Fine-structure Constant

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Fine-structure Constant

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

:

\alpha = 7.297 352 568(24) 	imes 10^{-3} = rac{1}{137.035 999 11(46)}. \

It can be defined as

:

\alpha = rac{k_C e^2}{\hbar c} = rac{e^2}{\hbar c 4 \pi arepsilon_0} \ = rac{e^2}{2 arepsilon_0 h c}

where

k_C \

is the Coulomb Constant ,

e \

is the Elementary Charge ,

\hbar = h/(2 \pi) \

is the reduced Planck's Constant ,

c \

is the Speed Of Light in a vacuum, and

arepsilon_0 \

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,

4 \pi arepsilon_0 \

, is the dimensionless constant 1. Then the fine-structure constant becomes

:

\alpha = rac{e^2}{\hbar c}. \

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

:

\alpha = \left( rac{e}{q_P} 
ight)^2. \

PHYSICAL INTERPRETATION

For any arbitrary length

s \

, the fine-structure constant is the ratio of two energies: (i) the energy needed to bring two electrons from infinity to a distance of

s \

against their electrostatic repulsion, and (ii) the energy of a single photon of Wavenumber

k = rac{2 \pi}{\lambda} = rac{1}{s}

where

\lambda

is the photon's Wavelength .

:

\alpha = rac{k_C e^2}{s} \div rac{h c}{\lambda} = rac{k_C e^2}{s} \div rac{h c}{2 \pi s} = rac{k_C e^2}{\hbar c}

Historically, the first physical interpretation of the fine-structure constant,

\alpha \

, 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

\alpha \

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

\alpha \

, with higher orders of

\alpha \

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

\alpha \

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.

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 Line s of distant astronomical objects and of the Oklo Natural Nuclear Fission Reactor , have not hinted any changes.

Recent improvements in astronomical techniques brought first hints in 2001 that

\alpha \

in fact might change its value over time. (For a brief article see (1) ).
However in recent years several experiments have put increasing tighter limits on the variability of

\alpha \

over time.
In April 2004, the first set of new and more-detailed observations on quasars made using the UVES Spectrograph on Kueyen , one of the 8.2-m Telescope s of ESO 's Very Large Telescope array at Paranal ( Chile ), puts limits to any change in

\alpha \

at 0.6 parts per million over the past ten billion years. ESO press release (2) (3) ).
In April and July 2005, further theoretical (4) and experimental (5) studies have placed even tighter bounds on the variation in the fine structure constant.

ANTHROPIC EXPLANATION

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

''Note: within the scientific community, 'numerology' has a colloquial significance and it is mostly used to refer to any scientific theory that appears to be more mathematical than experimental in its approach.''

As a dimensionless constant which does not seem to be directly related to any constructed Numerological "proofs" that

1 / \alpha \

was an exact integer, even relating it to the Eddington Number , his estimate of the number of electrons in the Universe. It has been suggested that this was a spoof or hoax . Experiments have since shown that

1 / \alpha \

is definitely not an integer.

Along similar lines, mathematician James Gilson has suggested that the fine-structure constant,

\alpha,\,

can be mathematically determined to be

:

\alpha =  rac{\cos \left(\pi/137 
ight)}{137} \ rac{	an \left(\pi/(137 \cdot 29) 
ight)}{\pi/(137 \cdot 29)}  \approx 1/137.0359997867

to a very large degree of accuracy. 29 and 137 are respectively the 10^th and 33^rd prime numbers. While this was, before 2002 CODATA, within the standard uncertainty of measurement for

\alpha,\,

now it is 1.7 standard uncertainties from the experimental data, which is possible, but a bit improbable.

QUOTE

''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.'' Richard P. Feynman , QED - The strange theory of light and matter, Princeton University Press 1985, p. 129

SEE ALSO

Coupling Constant s

EXTERNAL LINKS

http://physics.nist.gov/cuu/Constants/alpha.html

http://scienceworld.wolfram.com/physics/FineStructureConstant.html

Scientific American: Inconstant Constants Barrow and Webb discuss their results in a ''Scientific American'' article

REFERENCES

(4) ¹

(5) ²