Fine-structure constant

Value of Template:Math
Template:Physconst
Value of Template:Math
Template:Physconst

Template:Quantum field theory In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by Template:Mvar (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 e with the electromagnetic field, by the formula Template:Math. Its numerical value is approximately Template:Nowrap, with a relative uncertainty of Template:Physconst

The constant was named by Arnold Sommerfeld, who introduced it in 1916 when extending the Bohr model of the atom. Template:Math 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.Template:Efn

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

Definition

In terms of other physical constants, Template:Mvar may be defined as: $$\alpha =\frac{e^2}{2{\epsilon }_{0}hc}=\frac{e^2}{4\pi {\epsilon }_0\hslash c},$$ where

• Template:Mvar is the elementary charge (Template:Physconst);
• Template:Mvar is the Planck constant (Template:Physconst);
• Template:Mvar is the reduced Planck constant, Template:Math (Template:Physconst)
• Template:Mvar is the speed of light (Template:Physconst);
• Template:MvarTemplate:Sub is the electric constant (Template:Physconst).

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

The electrostatic CGS system implicitly sets Template:Math, as commonly found in older physics literature, where the expression of the fine-structure constant becomes $$\alpha =\frac{e^2}{\hslash c}.$$

A nondimensionalised system commonly used in high energy physics sets Template:Math, where the expression for the fine-structure constant becomes$$\alpha =\frac{e^2}{4\pi }.$$As such, the fine-structure constant is chiefly a quantity determining (or determined by) the elementary charge: Template:Math in terms of such a natural unit of charge.

In the system of atomic units, which sets Template:Math, the expression for the fine-structure constant becomes $$\alpha =\frac{1}{c}.$$

Measurement

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 Template:Math isTemplate:Physconst Template:Block indent This has a relative standard uncertainty of Template:Physconst

This value for Template:Math gives Template:Nowrap, 0.8 times the standard uncertainty away from its old defined value, with the mean differing from the old value by only 0.13 parts per billion.

Historically the value of the reciprocal of the fine-structure constant is often given. The CODATA recommended value is Template:Physconst Template:Block indent

While the value of Template:Mvar 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 Template:Mvar directly using the quantum Hall effect or the anomalous magnetic moment of the electron. Other methods include the A.C. Josephson effect and photon recoil in atom interferometry. There is general agreement for the value of Template:Mvar, 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. The theory of QED predicts a relationship between the dimensionless magnetic moment of the electron and the fine-structure constant Template:Mvar (the magnetic moment of the electron is also referred to as the [[g-factor (physics)|electron Template:Mvar-factor]] Template:Math). One of the most precise values of Template:Mvar obtained experimentally (as of 2023) is based on a measurement of Template:Math using a one-electron so-called "quantum cyclotron" apparatus, together with a calculation via the theory of QED that involved Template:Val tenth-order Feynman diagrams: Template:Block indent

This measurement of Template:Mvar has a relative standard uncertainty of Template:Val. This value and uncertainty are about the same as the latest experimental results.

Further refinement of the experimental value was published by the end of 2020, giving the value Template:Block indent with a relative accuracy of Template:Val, which has a significant discrepancy from the previous experimental value.

Physical interpretations

The fine-structure constant, Template:Mvar, has several physical interpretations. Template:Mvar is:Template:Unordered list

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

Variation with energy scale

In quantum electrodynamics, the more thorough quantum field theory underlying the electromagnetic coupling, the renormalization group dictates how the strength of the electromagnetic interaction grows logarithmically as the relevant energy scale increases. The value of the fine-structure constant Template:Mvar 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, Template:Sfrac 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 Template:Mvar ≈ 1/127.

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

Sommerfeld memorial at University of Munich

Based on the precise measurement of the hydrogen atom spectrum by Michelson and Morley in 1887,Template:Efn 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.Template:Efn The first physical interpretation of the fine-structure constant Template:Mvar 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. 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.Template:Rp

With the development of quantum electrodynamics (QED) the significance of Template:Math 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 Template:Math 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

Successive values determined for the fine-structure constantTemplate:Efn

Date	Template:Math	Template:Math	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)
2018 Dec	0.0072973525713(14)	137.035999046(27)	Parker, Yu, et al. (2018)
2019 May	0.0072973525693(11)	137.035999084(21)	CODATA 2018
2020 Dec	0.0072973525628(6)	137.035999206(11)	Morel et al. (2020)
2022 Dec	0.0072973525643(11)	137.035999177(21)	CODATA 2022
2023 Feb	0.0072973525649(8)	137.035999166(15)	Fan et al. (2023)Template:Efn

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

Potential variation over time

Template:Further Physicists have pondered whether the fine-structure constant is in fact constant, or whether its value differs by location and over time. A varying Template:Mvar has been proposed as a way of solving problems in cosmology and astrophysics. 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 Template:Mvar) actually vary.

