Dimensionless physical constant

In physics, a dimensionless physical constant (sometimes fundamental physical constant) is a universal physical constant. Because it is a dimensionless quantity, its numerical value is the same under all possible systems of units. Fundamental physical constant may also refer (as in NIST) to fundamental but dimensional physical constants such as the speed of light c, vacuum permittivity, and the gravitational constant G.

Introduction

While both mathematical constants and fundamental physical constants are dimensionless, the values of the physical constants cannot be calculated from any combination of purely mathematical ones, and are determined only by physical measurement.

Certain fundamental physical constants, such as the speed of light in a vacuum, the universal gravitational constant, and the constants of Planck and Boltzmann, are normalized to 1 if the units for time, length, mass, charge, and temperature are chosen appropriately. The resulting system of units is known as natural or Planck units. However, a handful of dimensionless physical constants cannot be eliminated in any system of units; their values must be determined experimentally. Such constants include:

α, the fine structure constant, the coupling constant for the electromagnetic interaction (≈1/137);
μ or β, the proton-to-electron mass ratio, the rest mass of the proton divided by that of the electron (≈1836). More generally, the rest masses of all elementary particles relative to that of the electron;
α_s, the coupling constant for the strong force (≈1);
α_G, the gravitational coupling constant.

The best known of the above constants is the fine structure constant:

\alpha ={\frac {e^{2}}{\hbar c\ 4\pi \varepsilon _{0}}}={\frac {1}{137.0359997}},

where $e\$ is the elementary charge, $\hbar \$ is the reduced Planck's constant, $c\$ is the speed of light in a vacuum, and $\varepsilon _{0}\$ is the permittivity of free space. At the risk of oversimplification, the fine structure constant determines the strength of the electromagnetic force. There is no accepted theory of why $\alpha$ has the value it does.

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 α_G is calculated from two protons, its value is ≈10⁻³⁸.

The list of fundamental physical constants increases when experiments measure new relationships between physical phenomena, and decreases when advances in physical theory show how some previously fundamental constant can be computed in terms of others. The reduction of chemistry to physics was a big step in this direction, since the properties of atoms and molecules can now be calculated from the Standard Model, at least in principle. A long-sought goal of theoretical physics is to find first principles from which some or all of the dimensionless constants can be calculated rather than empirically estimated. A successful Grand Unified Theory (colloquially termed a "Theory of Everything") might yield a further reduction in the number of fundamental constants, ideally to zero. To date, this goal has remained elusive.

The Standard Model

The Standard Model of particle physics contains 19 arbitrary dimensionless constants describing the masses of the particles and the strengths of the electroweak and strong forces. This was before neutrinos were discovered to have nonzero mass, and before a quantity called the theta angle was found to be indistinguishable from zero. After the discovery of neutrino mass, and omitting the theta angle, Baez (2002) noted that the new Standard Model requires 25 fundamental dimensionless constants, whose numerical values are, to the best of present understanding, arbitrary. These are the:

Fine structure constant;
Strong coupling constant;
Masses of the fundamental particles (in terms of the Planck mass or some other natural unit of mass), namely the six quarks, the six leptons, the Higgs boson, the W boson, and the Z boson;
Four parameters of the CKM matrix, describing how quarks oscillate between different forms;
Four parameters of the Maki-Nakagawa-Sakata matrix, which does the same thing for neutrinos.

One more fundamental constant is required for gravity, the:

Cosmological constant (in terms of Planck units) of Einstein's equations for general relativity.

Thus, there are 26 dimensionless fundamental physical constants. More constants will presumably be needed to describe the properties of dark matter. If the description of dark energy requires more than the cosmological constant, yet more constants will be needed.

Other

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.

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.

The study of some fundamental constants can border on numerology. For instance, the astrophysicist Arthur Eddington 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 at least six significant digits, the reciprocal of the fine-structure constant is 137.036.

