clocks, and Fundamental Constants
 |series=Lecture Notes in Physics
 |volume=648 |pages=167–185
 |isbn=978-3-540-21967-5
 |doi=10.1007/978-3-540-40991-5_11
}}
</ref><ref>{{cite journal |last=Petrov |first=Yu. V. |author2=Nazarov, A. I. |author3=Onegin, M. S. |author4=Sakhnovsky, E. G.  |year=2006 |title=Natural nuclear reactor at Oklo and variation of fundamental constants: Computation of neutronics of a fresh core |journal=Physical Review C |volume=74 |issue=6 |article-number=064610 |doi=10.1103/PHYSREVC.74.064610 |arxiv = hep-ph/0506186 |bibcode = 2006PhRvC..74f4610P |s2cid=118272311 }}</ref> The most recent constraint from Oklo, from Davis & Hamdan (2015), set an upper limit of 11&nbsp;ppb difference at 95% confidence level, a constraint comparable in strength to that from atomic-clock measurements.<ref>{{Cite journal|last1=Davis|first1=Edward D.|last2=Hamdan|first2=Leila|date=2015|title=Reappraisal of the limit on the variation in α implied by the Oklo natural fission reactors|journal=Physical Review C|volume=92|issue=1|article-number=014319|bibcode=2015PhRvC..92a4319D|doi=10.1103/physrevc.92.014319|arxiv=1503.06011|s2cid=119227720}}</ref>

Improved technology at the dawn of the 21st century made it possible to probe the value of {{mvar|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 {{mvar|α}}.<ref>
{{cite journal
 |last1=Webb      |first1=John K.           |last2=Flambaum   |first2=Victor V.
 |last3=Churchill |first3=Christopher W.    |last4=Drinkwater |first4=Michael J.
 |last5=Barrow    |first5=John D.
 |date=February 1999
 |title=Search for time variation of the fine structure constant
 |journal=[[Physical Review Letters]]
 |volume=82 |issue=5 |pages=884–887
 |doi=10.1103/PhysRevLett.82.884 |arxiv=astro-ph/9803165
 |bibcode=1999PhRvL..82..884W |s2cid=55638644
}}
</ref><ref>
{{cite journal
 |last1=Murphy    |first1=M.T.       |last2=Webb      |first2=J.K.
 |last3=Flambaum  |first3=V.V.       |last4=Dzuba     |first4=V.A.
 |last5=Churchill |first5=C.W.       |last6=Prochaska |first6=J.X.
 |last7=Barrow    |first7=J.D.       |last8=Wolfe     |first8=A.M.
 |display-authors=6
 |date=11 November 2001
 |title=Possible evidence for a variable fine-structure constant from QSO absorption lines: motivations, analysis and results
 |journal=[[Monthly Notices of the Royal Astronomical Society]]
 |volume=327 |issue=4 |pages=1208–1222
 |doi=10.1046/j.1365-8711.2001.04840.x |doi-access=free |arxiv=astro-ph/0012419
 |bibcode=2001MNRAS.327.1208M |s2cid=14294586
}}
</ref><ref>
{{cite journal
 |last1=Webb      |first1=J.K.       |last2=Murphy    |first2=M.T.
 |last3=Flambaum  |first3=V.V.       |last4=Dzuba     |first4=V.A.
 |last5=Barrow    |first5=J.D.       |last6=Churchill |first6=C.W.
 |last7=Prochaska |first7=J.X.       |last8=Wolfe     |first8=A.M.
 |display-authors=6
 |date=9 August 2001
 |title=Further evidence for cosmological evolution of the fine structure constant
 |journal=[[Physical Review Letters]]
 |volume=87 |issue=9 |article-number=091301
 |doi=10.1103/PhysRevLett.87.091301 |pmid=11531558
 |arxiv=astro-ph/0012539 |bibcode=2001PhRvL..87i1301W
 |s2cid=40461557
}}
</ref><ref>
{{cite journal
 |last1=Murphy   |first1=M.T.
 |last2=Webb     |first2=J.K.
 |last3=Flambaum |first3=V.V.
 |date=October 2003
 |title=Further evidence for a variable fine-structure constant from Keck/HIRES QSO absorption spectra
 |journal=[[Monthly Notices of the Royal Astronomical Society]]
 |volume=345 |issue=2 |pages=609–638
 |doi=10.1046/j.1365-8711.2003.06970.x |doi-access=free
 |arxiv=astro-ph/0306483
 |bibcode=2003MNRAS.345..609M |s2cid=13182756
}}
</ref>
Using the [[Keck telescopes]] and a data set of 128 [[quasar]]s at [[redshift]]s {{math|0.5 < ''z'' < 3}}, Webb ''et al.'' found that their spectra were consistent with a slight increase in {{mvar|α}} over the last 10–12&nbsp;billion years. Specifically, they found that
<math display="block">\frac{\ \Delta \alpha\ }{\alpha} ~~ \overset{\underset{\mathsf{~def~}}{}}{=} ~~ \frac{\ \alpha _\mathrm{prev}-\alpha _\mathrm{now}\ }{\alpha_\mathrm{now}} ~~=~~  \left(-5.7\pm 1.0 \right) \times 10^{-6} ~.</math>

