Physical constant

File:The sun1.jpg

Light from the Sun traverses space at a constant speed before it reaches the Earth. The speed of light through a vacuum is an important physical constant and is thought to be the maximum speed any object or radiation can attain in the universe.

In physics, a physical constant is a physical quantity with a value that is generally believed to be both universal in nature and to remain unchanged over time. By contrast, a mathematical constant, which also has a fixed value, does not directly involve any physical measurement.

There are many physical constants in science. Some of the most widely recognized are:

• the rationalized Planck's constant ħ,
• the gravitational constant G,
• the speed of light in a vacuum c,
• the electric constant ε₀,
• the elementary charge e, and
• the fine-structure constant α.

Some fundamental physical constants (such as α above) do not have dimensions. Physicists recognize that if these constants were significantly different from their current values, the universe would be so radically different that stars like our Sun would not be able to exist and intelligent life would not have emerged.

Physical constants with and without dimensional units

Physical constants may or may not have units of dimension. For example, the speed of light in a vacuum is thought to be the maximum speed limit for any object or radiation in the universe and is expressed in the dimensions of distance divided by time. On the other hand, the fine-structure constant α, which characterizes the strength of the electromagnetic interaction, is dimensionless.

For physical constants that have units of dimension, their numerical values depend on the unit system used, such as SI or cgs. Moreover, the numerical values for constants such as the speed of light c (299,792,458 meters per second) cannot be predicted by theory but need to be experimentally determined.

If two physical constants are expressed in the same units of dimension, their ratios are dimensionless numbers (because their units cancel each other). It is hoped that a future theory of physics will be able to predict these dimensionless constants. Moreover, theoretical physicists can use a process known as "nondimensionalization" to write equations describing the laws of physics with only dimensionless physical constants. On that basis, theoretical physicists tend to regard these dimensionless quantities as "fundamental physical constants".

Yet, researchers may use the phrase "fundamental physical constant" in other ways as well. For example, the National Institute of Standards and Technology (NIST) uses it to refer to any universal physical quantity believed to be constant, such as the speed of light, c, and the gravitational constant, G.

Perhaps the best-known dimensionless fundamental physical constant is the fine-structure constant α. Its value is currently measured as approximately 1/137.035999, but no one knows why it has this value. Many attempts have been made to derive this value theoretically, but so far none has succeeded. The same holds true for the dimensionless ratios of masses of fundamental particles, such as the ratio of the mass of the proton (m_p) to that of the electron (m_e), which is approximately 1836.152673. Nonetheless, with the development of quantum chemistry in the twentieth century, a large number of previously inexplicable dimensionless physical constants were successfully computed by theory. For this reason, some theoretical physicists hope for continued progress in explaining the values of dimensionless physical constants.

Dimensionless physical constants

In physics, dimensionless or fundamental physical constants are, in the strictest sense, universal physical constants that are independent of systems of units and hence are dimensionless quantities. However, the term may also be used (for example, by NIST) to refer to any dimensioned universal physical constant, such as the speed of light (free space) or the gravitational constant. While both mathematical constants and fundamental physical constants are dimensionless, the latter are determined only by physical measurement and not defined by any combination of pure mathematical constants. The list of fundamental physical constants decreases when physical theory advances and shows how some previously fundamental constant can be computed in terms of others. The list increases when experiments measure new effects.

Physicists try to make their theories simpler and more elegant by reducing the number of physical constants appearing in the mathematical expression of their theories. This is accomplished by defining the units of measurement in such a way that several of the most common physical constants, such as the speed of light, among others, are normalized to unity. The resulting system of units, known as natural units, has a fair following in the literature on advanced physics because it considerably simplifies many equations.

Some physical constants, however, are dimensionless numbers which cannot be eliminated in this way. Their values have to be ascertained experimentally. A classic example is the fine structure constant,

$\alpha ={\frac {e^{2}}{\hbar c\ 4\pi \epsilon _{0}}}={\frac {1}{137.03599911}},$

where $e\$ is the elementary charge, $\hbar \$ is the reduced Planck's constant, $c\$ is the speed of light in a vacuum, and $\epsilon _{0}\$ is the permittivity of free space. In simple terms, the fine structure constant determines how strong the electromagnetic force is. Nobody knows why it has the value it does.