In the experiments below, Template:Math represents the change in Template:Mvar over time, which can be computed by Template:Mvar_prev − Template:Mvar_now . If the fine-structure constant really is a constant, then any experiment should show that $$\frac{\mathrm{\Delta }\alpha }{\alpha }\stackrel{𝖽𝖾𝖿}{=}\frac{{\alpha }_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{v}}-{\alpha }_{\mathrm{n}\mathrm{o}\mathrm{w}}}{{\alpha }_{\mathrm{n}\mathrm{o}\mathrm{w}}}=0,$$ or as close to zero as experiment can measure. Any value far away from zero would indicate that Template:Mvar does change over time. So far, most experimental data is consistent with Template:Mvar being constant.

Past rate of change

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

Improved technology at the dawn of the 21st century made it possible to probe the value of Template:Mvar 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 Template:Mvar. Using the Keck telescopes and a data set of 128 quasars at redshifts Template:Math, Webb et al. found that their spectra were consistent with a slight increase in Template:Mvar over the last 10–12 billion years. Specifically, they found that $$\frac{\mathrm{\Delta }\alpha }{\alpha }\stackrel{𝖽𝖾𝖿}{=}\frac{{\alpha }_{\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{v}}-{\alpha }_{\mathrm{n}\mathrm{o}\mathrm{w}}}{{\alpha }_{\mathrm{n}\mathrm{o}\mathrm{w}}}=(-5.7\pm 1.0)\times 10^{-6}.$$

In other words, they measured the value to be somewhere between Template:Val and Template:Val. This is a very small value, but the error bars do not actually include zero. This result either indicates that Template:Mvar 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: $$\frac{\mathrm{\Delta }\alpha }{{\alpha }_{\mathrm{e}\mathrm{m}}}=(-0.6\pm 0.6)\times 10^{-6}.$$

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

King et al. have used Markov chain Monte Carlo methods to investigate the algorithm used by the UNSW group to determine Template:Sfrac from the quasar spectra, and have found that the algorithm appears to produce correct uncertainties and maximum likelihood estimates for Template:Sfrac for particular models. This suggests that the statistical uncertainties and best estimate for Template:Sfrac 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 Template:Mvar 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.

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. They proposed using this effect to measure the value of Template:Mvar 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 Template:Val (4 orders of magnitude better than the current quasar constraints). However, the constraint which can be placed on Template:Mvar is strongly dependent upon effective integration time, going as Template:Frac. The European LOFAR radio telescope would only be able to constrain Template:Sfrac to about 0.3%. The collecting area required to constrain Template:Sfrac 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

In 2008, Rosenband et al. used the frequency ratio of Template:Chem2 and Template:Chem2 in single-ion optical atomic clocks to place a very stringent constraint on the present-time temporal variation of Template:Mvar, namely Template:Sfrac = Template:Val 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 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

Researchers from Australia have said they had identified a variation of the fine-structure constant across the observable universe.

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 that the study may contain wrong data due to subtle differences in the two telescopes a totally different approach; he looks at the fine-structure constant as a scalar field and claims that if the telescopes are correct and the fine-structure constant varies smoothly over the universe, then the scalar field must have a very small mass. However, previous research has shown that the mass is not likely to be extremely small. Both of these scientists' early criticisms point to the fact that different techniques are needed to confirm or contradict the results, a conclusion Webb, et al., previously stated in their study.

Other research finds no meaningful variation in the fine structure constant.

Anthropic explanation

The anthropic principle is an argument about 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. One example is that, if modern grand unified theories are correct, then Template:Mvar needs to be between around 1/180 and 1/85 to have proton decay to be slow enough for life to be possible.

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. This led him in 1929 to conjecture that the reciprocal of the fine-structure constant was not approximately but precisely the integer 137. By the 1940s experimental values for Template:Sfrac deviated sufficiently from 137 to refute Eddington's arguments.

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. This constant so intrigued him that he collaborated with psychoanalyst Carl Jung in a quest to understand its significance. Similarly, Max Born believed that if the value of Template:Mvar differed, the universe would degenerate, and thus that Template:Mvar = Template:Sfrac is a law of nature.Template:Efn

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:

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.Template:Efn

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.

See also

• Dimensionless physical constant
• Hyperfine structure

External links

• Template:Cite book
• Template:Cite web (adapted from the Encyclopædia Britannica, 15th ed. by NIST)
• Template:Cite web
• 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.
• Template:Cite web
• Template:Cite web
• Template:Cite magazine
• Template:Cite web