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

In 2004, a smaller study of 23&nbsp;absorption systems by Chand ''et al.'', using the [[Very Large Telescope]], found no measurable variation:<ref>
{{cite journal
 |last1=Chand     |first1=H.       |last2=Srianand |first2=R.
 |last3=Petitjean |first3=P.       |last4=Aracil   |first4=B.
 |date=April 2004
 |title=Probing the cosmological variation of the fine-structure constant: Results based on VLT-UVES sample
 |journal=[[Astronomy & Astrophysics]]
 |volume=417 |issue=3 |pages=853–871
 |doi=10.1051/0004-6361:20035701 |arxiv=astro-ph/0401094
 |bibcode=2004A&A...417..853C |s2cid=17863903
}}
</ref><ref>
{{cite journal
 |last1=Srianand  |first1=R.   |last2=Chand  |first2=H.
 |last3=Petitjean |first3=P.   |last4=Aracil |first4=B.
 |date=26 March 2004
 |title=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
 |journal=[[Physical Review Letters]]
 |volume=92 |issue=12 |article-number=121302
 |doi=10.1103/PhysRevLett.92.121302 |pmid=15089663
 |arxiv=astro-ph/0402177 |bibcode=2004PhRvL..92l1302S
 |s2cid=29581666
}}
</ref>
<math display="block"> \frac{\Delta \alpha}{\alpha_\mathrm{em}}\ =\ \left(-0.6\pm 0.6\right) \times 10^{-6} ~.</math>

However, in 2007 simple flaws were identified in the analysis method of Chand ''et al.'', discrediting those results.<ref>
{{cite journal
 |last1=Murphy   |first1=M.T.
 |last2=Webb     |first2=J.K.
 |last3=Flambaum |first3=V.V.
 |date=6 December 2007
 |title=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'
 |journal=[[Physical Review Letters]]
 |volume=99 |issue=23 |article-number=239001
 |doi=10.1103/PhysRevLett.99.239001 |pmid=18233422
 |arxiv=0708.3677 |bibcode=2007PhRvL..99w9001M
 |s2cid=29266168
}}
</ref><ref>
{{cite journal
 |last1=Murphy   |first1=M.T.
 |last2=Webb     |first2=J.K.
 |last3=Flambaum |first3=V.V.
 |date=March 2008
 |title=Revision of VLT/UVES constraints on a varying fine-structure constant
 |journal=[[Monthly Notices of the Royal Astronomical Society]]
 |volume=384 |issue=3 |pages=1053–1062
 |doi=10.1111/j.1365-2966.2007.12695.x |doi-access=free
 |arxiv=astro-ph/0612407
 |bibcode=2008MNRAS.384.1053M |s2cid=10476451
}}
</ref>

King ''et al.'' have used [[Markov chain Monte Carlo]] methods to investigate the algorithm used by the UNSW group to determine {{sfrac|{{math|Δ''α''}} | {{mvar|α}} }} from the quasar spectra, and have found that the algorithm appears to produce correct uncertainties and maximum likelihood estimates for {{sfrac|{{math|Δ''α''}} | {{mvar|α}} }} for particular models.<ref>
{{cite journal
 |last1=King |first1=J.A.    |last2=Mortlock |first2=D.J.
 |last3=Webb |first3=J.K.    |last4=Murphy   |first4=M.T.
 |year=2009
 |title=Markov chain Monte Carlo methods applied to measuring the fine structure constant from quasar spectroscopy
 |journal=Memorie della Societa Astronomica Italiana
 |volume=80 |page=864
 |bibcode=2009MmSAI..80..864K
 |arxiv=0910.2699
}}
</ref> This suggests that the statistical uncertainties and best estimate for {{sfrac|{{math|Δ''α''}} | {{mvar|α}} }} stated by Webb ''et al.'' and Murphy ''et al.'' are robust.