A long-sought goal of theoretical physics is to reduce the number of fundamental constants that need to be put in by hand, by calculating some from first principles. The reduction of chemistry to physics was an enormous step in this direction, since properties of atoms and molecules can now be calculated from the Standard Model, at least in principle. A successful Grand Unified Theory or Theory of Everything might reduce the number of fundamental constants further, ideally to zero. However, this goal remains elusive.

According to Michio Kaku (1994: 124-27), the Standard Model of particle physics contains 19 arbitrary dimensionless constants that describe the masses of the particles and the strengths of the various interactions. This was before it was discovered that neutrinos can have nonzero mass, and his list includes a quantity called the theta angle which seems to be zero. After the discovery of neutrino mass, and leaving out the theta angle, John Baez (2002) noted that the new Standard Model requires 25 arbitrary fundamental constants, namely:

• the fine structure constant,
• the strong coupling constant,
• the masses of the fundamental particles (normalized to the mass of some natural unit of mass), namely the 6 quarks, the 6 leptons, the Higgs boson, the W boson and the Z boson,
• the 4 parameters of the CKM matrix, which describe how quarks can oscillate between different forms,
• the 4 parameters of the Maki-Nakagawa-Sakata matrix, which does the same thing for neutrinos.

If we take gravity into account we need at least one more fundamental constant, namely

• the cosmological constant of Einstein's equations, which describe general relativity.

This gives a total of 26 fundamental physical constants. There are presumably more constants waiting to be discovered which describe the properties of dark matter. If dark energy turns out to be more complicated than a mere cosmological constant, even more constants will be needed.

In his book Just Six Numbers, Martin Rees considers the following numbers:

• Nu: ratio of the electroweak to the gravitational force (also see gravitational coupling constant);
• Epsilon: related to the strong force;
• Omega: the number of electrons and protons in the observable universe;
• Lambda: cosmological constant;
• Q: ratio of fundamental energies;
• Delta: number of spatial dimensions.

These constants constrain any plausible fundamental physical theory, which must either be able to produce these values from basic mathematics, or accept these constants as arbitrary. The question then arises: how many of these constants emerge from pure mathematics, and how many represent degrees of freedom for multiple possible valid physical theories, only some of which can be valid in our Universe? This leads to a number of interesting possibilities, including the possibility of multiple universes with different values of these constants, and the relation of these theories to the anthropic principle.

Note that Delta = 3; being simply an integer, most physicists would not consider this a dimensionless physical constant of the usual sort.

Some study of the fundamental constants has bordered on numerology. For instance, the physicist Arthur Eddington argued that for several mathematical reasons, the fine structure constant had to be exactly 1/136. When its value was discovered to be closer to 1/137, he changed his argument to match that value. Experiments since his day have shown that his arguments are still wrong; the constant is about 1/137.036.

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

How constant are the physical constants?

Beginning with Paul Dirac in 1937, some scientists have speculated that physical constants might decrease in proportion to the age of the universe. Scientific experiments have not yet provided any definite evidence for this, but they have indicated that such changes, if any, would be very small, and the experimental results have placed uppermost limits on some putative changes. For example, the maximum possible relative change has been estimated at roughly 10^-5 per year for the fine structure constant α, and 10^-11 for the gravitational constant G.

There is currently a debate [1] [2] about whether changes in physical constants that have dimensions—such as G, c, ħ, or ε₀—would be operationally meaningful. It is, however, generally agreed that a sufficient change in a dimensionless constant (such as α) would definitely be noticed. John D. Barrow (2002) gives the following explanation.

"[An] important lesson we learn from the way that pure numbers like α define the world is what it really means for worlds to be different. The pure number we call the fine structure constant and denote by α is a combination of the electron charge, e, the speed of light, c, and Planck's constant, ħ. At first we might be tempted to think that a world in which the speed of light was slower would be a different world. But this would be a mistake. If c, ħ, and e were all changed so that the values they have in metric (or any other) units were different when we looked them up in our tables of physical constants, but the value of α remained the same, this new world would be observationally indistinguishable from our world. The only thing that counts in the definition of worlds are the values of the dimensionless constants of Nature. If all masses were doubled in value you cannot tell because all the pure numbers defined by the ratios of any pair of masses are unchanged."

