Dimensionless physical constant

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In physics, a dimensionless physical constant, sometimes called a fundamental physical constant, is a physical constant that is dimensionless. It has no units attached, but has a numerical value that is independent of the system of units used. Perhaps the best-known example is the fine-structure constant α, which has approximate value of 1137.036.

The term fundamental physical constant has also been used to refer to universal but dimensioned physical constants such as the speed of light c, vacuum permittivity ε0, Planck's constant h, and the gravitational constant G.[1] However, the numerical values of these constants are not fundamental, since they depend on the units used to express them. Increasingly, physicists reserve the use of the term fundamental physical constant for dimensionless physical constants that cannot be derived from any other source.[citation needed]


The numerical values of dimensional physical constants depend on the units used to express them. So one can define a basis set of units so that selected dimensional physical constants have numerical value 1. The basis set may consist of units of time, length, mass, charge, and temperature, or an equivalent set. A choice of units is called a system of units.

For example, the International System of Units (SI) is the most generally used system of units today. The numerical values of dimensional physical constants are arbitrary and have no natural significance. (An exception is the vacuum permeability constant µ0, whose numerical value of 4π×10−7 is a mathematical constant determined by the definition of the ampere in the SI system.)

The Planck units constitute another system of units. It is a system of natural units chosen so that the numerical values of the vacuum speed of light, the universal gravitational constant, and the constants of Planck, Coulomb, and Boltzmann are unity. These five dimensional physical constants then disappear from equations of physical laws, but as this occurs merely from the choice of units, these constants are considered not fundamental in an operationally distinguishable sense.[2][3]

In contrast, the numerical values of dimensionless physical constants are independent of the units used. These constants cannot be eliminated by any choice of a system of units. Such constants include:

Unlike mathematical constants, the values of the dimensionless fundamental physical constants cannot be calculated; they are determined only by physical measurement. This is one of the unsolved problems of physics.

One of the dimensionless fundamental constants is the fine structure constant:

 \alpha = \frac{e^2}{\hbar c \ 4 \pi \varepsilon_0}  \approx \frac{1}{137.03599908},

where e is the elementary charge, ħ is the reduced Planck's constant, c is the speed of light in a vacuum, and ε0 is the permittivity of free space. The fine structure constant is fixed to the strength of the electromagnetic force. At low energies, α ≈ 1/137, whereas at the scale of the Z boson, about 90 GeV, one measures α ≈ 1/127. There is no accepted theory explaining the value of α; Richard Feynman elaborates:

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 about 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 man. 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!

The analog of the fine structure constant for gravitation is the gravitational coupling constant. This constant requires the arbitrary choice of a pair of objects having mass. The electron and proton are natural choices because they are stable, and their properties are well measured and well understood. If αG is calculated from two protons, its value is ≈10−38.

The list of dimensionless physical constants increases in length whenever experiments measure new relationships between physical phenomena. The list of fundamental dimensionless constants, however, decreases when advances in physics show how some previously known constant can be computed in terms of others. A long-sought goal of theoretical physics is to find first principles from which all of the fundamental dimensionless constants can be calculated and compared to the measured values. A successful "Theory of Everything" would allow such a calculation, but so far, this goal has remained elusive.

Constants in the standard model and in cosmology[edit]

The original standard model of particle physics from the 1970s contained 19 fundamental dimensionless constants describing the masses of the particles and the strengths of the electroweak and strong forces. In the 1990s, neutrinos were discovered to have nonzero mass, and a quantity called the vacuum angle was found to be indistinguishable from zero.

The complete standard model requires 25 fundamental dimensionless constants (Baez, 2011). At present, their numerical values are not understood in terms of any widely accepted theory and are determined only from measurement. These 25 constants are:

One constant is required for cosmology:

Thus, currently there are 26 known fundamental dimensionless physical constants. However, this number may not be the final one. For example, if neutrinos turn out to be Majorana fermions, the Maki-Nakagawa-Sakata matrix has two additional parameters. Secondly, if dark matter is discovered, or if the description of dark energy requires more than the cosmological constant, further fundamental constants will be needed.

Well-known subsets[edit]

Certain dimensionless constants are discussed more frequently than others.

Barrow and Tipler[edit]

Barrow and Tipler (1986) anchor their broad-ranging discussion of astrophysics, cosmology, quantum physics, teleology, and the anthropic principle in the fine structure constant, the proton-to-electron mass ratio (which they, along with Barrow (2002), call β), and the coupling constants for the strong force and gravitation.

Martin Rees's Six Numbers[edit]

Martin Rees, in his book Just Six Numbers, mulls over the following six dimensionless constants, whose values he deems fundamental to present-day physical theory and the known structure of the universe:

N and ε govern the fundamental interactions of physics. The other constants (D excepted) govern the size, age, and expansion of the universe. These five constants must be estimated empirically. D, on the other hand, is necessarily a nonzero natural number and cannot be measured. Hence most physicists would not deem it a dimensionless physical constant of the sort discussed in this entry.

Any plausible fundamental physical theory must be consistent with these six constants, and must either derive their values from the mathematics of the theory, or accept their values as empirical.

Calculation attempts[edit]

No formulae for the fundamental physical constants are known to this day.

The mathematician Simon Plouffe has made an extensive search of computer databases of mathematical formulae, seeking formulae for the mass ratios of the fundamental particles.

One example of numerology is by the astrophysicist Arthur Eddington. He set out alleged mathematical reasons why the reciprocal of the fine structure constant had to be exactly 136. When its value was discovered to be closer to 137, he changed his argument to match that value. Experiments have since shown that Eddington was wrong; to six significant digits, the reciprocal of the fine-structure constant is 137.036.

An empirical relation between the masses of the electron, muon and tau has been discovered by physicist Yoshio Koide, but this formula remains unexplained.

See also[edit]


  1. ^ http://physics.nist.gov/cuu/Constants/ NIST
  2. ^ Michael Duff (2002) "Comment on time-variation of fundamental constants"
  3. ^ Michael Duff, O. Okun and Gabriele Veneziano (2002) "Trialogue on the number of fundamental constants," Journal of High Energy Physics 3: 023.
  4. ^ a b Rees, M. (2000), p. .
  5. ^ Rees, M. (2000), p. 53.
  6. ^ Rees, M. (2000), p. 110.
  7. ^ Rees, M. (2000), p. 118.


External articles[edit]

Articles on variance of the fundamental constants