# Fine-structure constant

The fine-structure constant or Sommerfeld fine-structure constant, usually denoted [itex]\alpha \ [itex], is the fundamental physical constant characterizing the strength of the electromagnetic interaction. It was originally introduced into physics in 1916 by Arnold Sommerfeld, as a measure of the relativistic deviations in atomic spectral lines from the predictions of the Bohr model.

The fine-structure constant is a dimensionless quantity, and its numerical value is independent of the system of units used. The value recommended by 2002 CODATA is

[itex] \alpha = 7.297 352 568(24) \times 10^{-3} = \frac{1}{137.035 999 11(46)} \ [itex] .

It can be defined as

[itex]\alpha = \frac{e^2}{\hbar c 4 \pi \epsilon_0} \ = \frac{e^2}{2 \epsilon_0 h c}[itex]

where [itex] e \ [itex] is the elementary charge, [itex]\hbar = h/(2 \pi) \ [itex] is the reduced Planck's constant, [itex] c \ [itex] is the speed of light in a vacuum, and [itex] \epsilon_0 \ [itex] is the permittivity of free space.

In electrostatic cgs units, the unit of electric charge (the Statcoulomb or esu of charge) is defined in such a way that the permittivity factor, [itex] 4 \pi \epsilon_0 \ [itex], is the dimensionless one. Then the fine-structure constant becomes

[itex]\alpha = \frac{e^2}{\hbar c} \ [itex] .

The fine-structure constant can also be thought of as the square of the ratio of the elementary charge to the Planck charge.

[itex]\alpha = \left( \frac{e}{q_P} \right)^2 \ [itex].
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## Physical interpretation

For any arbitrary length [itex] s \ [itex], the fine-structure constant is the ratio of two energies: (i) the energy needed to bring two electrons from infinity to a distance of [itex] s \ [itex]against their electrostatic repulsion, and (ii) the energy of a single photon of wavelength [itex] 2 \pi s \ [itex].

Historically, the first physical interpretation of the fine-structure constant, [itex] \alpha \ [itex], was the ratio of the velocity of the electron in the first circular orbit of the Bohr atom to the speed of light in vacuum. It appears naturally in Sommerfeld's analysis and determines the size of the splitting or fine-structure of the hydrogenic spectral lines.

In the theory of quantum electrodynamics, the fine structure constant plays the role of a coupling constant, representing the strength of the interaction between electrons and photons. Its value cannot be predicted by the theory, and has to be inserted based on experimental results. In fact, it is one of the twenty-odd "external parameters" in the Standard Model of particle physics.

The fact that [itex]\alpha \ [itex] is much less than 1 allows the use of perturbation theory in quantum electrodynamics. Physical results in this theory are expressed as power series in [itex]\alpha \ [itex], with higher orders of [itex]\alpha \ [itex] increasingly unimportant. In contrast, the large value of the corresponding factors in quantum chromodynamics makes calculations involving the strong force extremely difficult.

In the electroweak theory, one that unifies the weak interaction with electromagnetism, the fine-structure constant 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.

According to the theory of renormalization group, the value of the fine-structure constant (the strength of the electromagnetic interaction) depends on the energy scale. In fact, it grows logarithmically as the energy is increased. The observed value of [itex]\alpha \ [itex] is associated with the energy scale of the electron mass; the energy scale does not run below this because the electron (and the positron) is the lightest charged object whose quantum loops can contribute to the running. Therefore, we can say that 1/137.036 is the value of the fine-structure constant at zero energy. Moreover, as the energy scale increases, the electromagnetic interaction approaches the strength of the other two interactions, which is important for the theories of grand unification. If quantum electrodynamics were an exact theory, the fine-structure constant would actually diverge at an energy known as the Landau pole. This fact makes quantum electrodynamics inconsistent beyond the perturbative expansions.

One controversial explanation of the value of the fine-structure constant invokes the anthropic principle and argues that the value of the fine-structure is what it is because stable matter and therefore life and intelligent beings could not exist if the value were anything else. For instance, were α to change by 4%, carbon would no longer be produced in stellar fusion. If α were greater than 0.1, fusion would no longer occur in stars.

## Numerological explanations

As a dimensionless constant which does not seem to be directly related to any mathematical constant, the fine-structure constant has long been an object of fascination to physicists. Richard Feynman, one of the founders of quantum electrodynamics, referred to it as "one of the greatest damn mysteries of physics: a magic number that comes to use with no understanding by man." Towards the end of his life, the physicist Arthur Eddington constructed numerological "proofs" that [itex]1 / \alpha \ [itex] was an exact integer, even relating it to the Eddington number, his estimate of the number of electrons in the Universe. Experiments have since shown that [itex]1 / \alpha \ [itex] is definitely not an integer.

Along similar lines, mathematician James Gilson has suggested that the fine-structure constant, [itex]\alpha \ [itex] , can be mathematically determined to be

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

to a very large degree of accuracy. 29 and 137 are respectively the 10th and 33rd prime numbers. While this was, before 2002 CODATA, within the standard uncertainty of measurement for [itex]\alpha \ [itex], it is no longer as the CODATA value has been revised sufficiently to exclude this mathematical value.

## Is the fine structure constant really constant?

Physicists have been wondering whether the fine structure constant is really a constant, i.e. whether it always had the same value over the history of the universe, as some theories had been suggested which implied this not to be the case. First experimental tests of this question, most notably examination of spectral lines of distant astronomical objects and of the Oklo natural fission reactor, have not hinted any changes.

Recent improvements in astronomical techniques brought first hints in 2001 that [itex]\alpha \ [itex] in fact might change its value over time. (For a brief article see (1) (http://focus.aps.org/story/v8/st9) ). However in April 2004, new and more-detailed observations on quasars made using the UVES spectrograph on Kueyen, one of the 8.2-m telescopes of ESO's Very Large Telescope array at Paranal (Chile), puts limits to any change in [itex]\alpha \ [itex] at 0.6 parts per million over the past ten thousand million years. (See ESO press release (http://www.eso.org/outreach/press-rel/pr-2004/pr-05-04.html) or (2) (http://www.spacedaily.com/news/cosmology-04i.html) ).

As this limit contradicts the 2001 results, the question on whether [itex]\alpha \ [itex] is constant or not is open again and the correctness of the contradicting experiments is currently (as of 2004) hotly debated by the scientists involved.

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