Some philosophical ramifications

Some physicists have explored the notion that if the dimensionless fundamental physical constants differed sufficiently from their current values, the universe would have taken a very different form. For example, a change in the value of the fine-structure constant (α) by a few percent would be enough to eliminate stars like our Sun and to prevent the emergence of intelligent living organisms. It therefore appears that our universe is fine-tuned for intelligent life.

Those who endorse the "weak anthropic principle" argue that it is because these fundamental constants have their respective values, there was sufficient order and richness in elemental diversity for life to have formed, subsequently evolving the intelligence necessary to determine the values for these constants.

Table of universal constants

Quantity	Symbol	Value	Relative Standard Uncertainty
characteristic impedance of vacuum	$Z_{0}=\mu _{0}c\,$	376.730 313 461... Ω	defined
electric constant (permittivity of free space)	$\epsilon _{0}=1/(\mu _{0}c^{2})\,$	8.854 187 817... × 10-12F·m-1	defined
magnetic constant (permeability of free space)	$\mu _{0}\,$	4π × 10-7 N·A-2 = 1.2566 370 614... × 10-6 N·A-2	defined
Newtonian constant of gravitation	$G\,$	6.6742(10) × 10-11m3·kg-1·s-2	1.5 × 10-4
Planck's constant	$h\,$	6.626 0693(11) × 10-34 J·s	1.7 × 10-7
Dirac's constant	$\hbar =h/(2\pi )$	1.054 571 68(18) × 10-34 J·s	1.7 × 10-7
speed of light in vacuum	$c\,$	299 792 458 m·s-1	defined

Table of electromagnetic constants

Quantity	Symbol	Value1 (SI units)	Relative Standard Uncertainty
Bohr magneton	$\mu _{B}=e\hbar /2m_{e}$	927.400 949(80) × 10-26 J·T-1	8.6 × 10-8
conductance quantum	$G_{0}=2e^{2}/h\,$	7.748 091 733(26) × 10-5 S	3.3 × 10-9
Coulomb's constant	$\kappa =1/4\pi \epsilon _{0}\,$	8.987 742 438 × 109 N·m2C-2	defined
elementary charge	$e\,$	1.602 176 53(14) × 10-19 C	8.5 × 10-8
Josephson constant	$K_{J}=2e/h\,$	483 597.879(41) × 109 Hz· V-1	8.5 × 10-8
magnetic flux quantum	$\phi _{0}=h/2e\,$	2.067 833 72(18) × 10-15 Wb	8.5 × 10-8
nuclear magneton	$\mu _{N}=e\hbar /2m_{p}$	5.050 783 43(43) × 10-27 J·T-1	8.6 × 10-8
resistance quantum	$R_{0}=h/2e^{2}\,$	12 906.403 725(43) Ω	3.3 × 10-9
von Klitzing constant	$R_{K}=h/e^{2}\,$	25 812.807 449(86) Ω	3.3 × 10-9

Table of atomic and nuclear constants

Quantity		Symbol	Value1 (SI units)	Relative Standard Uncertainty
Bohr radius		$a_{0}=\alpha /4\pi R_{\infty }\,$	0.529 177 2108(18) × 10-10 m	3.3 × 10-9
Fermi coupling constant		$G_{F}/(\hbar c)^{3}$	1.166 39(1) × 10-5 GeV-2	8.6 × 10-6
fine-structure constant		$\alpha =\mu _{0}e^{2}c/(2h)=e^{2}/(4\pi \epsilon _{0}\hbar c)\,$	7.297 352 568(24) × 10-3	3.3 × 10-9
Hartree energy		$E_{h}=2R_{\infty }hc\,$	4.359 744 17(75) × 10-18 J	1.7 × 10-7
quantum of circulation		$h/2m_{e}\,$	3.636 947 550(24) × 10-4 m2 s-1	6.7 × 10-9
Rydberg constant		$R_{\infty }=\alpha ^{2}m_{e}c/2h\,$	10 973 731.568 525(73) m-1	6.6 × 10-12
Thomson cross section		$(8\pi /3)r_{e}^{2}$	0.665 245 873(13) × 10-28 m2	2.0 × 10-8
weak mixing angle		$\sin ^{2}\theta _{W}=1-(m_{W}/m_{Z})^{2}\,$	0.222 15(76)	3.4 × 10-3

