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Fine-structure constant
Fine-structure constant
Fine-structure constant
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Value of α
0.0072973525643(11)
Value of α−1
137.035999177(21)
Quantum field theory
Feynman diagram
History
Background
Symmetries
Tools
Equations
Standard Model
Incomplete theories
Scientists

In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter alpha), is a 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 with the electromagnetic field, by the formula ⁠⁠. Its numerical value is approximately 0.0072973525643 ≈ ⁠1/137.035999177⁠, with a relative uncertainty of 1.6×10−10.[1]

The constant was named by Arnold Sommerfeld, who introduced it in 1916[2] when extending the Bohr model of the atom. quantified 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.[a]

Why the constant should have this value is not understood,[3] but there are a number of ways to measure its value.

Definition

[edit]

In terms of other physical constants, may be defined as:[4]

where

is the elementary charge (1.602176634×10−19 C‍[5]);
is the Planck constant (6.62607015×10−34 J⋅Hz−1‍[6]);
is the reduced Planck constant, (1.054571817...×10−34 J⋅s‍[7])
is the speed of light (299792458 m⋅s−1‍[8]);
is the electrical permittivity of space (8.8541878188(14)×10−12 F⋅m−1‍[9]).

Since the 2019 revision of the SI, the only quantity in this list that does not have an exact value in SI units is the electric constant (vacuum permittivity).

Alternative systems of units

[edit]

The electrostatic CGS system implicitly sets ⁠⁠, as commonly found in older physics literature, where the expression of the fine-structure constant becomes

A normalised system of units commonly used in high energy physics selects artificial units for mass, distance, time, and electrical charge which cause in such a system of "natural units" the expression for the fine-structure constant becomes[10]

As such, the fine-structure constant is chiefly a quantity determining (or determined by) the elementary charge: 0.30282212  in terms of such a natural unit of charge.

In the system of atomic units, which sets ⁠⁠, the expression for the fine-structure constant becomes

Measurement

[edit]
Eighth-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 α is[1]

α = ⁠e2/ 4πε0ħc⁠ = 0.0072973525643(11).

This has a relative standard uncertainty of 1.6×10−10.[1]

This value for α gives the following value for the vacuum magnetic permeability (magnetic constant): µ0 = 4π × 0.99999999987(16)×10−7 H⋅m−1, with the mean differing from the old defined value by only 0.13 parts per billion, 0.8 times the standard uncertainty (0.16 parts per billion) of its recommended measured value.

Historically, the value of the reciprocal of the fine-structure constant is often given. The CODATA recommended value is [11]

⁠1/α⁠ = 137.035999177(21).

While the value of 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 α directly using the quantum Hall effect or the anomalous magnetic moment of the electron.[12] Other methods include the A.C. Josephson effect and photon recoil in atom interferometry.[13] 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.[13] 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 the electron g-factor ge). One of the most precise values of α obtained experimentally (as of 2023) is based on a measurement of ge using a one-electron so-called "quantum cyclotron" apparatus,[12] together with a calculation via the theory of QED that involved 12672 tenth-order Feynman diagrams:[14]

137.035999166(15) .

This measurement of α has a relative standard uncertainty of 1.1×10−10. This value and uncertainty are about the same as the latest experimental results.[15]

Further refinement of the experimental value was published by the end of 2020, giving the value

⁠1/α⁠ = 137.035999206(11),

with a relative accuracy of 8.1×10−11, which has a significant discrepancy from the previous experimental value.[16]

Physical interpretations

[edit]

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

  • The ratio of two energies:
    1. the energy needed to overcome the electrostatic repulsion between two electrons a distance of d apart, and
    2. the energy of a single photon of wavelength λ = 2πd (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 ⁠1/4πε0⁠⁠e2/ħ⁠, to the speed of light in vacuum, c.[17] This is Sommerfeld's original physical interpretation.
  • 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 mec2 equivalent to the mass of an electron. Using the virial theorem in the Bohr model of the atom ⁠⁠, which means that ⁠⁠. Essentially this ratio follows from the electron's velocity being ⁠⁠.
  • The two ratios of three characteristic lengths: the classical electron radius re, the reduced Compton wavelength of the electron ƛe, and the Bohr radius a0: re = αƛe = α2a0.
  • In quantum electrodynamics, α is directly related to the coupling constant determining the strength of the interaction between electrons and photons.[18] 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, and the conductance quantum, The optical conductivity of graphene for visible frequencies is theoretically given by ⁠ π /4⁠G0, and as a result its light absorption and transmission properties can be expressed in terms of the fine-structure constant alone.[19] 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 reflection would then be given by ⁠ π2 α2/ 4 (1 + πα/2)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).[20] For an electron orbiting an atomic nucleus with atomic number Z the relation is ⁠mv2/r⁠ = ⁠1/ 4πε0⁠ ⁠Ze2/r2⁠ . 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.[21]

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

[edit]

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's mass gives 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/ 137.03600 ⁠ 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 GeV, one instead measures an effective α ≈ 1/127.[22]

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

[edit]
Sommerfeld memorial at University of Munich

Based on the precise measurement of the hydrogen atom spectrum by Michelson and Morley in 1887,[b] 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.[c] 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.[26] 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.[27]: 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.