In 2007, Khatri and [[Benjamin D. Wandelt|Wandelt]] of the University of Illinois at Urbana-Champaign realized that the [[hydrogen line|21&nbsp;cm hyperfine transition in neutral hydrogen]] of the early universe leaves a unique absorption line imprint in the [[cosmic microwave background]] radiation.<ref name=Khatri>
{{cite journal
 |last1=Khatri  |first1=Rishi
 |last2=Wandelt |first2=Benjamin D.
 |date=14 March 2007
 |title=21&nbsp;cm radiation: A new probe of variation in the fine-structure constant
 |journal=[[Physical Review Letters]]
 |volume=98 |issue=11 |article-number=111301
 |doi=10.1103/PhysRevLett.98.111301 |pmid=17501040
 |arxiv=astro-ph/0701752 |bibcode=2007PhRvL..98k1301K
 |s2cid=43502450
}}
</ref> They proposed using this effect to measure the value of {{mvar|α}} during the epoch before the formation of the first stars. In principle, this technique provides enough information to measure a variation of 1&nbsp;part in {{val|e=9}} (4&nbsp;orders of magnitude better than the current quasar constraints). However, the constraint which can be placed on {{mvar|α}} is strongly dependent upon effective integration time, going as {{frac|{{sqrt|{{mvar|t}} }} }}. The European [[Low-Frequency Array (LOFAR)|LOFAR]] [[radio telescope]] would only be able to constrain {{sfrac| {{math|Δ}}{{mvar|α}} | {{mvar|α}} }} to about 0.3%.<ref name=Khatri/> The collecting area required to constrain {{sfrac| {{math|Δ}}{{mvar|α}} | {{mvar|α}} }} to the current level of quasar constraints is on the order of 100&nbsp;square kilometers, which is economically impracticable at present.