Table of physicochemical constants

Quantity		Symbol	Value1 (SI units)	Relative Standard Uncertainty
atomic mass unit (unified atomic mass unit)		$m_{u}=1\ u\,$	1.660 538 86(28) × 10-27 kg	1.7 × 10-7
Avogadro's number		$N_{A},L\,$	6.022 1415(10) × 1023	1.7 × 10-7
Boltzmann constant		$k=R/N_{A}\,$	1.380 6505(24) × 10-23 J·K-1	1.8 × 10-6
Faraday constant		$F=N_{A}e\,$	96 485.3383(83)C·mol-1	8.6 × 10-8
first radiation constant		$c_{1}=2\pi hc^{2}\,$	3.741 771 38(64) × 10-16 W·m2	1.7 × 10-7
	for spectral radiance	$c_{1L}\,$	1.191 042 82(20) × 10-16 W · m2 sr-1	1.7 × 10-7
Loschmidt constant	at $T$=273.15 K and $p$=101.325 kPa	$n_{0}=N_{A}/V_{m}\,$	2.686 7773(47) × 1025 m-3	1.8 × 10-6
gas constant		$R\,$	8.314 472(15) J·K-1·mol-1	1.7 × 10-6
molar Planck constant		$N_{A}h\,$	3.990 312 716(27) × 10-10 J · s · mol-1	6.7 × 10-9
molar volume of an ideal gas	at $T$=273.15 K and $p$=100 kPa	$V_{m}=RT/p\,$	22.710 981(40) × 10-3 m3 ·mol-1	1.7 × 10-6
	at $T$=273.15 K and $p$=101.325 kPa		22.413 996(39) × 10-3 m3 ·mol-1	1.7 × 10-6
Sackur-Tetrode constant	at $T$=1 K and $p$=100 kPa	$S_{0}/R={\frac {5}{2}}$ $+\ln \left[(2\pi m_{u}kT/h^{2})^{3/2}kT/p\right]$	-1.151 7047(44)	3.8 × 10-6
	at $T$=1 K and $p$=101.325 kPa		-1.164 8677(44)	3.8 × 10-6
second radiation constant		$c_{2}=hc/k\,$	1.438 7752(25) × 10-2 m·K	1.7 × 10-6
Stefan-Boltzmann constant		$\sigma =(\pi ^{2}/60)k^{4}/\hbar ^{3}c^{2}$	5.670 400(40) × 10-8 W·m-2·K-4	7.0 × 10-6
Wien displacement law constant		$b=(hc/k)/\,$ 4.965 114 231...	2.897 7685(51) × 10-3 m · K	1.7 × 10-6

Table of adopted values

Quantity		Symbol	Value (SI units)	Relative Standard Uncertainty
conventional value of Josephson constant2		$K_{J-90}\,$	483 597.9 × 109 Hz · V-1	defined
conventional value of von Klitzing constant3		$R_{K-90}\,$	25 812.807 Ω	defined
molar mass	constant	$M_{u}=M(\,^{12}{\mbox{C}})/12$	1 × 10-3 kg · mol-1	defined
	of carbon-12	$M(\,^{12}{\mbox{C}})=N_{A}m(\,^{12}{\mbox{C}})$	12 × 10-3 kg · mol−1	defined
standard acceleration of gravity (gee, free fall on Earth)		$g_{n}\,\!$	9.806 65 m·s-2	defined
standard atmosphere		${\mbox{atm}}\,$	101 325 Pa	defined

Notes

¹The values are given in the so-called concise form; the number in brackets is the standard uncertainty, which is the value multiplied by the relative standard uncertainty.
²This is the value adopted internationally for realizing representations of the volt using the Josephson effect.
³This is the value adopted internationally for realizing representations of the ohm using the quantum Hall effect.

See also

• Fundamental physical constant
• Fine-tuned universe
• Light
• Physical law
• CODATA
• Natural units
• Atomic units
• Planck units

References

ISBN links support NWE through referral fees

• CODATA Recommendations - 2002 CODATA Internationally recommended values of the Fundamental Physical Constants, U.S. National Institute of Standards and Technology.
• 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.
• Brian William Petley, 1985. Fundamental Physical Constants and the Frontier of Measurement, Adam Hilger. ISBN 0852744277.
• "How Many Fundamental Constants Are There?" by John Baez Mathematics Department, University of California, Riverside. Accessed on December 3, 2006.

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