History of measurements

[edit]
Successive values determined for the fine-structure constant[28][d]
Date α 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)[29]
2018 Dec 0.0072973525713(14) 137.035999046(27) Parker, Yu, et al. (2018)[30]
2019 May 0.0072973525693(11) 137.035999084(21) CODATA 2018
2020 Dec 0.0072973525628(6) 137.035999206(11) Morel et al. (2020)[16]
2022 Dec 0.0072973525643(11) 137.035999177(21) CODATA 2022
2023 Feb 0.0072973525649(8) 137.035999166(15) Fan et al. (2023)[12][e]

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

Potential variation over time

[edit]
Further information: 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 α has been proposed as a way of solving problems in cosmology and astrophysics.[31][32][33][34] 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, Δα represents the change in α over time, which can be computed by αpast − αnow . 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, up to 10 digits of accuracy.

Past rate of change

[edit]

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.[35][36][37][38][39][40]

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.[41][42][43][44] 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:[45][46]

However, in 2007 simple flaws were identified in the analysis method of Chand et al., discrediting those results.[47][48]

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.[49] 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 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 yet to be verified.[50][51][52][53]

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.[54] 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 1⁄√t . The European LOFAR radio telescope would only be able to constrain ⁠Δα/ α ⁠ to about 0.3%.[54] The collecting area required to constrain ⁠Δα/ α ⁠ to the current level of quasar constraints is on the order of 100 square kilometers, which is economically impracticable at present.

Present rate of change

[edit]

In 2008, Rosenband et al.[55] 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 ⁠Δα/ α ⁠ = (−1.6±2.3)×10−17 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[56] 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

[edit]

Researchers from Australia have said they had identified a variation of the fine-structure constant across the observable universe.[57][58][59][60][61][62]

These results have not been replicated by other researchers. In September and October 2010, after released research by Webb et al., physicists C. Orzel and S.M. Carroll separately suggested various approaches of how Webb's observations may be wrong. Orzel argues[63] that the study may contain wrong data due to subtle differences in the two telescopes.[64] Carroll takes an altogether 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, a conclusion Webb, et al., previously stated in their study.[60]

Other research finds no meaningful variation in the fine-structure constant.[65][66]

Anthropic explanation

[edit]

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 α needs to be between around 1/180 and 1/85 to have proton decay to be slow enough for life to be possible.[67]

Numerological explanations

[edit]

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.[68] This led him in 1929 to conjecture that the reciprocal of the fine-structure constant was not approximately but precisely the integer 137.[69] By the 1940s experimental values for ⁠1/ α ⁠ deviated sufficiently from 137 to refute Eddington's arguments.[27]

Physicist Wolfgang Pauli commented on the appearance of certain numbers in physics, including the fine-structure constant, which he also noted approximates reciprocal of the prime number 137.[70] This constant so intrigued him that he collaborated with psychoanalyst Carl Jung in a quest to understand its significance.[71] Similarly, Max Born believed that if the value of α differed, the universe would degenerate, and thus that α = ⁠1/137⁠ is a law of nature.[72][f]

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 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!

— R. P. Feynman[3]

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.[g]

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 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.[74]

Quotes

[edit]

For historical reasons, 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 α must be regarded as a variable, too. The value ⁠1/ 137 ⁠ is the asymptotic value of α shown in Fig. 1.5a.[76]

— F. Halzen & A. Martin (1984)[75]


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.

— M.H. MacGregor (2007)[77]


When I die my first question to the Devil will be: What is the meaning of the fine structure constant?