=== Present rate of change ===
In 2008, Rosenband ''et al.''<ref>
{{cite journal
 |last1 =Rosenband |first1 =T.           |last2 =Hume       |first2 =D.B.
 |last3 =Schmidt   |first3 =P.O.         |last4 =Chou       |first4 =C.W.
 |last5 =Brusch    |first5 =A.           |last6 =Lorini     |first6 =L.
 |last7 =Oskay     |first7 =W.H.         |last8 =Drullinger |first8 =R.E.
 |last9 =Fortier   |first9 =T.M.         |last10=Stalnaker  |first10=J.E.
 |last11=Diddams   |first11=S.A.         |last12=Swann      |first12=W.C.
 |last13=Newbury   |first13=N.R.         |last14=Itano      |first14=W.M.
 |last15=Wineland  |first15=D.J.         |last16=Bergquist  |first16=J.C.
 |display-authors=6
 |date=28 March 2008
 |title=Frequency ratio of Al{{sup|+}} and Hg{{sup|+}} single-ion optical clocks; metrology at the 17th&nbsp;decimal place
 |journal=Science
 |volume=319 |issue=5871 |pages=1808–1812
 |doi=10.1126/science.1154622 |pmid=18323415
 |bibcode=2008Sci...319.1808R |s2cid=206511320
 |url=https://zenodo.org/record/1230892
 |doi-access=free
}}
</ref> used the frequency ratio of {{chem2|Al+}} and {{chem2|Hg+}} in single-ion optical atomic clocks to place a very stringent constraint on the present-time temporal variation of {{mvar|α}}, namely {{sfrac| {{math|Δ}}{{mvar|α}} | {{mvar|α}} }} = {{val|-1.6|2.3|e=-17}} per year. A present day null constraint on the time variation of alpha does not necessarily rule out time variation in the past. Indeed, some theories<ref>
{{cite journal
 |last1=Barrow   |first1=John D.
 |last2=Sandvik  |first2=Håvard Bunes
 |last3=Magueijo |first3=João
 |date=21 February 2002
 |title=Behavior of varying-alpha cosmologies
 |journal=[[Physical Review D]]
 |volume=65  |issue=6  |article-number=063504
 |doi=10.1103/PhysRevD.65.063504  |arxiv=astro-ph/0109414
 |bibcode=2002PhRvD..65f3504B  |s2cid=118077783
}}
</ref> 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 ==
Measuring the tiny spatial variation of the fine-structure constant with astronomical observations is a challenging project. 
In 2011, evidence for spatial variation across the observable universe was obtained by reanalyzing an existing spectroscopy data set.<ref name="Kumar-2023">{{Cite journal |last=Kumar Aluri |first=Pavan |last2=Cea |first2=Paolo |last3=Chingangbam |first3=Pravabati |last4=Chu |first4=Ming-Chung |last5=Clowes |first5=Roger G |last6=Hutsemékers |first6=Damien |last7=Kochappan |first7=Joby P |last8=Lopez |first8=Alexia M |last9=Liu |first9=Lang |last10=Martens |first10=Niels C M |last11=Martins |first11=C J A P |last12=Migkas |first12=Konstantinos |last13=Ó Colgáin |first13=Eoin |last14=Pranav |first14=Pratyush |last15=Shamir |first15=Lior |date=May 4, 2023 |title=Is the observable Universe consistent with the cosmological principle? |url=https://iopscience.iop.org/article/10.1088/1361-6382/acbefc |journal=Classical and Quantum Gravity |volume=40 |issue=9 |page=094001 |doi=10.1088/1361-6382/acbefc |issn=0264-9381|arxiv=2207.05765 }}</ref><ref>
{{cite news
 |last=Johnston |first=H.
 |date=2 September 2010
 |url=https://physicsworld.com/a/changes-spotted-in-fundamental-constant/
 |title=Changes spotted in fundamental constant
 |website=[[Physics World]]
 |access-date=11 September 2010
}}
</ref><ref name=Webb-King-etal-2011>
{{cite journal
 |last1=Webb     |first1=J.K.       |last2=King       |first2=J.A.
 |last3=Murphy   |first3=M.T.       |last4=Flambaum   |first4=V.V.
 |last5=Carswell |first5=R.F.       |last6=Bainbridge |first6=M.B.
 |date=31 October 2011
 |title=Indications of a spatial variation of the fine structure constant
 |journal=[[Physical Review Letters]]
 |volume=107  |issue=19  |article-number=191101
 |doi=10.1103/PhysRevLett.107.191101  |pmid=22181590
 |arxiv=1008.3907  |bibcode=2011PhRvL.107s1101W
 |hdl=1959.3/207294  |hdl-access=free |s2cid=23236775
}}
</ref> A second set of measurements confirmed some effect, at the level of a few part per million with a large uncertainty, {{val|2e-5|7}}.<ref>
{{cite journal
 |last1 =Wilczynska  |first1 =Michael R.     |last2 =Webb     |first2 =John K.
 |last3 =Bainbridge  |first3 =Matthew        |last4 =Barrow   |first4 =John D.
 |last5 =Bosman      |first5 =Sarah E.I.     |last6 =Carswell |first6 =Robert F.
 |last7 =Dąbrowski   |first7 =Mariusz P.     |last8 =Dumont   |first8 =Vincent
 |last9 =Lee         |first9 =Chung-Chi      |last10=Leite    |first10=Ana Catarina
 |last11=Leszczyńska |first11=Katarzyna      |last12=Liske    |first12=Jochen
 |last13=Marosek     |first13=Konrad         |last14=Martins  |first14=Carlos J.A.P.
 |last15=Milaković   |first15=Dinko          |last16=Molaro   |first16=Paolo
 |last17=Pasquini    |first17=Luca
 |display-authors=6
 |date=1 April 2020
 |title=Four direct measurements of the fine-structure constant 13 billion years ago
 |journal=[[Science Advances]]
 |volume=6 |issue=17 |article-number=eaay9672
 |doi=10.1126/sciadv.aay9672 |pmid=32917582
 |pmc=7182409 |arxiv=2003.07627 |bibcode=2020SciA....6.9672W
}}
</ref>
Subsequent and more sophisticated analysis found no variation down the parts per million level.<ref>
{{cite journal
 |last1=Milaković |first1=Dinko          |last2=Lee      |first2=Chung-Chi
 |last3=Carswell  |first3=Robert F.      |last4=Webb     |first4=John K.
 |last5=Molaro    |first5=Paolo          |last6=Pasquini |first6=Luca
 |date=5 March 2021
 |title=A new era of fine structure constant measurements at high redshift
 |journal=[[Monthly Notices of the Royal Astronomical Society]]
 |volume=500 |pages=1–21
 |doi=10.1093/mnras/staa3217 |doi-access=free |arxiv=2008.10619
}}
</ref><ref>{{Cite journal |last=Perivolaropoulos |first=L. |last2=Skara |first2=F. |date=December 2022 |title=Challenges for Λ CDM: An update |url=https://linkinghub.elsevier.com/retrieve/pii/S1387647322000185 |journal=New Astronomy Reviews |language=en |volume=95 |article-number=101659 |doi=10.1016/j.newar.2022.101659|url-access=subscription }}</ref>

