Fine-structure constant

The definition of α in terms of other fundamental physical constants is:[2]

$${\displaystyle \alpha ={\frac {e^{2}}{2\varepsilon _{0}hc}},}$$

where

• e is the elementary charge (1.602176634×10⁻¹⁹ C);
• h is the Planck constant (6.62607015×10⁻³⁴ J⋅s);
• c is the speed of light in vacuum (299792458 m/s);
• ε₀ is the electric constant or permittivity of vacuum or free space (8.8541878128×10⁻¹² F/m).

Since the 2019 redefinition of the SI base units, the only measured quantity in this list is the electric constant; the elementary charge, Planck constant and speed of light in vacuum are given defined values in SI units.

There are several equivalent expressions for α, for example,

$${\displaystyle \alpha ={\frac {e^{2}}{2\varepsilon _{0}hc}}={\frac {e^{2}\mu _{0}c}{2h}}={\frac {e^{2}}{4\pi \varepsilon _{0}\hbar c}},}$$

where

• μ₀ is the vacuum permeability, μ₀ = 1/ε₀c²
• ħ is the reduced Planck constant, ħ = h/2π

Alternative systems of units

Where nondimensionalisation sets ${\displaystyle c=h=e=1,}$ the magnetic permeability and electric permittivity of vacuum are ${\displaystyle \mu _{0}={\frac {1}{\varepsilon _{0}}}=2\alpha \ .}$

The electrostatic cgs system sets the Coulomb constant k_e = 1, as commonly found in older physics literature, where the expression of the fine-structure constant becomes

$${\displaystyle \alpha ={\frac {e^{2}}{\hbar c}}.}$$

A nondimensionalised system commonly used in high energy physics sets ${\displaystyle \ \varepsilon _{0}=c=\hbar =1\ ,}$ where the expressions for the fine-structure constant becomes[3]

$${\displaystyle \alpha ={\frac {e^{2}}{4\pi }}.}$$

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

In the system of Hartree atomic units, which sets e = m_e = ħ = 4πε₀ = 1, the expression for the fine-structure constant becomes

$${\displaystyle \alpha ={\frac {1}{c}}.}$$

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 2018 CODATA recommended value of α is[4]

α = e²/ 4πε₀ħc = 0.0072973525693(11).

This has a relative standard uncertainty of 1.5×10⁻¹⁰.[4]

This value for α gives µ₀ = 4π × 1.00000000054(15)×10⁻⁷ H⋅m⁻¹, 3.6 standard deviations away from its old defined value, but with the mean differing from the old value by only 0.54 parts per billion.

Historically the value of the reciprocal of the fine-structure constant is often given. The 2018 CODATA recommended value is[5]

1/α = 137.035999084(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. Other methods include the A.C. Josephson effect and photon recoil in atom interferometry.[6] 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.[6] The theory of QED predicts a relationship between the dimensionless magnetic moment of the electron and the fine-structure constant α (the magnetic moment of the electron is also referred to as "Landé g-factor" and symbolized as g). The most precise value of α obtained experimentally (as of 2012) is based on a measurement of g using a one-electron so-called "quantum cyclotron" apparatus, together with a calculation via the theory of QED that involved 12672 tenth-order Feynman diagrams:[7]

1/α = 137.035999174(35).

This measurement of α has a relative standard uncertainty of 2.5×10⁻¹⁰. This value and uncertainty are about the same as the latest experimental results.[8] Further refinement of this work were published by the end of 2020, giving the value

1/α = 137.035999206(11),

with a relative accuracy of 8.1×10⁻¹¹.[9]

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

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.

In quantum electrodynamics, the more thorough quantum field theory underlying the electromagnetic coupling, the renormalization group dictates how the strength of the electromagnetic interaction grows logarithmically as the relevant energy scale increases. The value of the fine-structure constant α is linked to the observed value of this coupling associated with the energy scale of the electron mass: the electron is a lower bound for this energy scale, because it (and the positron) is the lightest charged object whose quantum loops can contribute to the running. Therefore, 1/ 137.03600  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.[17]

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.

Sommerfeld memorial at University of Munich

Based on the precise measurement of the hydrogen atom spectrum by Michelson and Morley in 1887,[lower-alpha 3]^(p430) </ref> 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.[lower-alpha 4] The first physical interpretation of the fine-structure constant α was as the ratio of the velocity of the electron in the first circular orbit of the relativistic Bohr atom to the speed of light in the vacuum.[20] Equivalently, it was the quotient between the minimum angular momentum allowed by relativity for a closed orbit, and the minimum angular momentum allowed for it by quantum mechanics. It appears naturally in Sommerfeld's analysis, and determines the size of the splitting or fine-structure of the hydrogenic spectral lines. This constant was not seen as significant until Paul Dirac's linear relativistic wave equation in 1928, which gave the exact fine structure formula.[21]^: 407

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

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.[26][27][28][29] 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 α_prev − α_now . If the fine-structure constant really is a constant, then any experiment should show that

${\displaystyle {\frac {\ \Delta \alpha \ }{\alpha }}~~{\overset {\underset {\mathsf {~def~}}{}}{=}}~~{\frac {\ \alpha _{\mathrm {prev} }-\alpha _{\mathrm {now} }\ }{\alpha _{\mathrm {now} }}}~~=~~0~,}$

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.

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. For instance, were α to change by 4%, stellar fusion would not produce carbon, so that carbon-based life would be impossible. If α were greater than 0.1, stellar fusion would be impossible, and no place in the universe would be warm enough for life as we know it.[62]

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.[63] This led him in 1929 to conjecture that the reciprocal of the fine-structure constant was not approximately but precisely the integer 137.[64] By the 1940s experimental values for 1/ α  deviated sufficiently from 137 to refute Eddington's arguments.[21]

The fine-structure constant so intrigued physicist Wolfgang Pauli that he collaborated with psychoanalyst Carl Jung in a quest to understand its significance.[65] 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.[66][lower-alpha 5]

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[67]

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.[lower-alpha 6]

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

In the early 21st century, multiple physicists, including Stephen Hawking in his book A Brief History of Time, began exploring the idea of a multiverse, and the fine-structure constant was one of several universal constants that suggested the idea of a fine-tuned universe.[69]

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)[70]

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

— Wolfgang Pauli

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

Wikiquote has quotations related to Fine-structure constant.

• Adler, Stephen L. (1973). "Theories of the fine structure constant α" (PDF). Atomic Physics. Vol. 3. pp. 73–84. doi:10.1007/978-1-4684-2961-9_4. ISBN 978-1-4684-2963-3.
• "The fine structure constant". Introduction to the constants for nonexperts. National Institute of Standards and Technology. (adapted from the Encyclopædia Britannica, 15th ed. by NIST)
• "CODATA recommended value of α" (PDF). 2010.
• 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.
• "Quotes about the fine structure constant". Good Reads.
• "Fine structure constant". Eric Weisstein's World of Physics – via scienceworld.wolfram.com.
• Barrow, J.D.; Webb, John K. (June 2005). "Inconstant constants". Scientific American.
• Eaves, Laurence (2009). "The fine structure constant". Sixty Symbols. Brady Haran for the University of Nottingham.