— Wolfgang Pauli[78]

See also

[edit]

Footnotes

[edit]

References

[edit]
[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Fine-structure constant

View on Grokipedia
from Grokipedia
The fine-structure constant, denoted by the Greek letter α, is a fundamental dimensionless physical constant that quantifies the strength of the electromagnetic interaction between elementary charged particles, such as electrons and photons. It is defined as α = e² / (4πε₀ ħ c), where e is the elementary charge, ε₀ is the vacuum permittivity, ħ is the reduced Planck constant, and c is the speed of light in vacuum. The current recommended value, based on the 2022 CODATA adjustment, is α = 7.2973525643(11) × 10⁻³, or equivalently, its inverse 1/α ≈ 137.035999177(21), with a relative uncertainty of about 1.5 × 10⁻¹⁰. This value is often approximated as 1/137, and the fine-structure constant has been described by physicist Richard Feynman as "one of the greatest damn mysteries of physics" due to its seemingly arbitrary yet fundamental nature. Its dimensionless nature makes it a pure number without units, intriguing physicists because it appears ubiquitously in atomic, molecular, and , influencing everything from the stability of to the of in materials. Introduced by Arnold Sommerfeld in 1916, the constant arose from his extension of the Bohr model of the hydrogen atom, incorporating relativistic effects to account for the observed fine splitting in atomic spectral lines. This "fine structure" refers to the small deviations from the energy levels predicted by the non-relativistic Bohr theory, which Sommerfeld explained by allowing elliptical orbits and considering the relativistic increase in electron mass. Over time, the fine-structure constant has become central to quantum electrodynamics (QED), the quantum field theory describing electromagnetic interactions, where it determines the magnitude of radiative corrections and governs phenomena like the Lamb shift and anomalous magnetic moments of particles. As one of the key coupling constants in the of , α highlights the fundamental scale of relative to other forces, such as and weak nuclear forces. For comparison, the analogous gravitational fine-structure constant, defined as α_G ≈ G m^2 / ℏ c for typical baryonic masses m (such as protons), has a value of approximately 10^{-40}, illustrating the relative weakness of gravity compared to electromagnetism at fundamental particle scales and making gravitational effects irrelevant for atomic and subatomic orbits. Although α is not predicted by theory and must be measured experimentally, its value runs with energy scale due to quantum effects like , increasing slightly at higher energies—for instance, α ≈ 1/128 near the Z boson mass. Precise measurements, often using techniques like the or g-2 experiments, continue to refine its value and test the consistency of QED predictions.

Definition and Fundamentals

Definition

The fine-structure constant, denoted by the symbol α\alpha, is a fundamental dimensionless quantity in physics that quantifies the strength of the electromagnetic interaction between elementary charged particles. In SI units, it is precisely defined as α=e24πϵ0ℏc,\alpha = \frac{e^2}{4\pi \epsilon_0 \hbar c}, where ee is the elementary electric charge, ϵ0\epsilon_0 is the vacuum permittivity, ℏ\hbar is the reduced Planck's constant, and cc is the speed of light in vacuum. This expression arises from combining the fundamental constants that govern electric charge, quantum mechanics, and relativity, providing a measure of how strongly charged particles couple to the electromagnetic field. The originates from the observation of in the spectral lines of atoms, particularly the closely spaced splittings in the atom's emission , which could not be explained by the non-relativistic alone. In 1916, introduced α\alpha while extending the Bohr atomic model to include relativistic corrections, showing that the splitting in hydrogen's spectral lines is proportional to α2\alpha^2. These splittings represent small deviations in levels due to the electron's relativistic motion and spin-orbit coupling. Being dimensionless, α\alpha is independent of any unit system, making it a universal constant that permeates diverse areas of physics without reliance on arbitrary scales. This property underscores its role as a pure number reflecting the intrinsic scale of electromagnetic interactions in nature. In the context of Paul Dirac's 1928 relativistic quantum mechanical treatment of the , the yields energy levels where the splitting factor is exactly α2/n3\alpha^2 / n^3 (with nn the principal quantum number), confirming Sommerfeld's earlier relativistic approximation and elevating α\alpha to a of .

Numerical Value and Dimensionless Nature

The fine-structure constant, denoted by α, has the recommended value of 7.2973525643(11) × 10^{-3}, or equivalently, its reciprocal 1/α = 137.035999177(21) (often approximated as ≈137 in physical discussions), as determined by the 2022 CODATA adjustment of fundamental physical constants. This value carries a relative standard uncertainty of approximately 1.5 × 10^{-10}, reflecting the high precision achieved through contemporary measurements. The constant is dimensionless, meaning it is a pure numerical value independent of the choice of units, which ensures its invariance across different physical systems and scales. In the Gaussian cgs unit system, α is explicitly given by the expression α=e2ℏc,\alpha = \frac{e^2}{\hbar c}, where e is the (in statcoulombs), ℏ is the reduced Planck's constant, and c is the . This formulation highlights its role as the coupling strength of , scaled by fundamental constants. The absence of any theoretical framework predicting the exact numerical value of α remains one of the unresolved mysteries in fundamental physics, with no derivation available from the or beyond; as physicist Richard Feynman remarked, it is "one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man." The dimensionless nature of α has profound implications for . It remains unchanged under unitary transformations or rescalings of physical quantities, facilitating comparisons across disparate energy scales. In , where ℏ = c = 1, the expression simplifies such that the square of the is e² = α, underscoring its direct interpretation as the strength of electromagnetic interactions without additional dimensional factors. This property positions α as a fundamental parameter that permeates diverse areas of physics, from atomic spectra to particle interactions, always manifesting as a fixed numerical factor.