== Anthropic explanation ==
The [[anthropic principle]] provides an argument as to the reason the fine-structure constant has the value it does: stable matter, and therefore life and intelligent beings, could not exist if its value were very different.  For instance, if modern grand unified theories are correct, then {{mvar|α}} needs to be between around 1/180 and 1/85 to have proton decay to be slow enough for life to be possible.<ref>
{{cite journal
 |last=Barrow |first=John D.
 |year=2001
 |title=Cosmology, life, and the anthropic principle
 |journal=[[Annals of the New York Academy of Sciences]]
 |volume=950  |issue=1  |pages=139–153
 |doi=10.1111/j.1749-6632.2001.tb02133.x
 |pmid=11797744  |bibcode=2001NYASA.950..139B
 |s2cid=33396683
}}</ref>

In the late 20th century, multiple physicists, including [[Stephen Hawking]] in his 1988 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]].<ref name=Hawking-1988>
{{cite book
 |last=Hawking |first=S. |author-link=Stephen Hawking 
 |year=1988
 |title=A Brief History of Time
 |url=https://archive.org/details/briefhistoryofti00step_1
 |url-access=registration
 |publisher=Bantam Books
 |isbn=978-0-553-05340-1
 |pages=[https://archive.org/details/briefhistoryofti00step_1/page/7 7], 125
}}</ref>

== 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.

[[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.<ref>
{{cite book
 |last=Eddington |first=A. S. |author-link=Arthur Eddington
 |year=1956
 |chapter=The constants of nature
 |editor-last=Newman |editor-first=J. R.
 |title=The World of Mathematics
 |volume=2 |pages=1074–1093
 |publisher=[[Simon & Schuster]]
}}</ref>
This led him in 1929 to conjecture that the reciprocal of the fine-structure constant was not approximately but precisely the [[integer]] [[137 (number)|137]].<ref>
{{cite journal
 |last=Whittaker |first=Edmund
 |date=1945
 |title=Eddington's theory of the constants of nature
 |journal=[[The Mathematical Gazette]]
 |volume=29 |issue=286 |pages=137–144
 |doi=10.2307/3609461 |jstor=3609461 |s2cid=125122360
}}</ref>
By the 1940s experimental values for {{sfrac|1| {{mvar|α}} }} deviated sufficiently from 137 to refute Eddington's arguments.<ref name=Kragh-2003>
{{cite journal
 |last=Kragh |first=Helge
 |date=July 2003
 |title=Magic number: A partial history of the fine-structure constant
 |journal=Archive for History of Exact Sciences
 |volume=57 |issue=5 |pages=395–431
 |doi=10.1007/s00407-002-0065-7
 |jstor=41134170 |s2cid=118031104
}}</ref>