Martin Rees's Six Numbers

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≈10³⁶: the ratio of the fine structure constant (the dimensionless coupling constant for electromagnetism) to the gravitational coupling constant, the latter defined using two protons. In Barrow and Tipler (1986) and elsewhere in Wikipedia, this ratio is denoted α/α_G. N governs the relative importance of gravity and electrostatic attraction/repulsion in explaining the properties of baryonic matter;^[1]
ε≈0.007: The fraction of the mass of four protons that is released as energy when fused into a helium nucleus. ε governs the energy output of stars, and is determined by the coupling constant for the strong force;^[2]
Ω ≈ 0.3: the ratio of the actual density of the universe to the critical (minimum) density required for the universe to expand forever. Ω determines the ultimate fate of the universe. If Ω>1, the universe will experience a Big Crunch. If Ω<1, the universe will expand forever;^[1]
λ ≈ 0.7: The ratio of the energy density of the universe, due to the cosmological constant, to the critical density of the universe. Others denote this ratio by $\Omega _{\Lambda }$ ;^[3]
Q ≈ 10^{– 5}: The energy required to break up and disperse an instance of the largest known structures in the universe, namely a galactic cluster or supercluster, expressed as a fraction of the energy equivalent to the rest mass m of that structure, namely mc²;^[4]
D = 3: the number of macroscopic spatial dimensions.

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. There are also compelling physical and mathematical reasons why D = 3.

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. The question then arises: how many values of these constants result from purely mathematical considerations, and how many represent degrees of freedom for possible valid physical theories, only some of which are possible in a universe with intelligent observers? This leads to a number of interesting possibilities, such as multiple universes, each with different values of these constants. Multiple universes give rise to selection effects and the anthropic principle.

References

^ ^a ^b Rees, M. (2000), p. .
^ Rees, M. (2000), p. 53.
^ Rees, M. (2000), p. 110.
^ Rees, M. (2000), p. 118.

John D. Barrow, 2002. The Constants of Nature; From Alpha to Omega - The Numbers that Encode the Deepest Secrets of the Universe. Pantheon Books. ISBN 0-375-42221-8.
Barrow, John D.; Tipler, Frank J. (1986). The Anthropic Cosmological Principle (1st ed.). Oxford University Press. ISBN 978-0-19-282147-8. LCCN 87028148.
Michio Kaku, 1994. Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension. Oxford University Press.
Martin Rees, 1999. Just Six Numbers: The Deep Forces that Shape the Universe. London: Weidenfeld & Nicolson. ISBN 0-7538-1022-0

External articles

General

Fundamental Physical Constants from NIST
Values of fundamental constants. CODATA, 2002.
John Baez, 2002, "How Many Fundamental Constants Are There?"
Plouffe. Simon, 2004, "A search for a mathematical expression for mass ratios using a large database."

Do the fundamental constants vary?

John Bahcall, Charles L. Steinhardt, and Schlegel, David, 2004, "Does the fine-structure constant vary with cosmological epoch?" Astrophys. J. 600: 520.
John Barrow and Webb, J. K., "Inconstant Constants - Do the inner workings of nature change with time?" Scientific American (June 2005).
Marion, H., et al. 2003, "A search for variations of fundamental constants using atomic fountain clocks," Phys.Rev.Lett. 90: 150801.
Martins, J.A.P. et al., 2004, "WMAP constraints on varying α and the promise of reionization," Phys.Lett. B585: 29-34.
Olive, K.A., et al., 2002, "Constraints on the variations of the fundamental couplings," Phys.Rev. D66: 045022.
Uzan, J-P, 2003, "The fundamental constants and their variation: observational status and theoretical motivations," Rev.Mod.Phys. 75: 403.
Webb, J.K. et al., 2001, "Further evidence for cosmological evolution of the fine-structure constant," Phys. Rev. Lett. 87: 091301.

External links

Murphy, Michael, Web page at the Swinburne University of Technology, Australia.
Webb, John K., Web page at the University of New South Wales, Australia.

[Rees,_M._2000,_p-1] Rees, M. (2000), p. .

[2] Rees, M. (2000), p. 53.

[3] Rees, M. (2000), p. 110.

[4] Rees, M. (2000), p. 118.

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