Measurement and Determination

Historical Measurements

The fine-structure constant was first estimated in 1916 by through his analysis of the observed in the lines of atoms, particularly the splitting in the . Using a relativistic modification to the Bohr atomic model, Sommerfeld derived the constant as a dimensionless parameter governing the strength of electromagnetic interactions at atomic scales, yielding an approximate value of α≈1/137\alpha \approx 1/137. This estimate was based on precise measurements of separations, such as the Hα\alpha line, which aligned closely with theoretical predictions from the relativistic fine-structure formula. A significant contribution to the numerical determination of α\alpha came from Robert A. Millikan's oil-drop experiment in 1917, which provided one of the earliest accurate measurements of the elementary charge e=(4.774±0.002)×10−10e = (4.774 \pm 0.002) \times 10^{-10} esu. Since α=e2/(ℏc)\alpha = e^2 / (\hbar c) in Gaussian units, Millikan's value of ee, combined with contemporaneous determinations of the speed of light cc (from interferometric methods) and the reduced Planck's constant ℏ\hbar (derived from Planck's constant hh via blackbody radiation or photoelectric effect data), enabled the first quantitative computations of α\alpha from fundamental constants. These early ℏ\hbar and cc values, however, carried uncertainties of several percent, limiting the precision of α\alpha to about four significant figures. By the mid-20th century, particularly in the , refinements in —such as high-resolution studies of and fine-structure splittings—and precision for cc improved the accuracy of α\alpha. These efforts yielded values around α−1≈137.04\alpha^{-1} \approx 137.04, marking a deviation from the 137 conjectured in some theoretical models and highlighting the constant's empirical nature. Challenges in achieving higher precision persisted due to incomplete knowledge of fundamental constants, including variations in hh measurements and the nascent understanding of quantum electrodynamic corrections to atomic spectra.

Modern Measurements and Precision

Since the 1980s, the and Josephson junctions have provided foundational high-precision measurements of the ee and Planck's constant ℏ\hbar, enabling indirect determinations of the fine-structure constant α=e2/(4πϵ0ℏc)\alpha = e^2 / (4\pi \epsilon_0 \hbar c) through metrological standards. These techniques achieved relative precisions on the order of 10−910^{-9} for α\alpha by the late , establishing a benchmark for linking electrical units to fundamental constants before the 2019 SI redefinition fixed ee and ℏ\hbar. Advancements in atomic recoil experiments using laser-cooled atoms have significantly improved precision in the 21st century. By measuring the recoil velocity of atoms from photon absorption, such as in or cesium, the ratio h/mh/m (where mm is the ) is determined, which combines with spectroscopic data to yield α\alpha. A notable 2008 experiment at LKB (Laboratoire Kastler Brossel) using Bloch oscillations in an accelerated optical lattice for 87^{87}Rb atoms reported α−1=137.03599945(62)\alpha^{-1} = 137.03599945(62), corresponding to a relative of approximately 4.5×10−94.5 \times 10^{-9}. Subsequent refinements, including NIST contributions to , have pushed uncertainties to around 10−1010^{-10}, making recoil methods competitive with other approaches. Additionally, 2024 electron g-factor measurements using Penning traps have refined α\alpha via the anomalous magnetic moment ae=(g−2)/2a_e = (g-2)/2, with an updated value from Harvard improving agreement with predictions while highlighting minor tensions (around 1-2σ\sigma) with some higher-order theoretical terms. The current CODATA 2022 recommended value is α=7.2973525643(11)×10−3\alpha = 7.2973525643(11) \times 10^{-3}, or α−1=137.035999177(21)\alpha^{-1} = 137.035999177(21), with a relative of 1.6 \times 10^{-10}, dominated by inputs from gg-factor and data. This precision reveals subtle discrepancies with certain theoretical predictions in , such as those involving hadronic contributions, motivating ongoing refinements.