Physicist [[Wolfgang Pauli]] commented on the appearance of [[Numerology#Related uses|certain numbers in physics]], including the fine-structure constant, which he also noted approximates reciprocal of the prime number [[137 (number)#Physics|137]].<ref>{{cite journal |url=https://www.newscientist.com/article/mg20227051.800-cosmic-numbers-pauli-and-jungs-love-of-numerology.html |title=Cosmic numbers: Pauli and Jung's love of numerology |first=Dan |last=Falk |issue=2705 |date=24 April 2009 |journal=New Scientist}}</ref> This constant so intrigued him that he collaborated with psychoanalyst [[Carl Jung]] in a quest to understand its significance.<ref>
{{cite journal
 |last1=Várlaki |first1=Péter
 |last2=Nádai   |first2=László
 |last3=Bokor   |first3=József
 |title=Number archetypes and 'background' control theory concerning the fine structure constant
 |journal=Acta Polytechica Hungarica
 |date=2008
 |volume=5 |issue=2 |pages=71–104
 |url=http://eprints.sztaki.hu/id/eprint/4822
}}</ref> Similarly, [[Max Born]] believed that if the value of {{mvar|α}} was larger, then it would not be possible to distinguish matter from ether, and thus that {{mvar|α}} = {{sfrac|1|137}} is a law of nature.<ref name=Miller-2009>
{{cite book
 |last = Miller |first=A. I.
 |year = 2009
 |title = Deciphering the Cosmic Number: The Strange Friendship of Wolfgang Pauli and Carl Jung
 |page = [https://archive.org/details/isbn_9780393065329/page/253 253]
 |publisher = [[W. W. Norton & Co.]]
 |isbn = 978-0-393-06532-9
 |url = https://archive.org/details/isbn_9780393065329/page/253
}}</ref>{{efn|"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 {{sfrac|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]]<ref name=Miller-2009/>
}}

[[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:

{{blockquote|
There is a most profound and beautiful question associated with the observed coupling constant, {{math|''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 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 humans. 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 Feynman|R. P. Feynman]]<ref name=Feynman1985>
 {{cite book
  |last=Feynman |first=R. P. |author-link=Richard Feynman
  |year=1985
  |title=QED: The Strange Theory of Light and Matter
  |publisher=[[Princeton University Press]]
  |isbn=978-0-691-08388-9
  |title-link=QED: The Strange Theory of Light and Matter
  |page=[https://archive.org/details/qedstrangetheory00feyn_822/page/n133 129]
 }}</ref>
}}

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]].{{efn|"There have been a few examples of numerology that have led to theories that transformed society: See the mention of [[Gustav Kirchhoff|Kirchhoff]] and [[Johann Balmer|Balmer]] in [[I. J. Good|Good]] (1962) p.&nbsp;316 ... and one can well include [[Johannes Kepler|Kepler]] on account of [[Kepler's third law|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." — [[I. J. Good]] (1990)<ref>
 {{cite book
  |contributor-last=Good |contributor-first=I. J. |contributor-link=I. J. Good
  |year=1990
  |contribution=A quantal hypothesis for hadrons and the judging of physical numerology
  |last1=Grimmett |first1=G. R. 
  |last2=Welsh |first2=D. J. A. 
  |title=Disorder in Physical Systems
  |publisher=[[Oxford University Press]]
  |page=141
  |isbn=978-0-19-853215-6
  |chapter-url=https://www.statslab.cam.ac.uk/~grg/books/hammfest/9-jg.ps
 }}</ref>
}}

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.

== Quotes ==
{{blockquote|
For historical reasons, {{mvar|α}} is known as the fine structure constant. Unfortunately, this name conveys a false impression. We have seen that the charge of an electron is not strictly constant but varies with distance because of quantum effects; hence {{mvar|α}} must be regarded as a variable, too. The value {{sfrac|1|&thinsp;137&thinsp;}} is the asymptotic value of {{mvar|α}} shown in Fig.&nbsp;1.5a.{{refn|
The asymptotic value of {{mvar|α}} ''for larger observation distances'', is intended here.<br/>
Caption: Fig&nbsp;1.5. Screening of the (a)&nbsp;electric charge and (b)&nbsp;the color charge in quantum field theory. Graph of electron charge versus distance from the bare e<sup>−</sup>&nbsp;charge.<ref name=Halzen-Martin-1984>
{{cite book
 |last1=Halzen |first1=F.   |author-link1=Francis Halzen
 |last2=Martin |first2=A.D. |author-link2=Alan Martin (physicist)
 |year=1984
 |title=Quarks and Leptons: An introductory course in modern particle physics
 |publisher=John Wiley & Sons
 |isbn=978-0-471-88741-6
 |page=13
 |url=https://archive.org/details/quarksleptonsint0000halz
 |url-access=registration  |via=[[Internet Archive]] (archive.org)
}}
</ref>
}}  | [[Francis Halzen|F. Halzen]] & [[Alan Martin (physicist)|A. Martin]] (1984)<ref name=Halzen-Martin-1984/>
}}