Physical Interpretations

Role in Atomic and Molecular Physics

In , the fine-structure constant α governs the magnitude of relativistic corrections that cause the splitting of spectral lines in hydrogen-like atoms, known as . These corrections arise from the relativistic nature of the electron's motion and spin-orbit coupling, as captured by the . The energy shift ΔE for a given level n,j is ΔE = E_n \frac{\alpha^2}{n^2} \left( \frac{n}{j + \frac{1}{2}} - \frac{3}{4} \right), where E_n is the non-relativistic Bohr energy and j is the . This shift is on the order of α² ≈ 5 × 10^{-5} relative to the gross structure levels, making it a small but observable perturbation that refines the Bohr model's predictions. Beyond the Dirac-derived , quantum electrodynamic (QED) effects introduce additional shifts, such as the , which further splits the 2S_{1/2} and 2P_{1/2} levels in by about 1057 MHz. While the originates from and diagrams in QED, its magnitude is ordered by powers of α, scaling roughly as α³ times the fine-structure scale, emphasizing α's role in quantifying the relative strength of electromagnetic interactions at atomic scales. This correction, though smaller than the Dirac fine structure by a factor of about α, is crucial for matching experimental spectra precisely. The fine-structure constant also influences applications like the , where an external splits atomic levels; in the presence of , this results in the anomalous Zeeman effect, with splitting patterns depending on the that incorporates j and thus the α-induced fine-structure intervals. For instance, in atoms, the modulates Zeeman sublevels, altering selection rules and polarization in spectral lines. Similarly, —arising from electron-nuclear spin interactions—scales with α⁴ times the ratio m_e/m_p, as the interaction energy involves the Fermi contact term proportional to the at the nucleus, which itself depends on α³ from the . In , this yields the 21 cm hyperfine transition, with splitting ΔE_hf ∝ (8/3) g_p α⁴ (m_e/m_p) Ry, where Ry is the Rydberg energy. In broader atomic contexts, α determines the scale of ionization potentials through the , Ry ≈ (1/2) α² m_e c², which sets the for ic ions and influences multi-electron atoms via screening. For elements like and , relativistic fine-structure effects subtly affect chemical bonding; increasing α would weaken covalent bonds by enhancing spin-orbit coupling, potentially altering dissociation energies in H₂ by up to a few percent for α variations of order 1%. These influences highlight α's foundational role in ordering electromagnetic effects from atomic spectra to molecular stability in systems.

Significance in Quantum Electrodynamics

In (QED), the fine-structure constant α acts as the fundamental coupling parameter that governs the strength of electromagnetic interactions between charged particles, such as electrons and . Developed through the work of , , Sin-Itiro Tomonaga, and in the late 1940s, perturbative QED relies on an expansion series in powers of α (or more precisely α/π for loop contributions), which converges effectively due to α ≈ 1/137 being much less than 1 at low energies. This expansion is visualized through Feynman diagrams, where each vertex representing a emission or absorption is scaled by the e, related to α by e = √(4πα) in ; higher-order diagrams incorporate additional loops and vertices, allowing precise calculations of scattering amplitudes and decay rates. A key success of this perturbative framework is the prediction of the electron's anomalous , a_e = (g_e - 2)/2, where g_e is the electron's . In the Dirac theory, g_e = 2 exactly, but QED corrections yield a_e = α/(2π) at leading order, as first computed by Schwinger using proper-time methods in the . Higher-order terms include contributions like α²/(π²) and beyond, with the full QED prediction now known to over five loops, matching experimental measurements to 10 decimal places and serving as one of the most stringent tests of the theory; for instance, the tenth-order term contributes only about 10^{-13} to a_e. Despite its successes at accessible energies, QED exhibits nontrivial behavior at high scales due to the positive sign of its beta function, leading to a where α diverges. This theoretical singularity occurs around 10^{37} GeV in the context of the full , far beyond the electroweak scale, signaling that pure QED lacks and requires embedding in a larger theory for UV completion; the running of α from effects increases its value logarithmically with energy. At the electroweak scale (around the Z boson mass of 91 GeV), α(M_Z) ≈ 1/128, a value crucial for unification discussions, as it approaches the weak and strong couplings in grand unified theories, though discrepancies persist without new physics.

Dependencies and Variations

Variation with Energy Scale

In quantum electrodynamics (QED), the fine-structure constant α\alpha exhibits a scale dependence known as "running," arising from vacuum polarization effects where virtual fermion-antiparticle pairs screen the bare electromagnetic charge. This behavior is captured by the renormalization group equation (RGE) at one-loop order: dαdln⁡μ=α23π∑fQf2nc,\frac{d\alpha}{d \ln \mu} = \frac{\alpha^2}{3\pi} \sum_f Q_f^2 n_c, where μ\mu is the renormalization scale, the sum runs over active fermion flavors ff with electric charge QfQ_f (in units of the elementary charge ee), and nc=1n_c = 1 (3) for leptons (quarks) accounting for color degrees of freedom. The leading-logarithmic solution to this equation, valid away from mass thresholds, takes the approximate form α(μ)=α(0)1−α(0)3π(∑fQf2nc)ln⁡(μ2m2),\alpha(\mu) = \frac{\alpha(0)}{1 - \frac{\alpha(0)}{3\pi} \left( \sum_f Q_f^2 n_c \right) \ln\left( \frac{\mu^2}{m^2} \right)}, with α(0)≈1/137.036\alpha(0) \approx 1/137.036 the low-energy value (at μ∼me\mu \sim m_e, the electron mass) and mm a reference scale below the lightest threshold. The positive beta function coefficient in QED leads to an increase in α(μ)\alpha(\mu) with μ\mu, reflecting reduced screening at shorter distances. At the electroweak scale μ=MZ≈91\mu = M_Z \approx 91 GeV, α(MZ)≈1/128.9\alpha(M_Z) \approx 1/128.9. Below MZM_Z, lepton loop contributions dominate the running, with the three charged lepton generations (electron, muon, tau; each Qf=1Q_f = 1, nc=1n_c = 1) providing ∑Qf2nc=3\sum Q_f^2 n_c = 3, augmented by lighter quark effects through hadronic vacuum polarization. Above MZM_Z, heavier particles enter, including the top (Qf=2/3Q_f = 2/3, nc=3n_c = 3) whose large mass enhances its impact near the threshold, and electroweak bosons (, ). At even higher scales, (QCD) influences the evolution via gluon-mediated corrections to loops, slowing the hadronic contribution's growth. These multi-loop and threshold effects are incorporated in higher-order calculations for precision. The predicted running has been experimentally verified through e+e−e^+ e^- scattering cross-sections at the Large Electron-Positron Collider (LEP), where the effective α\alpha at the Z-pole resonance was extracted from the hadronic and leptonic event rates, yielding α(MZ)−1=128.89±0.07\alpha(M_Z)^{-1} = 128.89 \pm 0.07 in agreement with . Complementary confirmation arises from global electroweak precision fits to LEP/SLD , which constrain α(MZ)\alpha(M_Z) to better than 0.1% accuracy and test QED consistency within the . Lower-energy running, including hadronic effects, has been measured in processes like e+e−→μ+μ−γe^+ e^- \to \mu^+ \mu^- \gamma by experiments such as KLOE and .

Temporal and Spatial Variations

The fine-structure constant has been scrutinized for possible temporal variations over cosmological timescales using absorption spectra, which probe distant intervening gas clouds. Early analyses by Webb et al. in 1999 suggested a potential change of Δα/α ≈ (-0.72 ± 0.18) × 10^{-5} at redshifts z ≈ 0.5–1.6, corresponding to roughly 10 billion years ago. Subsequent studies expanded the dataset, with claims of a dipole-like pattern implying variations up to ~10^{-5} over similar epochs. However, comprehensive reanalyses, including a 2024 study incorporating convergence properties of measurements from multiple systems, indicate no statistically significant deviation, with Δα/α consistent with zero at the 10^{-6} level or better. Tighter constraints on past variations come from natural nuclear reactors and primordial . The Oklo natural fission reactor in , operating approximately 2 billion years ago, provides isotopic ratios sensitive to α; analyses yield |Δα/α| < 5.8 × 10^{-8} over that interval. Big Bang (BBN) models, calibrated against observed primordial abundances of light elements like deuterium and helium, impose even stricter limits from the early universe at z ≈ 10^9; recent evaluations constrain |Δα/α| < 10^{-3} relative to the present value, though some specialized BBN studies suggest bounds as tight as <10^{-10} when incorporating correlated variations in other constants. At present epochs, laboratory tests using atomic clocks and lunar laser ranging set the most stringent limits on the rate of temporal change. Comparisons of optical atomic clocks, such as those based on ytterbium and dysprosium transitions, over intervals of years detect no variation exceeding |dα/α/dt| < 10^{-18} yr^{-1}. Lunar laser ranging experiments, monitoring Earth-Moon dynamics, complement these by constraining coupled variations in fundamental constants, yielding no detectable drift beyond |dα/α/dt| < 10^{-17} yr^{-1} up to 2025. These results affirm the constancy of α on human timescales. Spatial variations in α have been proposed based on directional dependencies in quasar data. Webb et al. in 2010 reported evidence for a dipole anisotropy, with Δα/α ≈ 10^{-6} aligned toward the southern sky, using Keck and VLT observations of ~300 absorption systems. This claim suggested possible spatial inhomogeneity at the 4σ level. However, independent analyses, including 2023–2024 studies leveraging large galaxy samples and DESI survey data, refute the dipole, finding no confirmed anisotropy and constraining spatial fluctuations to <10^{-6} across the observable universe.