{{blockquote|
The mystery about {{mvar|α}} is actually a double mystery: The first mystery – the origin of its numerical value {{mvar|α}} ≈ {{sfrac| 1 |&thinsp;137&thinsp;}} – has been recognized and discussed for decades. The second mystery – the range of its domain – is generally unrecognized. | M.H. MacGregor (2007)<ref>
  {{cite book
    |last = MacGregor |first=M.H.
    |year = 2007
    |title = The Power of Alpha
    |publisher = [[World Scientific]]
    |isbn = 978-981-256-961-5
    |page = [https://books.google.com/books?id=jdloDQAAQBAJ&lpg=PP1&pg=PA69 69]
  }}</ref>
}}


{{blockquote|
When I die my first question to the Devil will be: What is the meaning of the fine structure constant? | Wolfgang Pauli<ref>
{{cite magazine
 |title=137 – the fine structure constant
 |department=Physics
 |magazine=Ars Magine
 |quote=''Umetnost promišljanja i uobrazilje'' [The art of thinking] –  אהיה  [I will be]
 |lang=en, sr, he
 |url=https://arsmagine.com/others/fine-structure-constant/
 |via=ArsMagine.com |access-date=2024-06-28
}}
</ref> }}

== See also ==
* [[Dimensionless physical constant]]
* [[Hyperfine structure]]

== Footnotes ==
{{notelist}}

== References ==
{{reflist|25em}}

== External links ==
{{wikiquote}}
* {{cite book
 |last1=Adler |first1=Stephen L. |author-link1=Stephen L. Adler
 |year=1973
 |chapter=Theories of the fine structure constant {{mvar|α}}
 |title=Atomic Physics |volume=3
 |pages=73–84
 |doi=10.1007/978-1-4684-2961-9_4
 |isbn=978-1-4684-2963-3
 |chapter-url=http://lss.fnal.gov/archive/1972/pub/Pub-72-059-T.pdf
}}
* {{cite web
 |title=The fine structure constant
 |series=Introduction to the constants for nonexperts
 |publisher=National Institute of Standards and Technology
 |url=http://physics.nist.gov/cuu/Constants/alpha.html
}} (adapted from the ''[[Encyclopædia Britannica]]'', 15th&nbsp;ed. by [[NIST]])
* {{cite web |url=http://physics.nist.gov/cuu/Constants/codata.pdf |archive-url=https://web.archive.org/web/20080216063410/http://physics.nist.gov/cuu/Constants/codata.pdf |archive-date=2008-02-16 |url-status=live
 |title=CODATA recommended value of {{mvar|α}}
 |year=2010
}} <!-- New CODATA 2018 is available, so maybe helpful to state here? -->
* [https://www.quantamagazine.org/physicists-measure-the-magic-fine-structure-constant-20201202/ Physicists Nail Down the 'Magic Number' That Shapes the Universe] (Natalie Wolchover, ''Quanta magazine,'' December 2, 2020). The value of this constant is given here as 1/137.035999206 (note the difference in the last three digits). It was determined by a team of four physicists led by Saïda Guellati-Khélifa at the Kastler Brossel Laboratory in Paris.
* {{cite web
 |title=Fine structure constant
 |website=Eric Weisstein's World of Physics
 |url=http://scienceworld.wolfram.com/physics/FineStructureConstant.html
 |via=scienceworld.wolfram.com
}}
* {{cite magazine
 |author1=Barrow, J.D. |author1-link=John D. Barrow
 |author2=Webb, John K. <!-- |author2-link=John K. Webb -->
 |date=June 2005
 |title=Inconstant constants
 |magazine=[[Scientific American]]
 |url=http://www.sciam.com/article.cfm?articleID=0005BFE6-2965-128A-A96583414B7F0000&ref=sciam
}}
* {{cite web
 |last=Eaves |first=Laurence |author-link=Laurence Eaves 
 |year=2009
 |title=The fine structure constant
 |website=Sixty Symbols
 |publisher=[[Brady Haran]] for the [[University of Nottingham]]
 |url=http://www.sixtysymbols.com/videos/finestructure.htm
}}

[[Category:Dimensionless constants]]
[[Category:Electromagnetism]]
[[Category:Fundamental constants]]
[[Category:Arnold Sommerfeld]]