Historical Development

Discovery and Early Interpretations

The fine-structure constant, denoted as α\alpha, was first introduced by in 1916 as part of his extension of the within the framework of . Sommerfeld sought to explain the observed fine splitting of spectral lines, such as the doublet in the hydrogen Hα\alpha line, which could not be accounted for by 's original 1913 model. By incorporating relativistic corrections to the electron's orbital motion, Sommerfeld derived that the splitting arises from the ratio of the electron's velocity vv in the innermost Bohr orbit to the speed of light cc, yielding v/c≈α≈1/137v/c \approx \alpha \approx 1/137. This dimensionless parameter encapsulated the strength of the electromagnetic interaction relative to relativistic effects, marking α\alpha as a fundamental constant in early . In 1928, Paul Dirac advanced this understanding through his relativistic wave equation for the electron, which precisely predicted the fine structure of hydrogen spectral lines without ad hoc assumptions. Dirac's equation incorporated both quantum mechanics and special relativity, naturally yielding the fine-structure splitting as a function of α\alpha, where the constant appeared as the coupling parameter governing electromagnetic interactions. This formulation elevated α\alpha from an empirical correction in the Bohr-Sommerfeld model to a core element of relativistic quantum theory, confirming Sommerfeld's approximate value while providing an exact theoretical basis. Prior to the 1940s, the fine-structure constant was primarily interpreted in the context of the Bohr-Sommerfeld atomic model as the characteristic ratio of the electron's orbital velocity to the speed of light in the ground state. This view framed α\alpha as a measure of how closely atomic electrons approach relativistic speeds, influencing the quantization of angular momentum and energy levels in hydrogen-like atoms. Early calculations, such as those for the hydrogen fine structure, relied on this velocity ratio to match experimental spectral data, underscoring α\alpha's role in bridging classical orbits with quantum discreteness. The numerical value of α≈1/137\alpha \approx 1/137 also sparked early numerological interest among physicists. In the 1930s, Arthur Eddington speculated that the inverse fine-structure constant was exactly 136, deriving this from his fundamental theory of the world, which emphasized epistemological and aesthetic principles over empirical measurement. Eddington's arguments, detailed in works like his 1931 paper, posited that α−1\alpha^{-1} must be an integer tied to the structure of physical laws, reflecting a broader fascination with the constant's seemingly arbitrary yet precise magnitude. These interpretations, while influential, remained speculative and were later refined by more rigorous quantum developments.

Evolution Through Quantum Theory

In the 1940s, the fine-structure constant α began to play a central role in quantum electrodynamics (QED) through calculations addressing discrepancies in atomic spectra. Hans Bethe calculated the in hydrogen in 1947, attributing the energy level splitting between the 2S_{1/2} and 2P_{1/2} states to radiative corrections involving virtual electron-positron pairs, with the shift scaling as α (up to logarithms) times the Dirac fine structure. This non-relativistic approximation incorporated α as the fundamental measure of electromagnetic coupling strength, resolving infinities via mass renormalization inspired by earlier work. Enrico Fermi contributed foundational QED techniques in the early 1940s, including self-energy evaluations that Bethe adapted for the Lamb shift computation during a train journey following the experimental announcement. The formalization of QED in the late 1940s elevated α to the status of the theory's primary expansion parameter. Sin-Itiro Tomonaga developed a covariant perturbation theory in 1946, enabling consistent handling of relativistic effects in electron-photon interactions. Julian Schwinger and Richard Feynman independently advanced this framework in 1948, introducing functional integrals and path-integral formulations, respectively, where scattering amplitudes expand in powers of α ≈ 1/137, quantifying loop corrections like vacuum polarization. Their work, unified by Freeman Dyson's equivalence proofs, demonstrated QED's predictive power for phenomena such as the anomalous magnetic moment, earning Tomonaga, Schwinger, and Feynman the 1965 Nobel Prize in Physics; α emerged as the dimensionless coupling dictating the theory's perturbative validity up to high energies. From the 1970s to the 2000s, efforts to embed α within broader unification schemes extended its theoretical significance in the Standard Model. Grand Unified Theories (GUTs), starting with the minimal SU(5) model proposed by Howard Georgi and Sheldon Glashow in 1974, aimed to unify the electromagnetic, weak, and strong couplings at a high-energy scale, but the minimal non-supersymmetric version predicted convergence around 10^{12}-10^{14} GeV with α_GUT ≈ 1/25, which was too low to evade proton decay limits. Supersymmetric extensions in the 1980s and 1990s raised the scale to around 10^{16} GeV, implying α evolves via renormalization group running to a unified value α_GUT ≈ 1/25 at that scale, though distinct from the Planck scale of 10^{19} GeV. Subsequent models like SO(10) refined these predictions, incorporating proton decay and neutrino masses, but discrepancies in low-energy coupling measurements—such as α_s at the Z boson mass—challenged minimal GUTs, prompting further supersymmetric adjustments that aligned the unification scale more closely with observations. These frameworks highlighted α's role in testing unification, with logarithmic running governed by β-functions involving particle content, though no exact derivation of its low-energy value emerged.

Theoretical Explanations

Anthropic Principle

The fine-structure constant, denoted as α≈1/137\alpha \approx 1/137, is often cited in fine-tuning arguments within the anthropic principle, which posits that the universe's physical parameters must permit the existence of observers to make such observations possible. Specifically, values of α\alpha significantly larger than its observed magnitude would destabilize atomic structures essential for chemistry; for instance, if α>1/95\alpha > 1/95, iron atoms become unstable, disrupting the formation of complex molecules necessary for life. Conversely, if α<1/205\alpha < 1/205, would favor nickel over iron as the endpoint of fusion processes, preventing the production of elements critical for stable stars and planetary systems. These narrow bounds highlight how α\alpha's value enables the electromagnetic interactions required for stable atoms and molecular bonds, underscoring the apparent tuning for . John D. Barrow and Frank J. Tipler, in their seminal 1986 work, formalized the anthropic cosmological principle, applying it to constants like α\alpha to argue that the universe's structure is constrained by the necessity of supporting intelligent life. They distinguish between the weak anthropic principle (observational selection effects) and the strong version (the universe must evolve observers), using α\alpha's role in atomic stability to illustrate how deviations would preclude biological complexity, such as the dipole moments in water molecules vital for life's solvent properties. This framework posits that α\alpha's specific value is not coincidental but required for the emergence of observers capable of measuring it. In scenarios, particularly the , α\alpha can vary across different vacua, with our universe's value selected anthropically because it allows for . The landscape encompasses approximately 1050010^{500} possible configurations of , where the vacuum expectation values of scalar fields determine coupling constants like α\alpha, enabling regions hospitable to observers while others remain barren. This resolves fine-tuning by invoking a vast ensemble where life-bearing universes are statistically inevitable, though only those permitting complex chemistry and are observed. Critics of the approach to α\alpha emphasize its limited testability, as predictions rely on unobservable multiverses, rendering it philosophically intriguing but empirically challenging; however, analogies to the —another finely tuned parameter with explanations in similar landscapes—bolster its plausibility by demonstrating consistent explanatory patterns across constants. Efforts to test it involve assessing whether life-permitting values occupy a significant of the parameter space, but current models suggest α\alpha's tuning remains a key example of in cosmic ensembles.

Numerological and Speculative Theories

In the early , proposed a combinatorial derivation for the fine-structure constant α\alpha, initially aiming to establish its inverse as exactly 136 based on the number of in fundamental particles and cosmological considerations. This approach, outlined in his work, relied on aesthetic and numerical principles rather than empirical derivation, predicting α=1/136\alpha = 1/136 through relations involving the Eddington number and properties. However, subsequent measurements indicated a value closer to 1/1371/137, prompting Eddington to adjust his combinatorial scheme to fit the new data, highlighting the speculative nature of the method. Religious and esoteric interpretations have also linked the approximate value of 1/1371/137 to numerological significance, particularly in Kabbalistic traditions where the Hebrew word for "Kabbalah" (קַבָּלָה) yields a gematria value of 137, suggesting a mystical connection to the constant's role in governing electromagnetic interactions. Proponents argue this alignment reflects a deeper, non-physical harmony between ancient mysticism and modern physics, with 137 symbolizing reception or parallel structures in creation. Such views, while intriguing, remain outside scientific discourse and lack testable predictions. Contemporary speculative theories continue this numerological tradition, proposing derivations of α≈1/137\alpha \approx 1/137 through geometric or mathematical constructs. For instance, models incorporating the ϕ≈1.618\phi \approx 1.618 relate it to α\alpha via scaling laws and geometries. Similarly, recent frameworks explore ratios involving π\pi and the Euler-Mascheroni constant γ≈0.577\gamma \approx 0.577 to resolve α\alpha within extended contexts, though these remain unverified. Classical electron models, reviving pre-quantum ideas, derive α≈1/137\alpha \approx 1/137 from the ratio of Coulomb to gravitational forces, where the gravitational analog αG=Gm2/ℏc≈5.9×10−39\alpha_G = G m^2 / \hbar c \approx 5.9 \times 10^{-39} for typical baryonic masses such as two protons, or from radius considerations. While used in some speculative derivations linking the 137 to primordial charge dynamics, this analog underscores the vast disparity in force strengths between electromagnetism and gravity without implying any established exotic roles. These numerological and speculative approaches face significant critiques for their lack of and reliance on post-hoc adjustments to match measurements, often prioritizing over falsifiable mechanisms. Unlike mainstream theories, they fail to integrate with or yield novel experimental tests, rendering them philosophically appealing but scientifically marginal.

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