Physical constant

A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that cannot be explained by a theory and therefore must be measured experimentally. It is distinct from a mathematical constant, which has a fixed numerical value, but does not directly involve any physical measurement.

There are many physical constants in science, some of the most widely recognized being the speed of light in vacuum c, the gravitational constant G, the Planck constant h, the electric constant ε₀, and the elementary charge e. Physical constants can take many dimensional forms: the speed of light signifies a maximum speed for any object and its dimension is length divided by time; while the proton-to-electron mass ratio, is dimensionless.

The term "fundamental physical constant" is sometimes used to refer to universal-but-dimensioned physical constants such as those mentioned above.^[1] Increasingly, however, physicists reserve the expression for the narrower case of dimensionless universal physical constants, such as the fine-structure constant α, which characterizes the strength of the electromagnetic interaction.

Physical constant, as discussed here, should not be confused with empirical constants, which are coefficients or parameters assumed to be constant in a given context without being fundamental.^[2] Examples include the characteristic time, characteristic length, or characteristic number (dimensionless) of a given system, or material constants (e.g., Madelung constant, electrical resistivity, and heat capacity) of a particular material or substance.

Characteristics

Physical constants are parameters in a physical theory that cannot be explained by that theory. This may be due to the apparent fundamental nature of the constant or due to limitations in the theory. Consequently, physical constants must be measured experimentally.^[3]: 9

The set of parameters considered physical constants change as physical models change and how fundamental they appear can change. For example, ${\displaystyle c}$  the speed of light was originally considered a property of light, a specific system. The discovery and verification of Maxwell's equations connect the same quantity an entire system, electromagnetism. When the theory of special relativity emerged, the quantity came to be understood as the basis of causality.^[3] The speed of light is so fundamental it now defines the international unit of length.

Relationship to units

Numerical values

Whereas the physical quantity indicated by a physical constant does not depend on the unit system used to express the quantity, the numerical values of dimensional physical constants do depend on choice of unit system. The term "physical constant" refers to the physical quantity, and not to the numerical value within any given system of units. For example, the speed of light is defined as having the numerical value of 299792458 when expressed in the SI unit metres per second, and as having the numerical value of 1 when expressed in the natural units Planck length per Planck time. While its numerical value can be defined at will by the choice of units, the speed of light itself is a single physical constant.

International System of Units

Main article: SI base unit

Since May 2019, all of the units in the International System of Units have been redefined in terms of fixed natural phenomena, including three fundamental constants: the speed of light in vacuum, c; the Planck constant, h; and the elementary charge, e.^[4]: 128

As a result of the new definitions, an SI unit like the kilogram can be written in terms of fundamental constants and one experimentally measured constant, Δν_Cs:^[4]: 131

1 kg = (299792458)²/(6.62607015×10⁻³⁴)(9192631770)hΔν_Cs/c².

Natural units

Main article: Natural units

It is possible to combine dimensional universal physical constants to define fixed quantities of any desired dimension, and this property has been used to construct various systems of natural units of measurement. Depending on the choice and arrangement of constants used, the resulting natural units may be convenient to an area of study. For example, Planck units, constructed from c, G, ħ, and k_B give conveniently sized measurement units for use in studies of quantum gravity, and atomic units, constructed from ħ, m_e, e and 4πε₀ give convenient units in atomic physics. The choice of constants used leads to widely varying quantities.

Number of fundamental constants

The number of fundamental physical constants depends on the physical theory accepted as "fundamental". Currently, this is the theory of general relativity for gravitation and the Standard Model for electromagnetic, weak and strong nuclear interactions and the matter fields. Between them, these theories account for a total of 19 independent fundamental constants. There is, however, no single "correct" way of enumerating them, as it is a matter of arbitrary choice which quantities are considered "fundamental" and which as "derived". Uzan^[3] lists 22 "fundamental constants of our standard model" as follows:

• the gravitational constant G,
• the speed of light c,
• the Planck constant h,
• the 9 Yukawa couplings for the quarks and leptons (equivalent to specifying the rest mass of these elementary particles),
• 2 parameters of the Higgs field potential,
• 4 parameters for the quark mixing matrix,
• 3 coupling constants for the gauge groups SU(3) × SU(2) × U(1) (or equivalently, two coupling constants and the Weinberg angle),
• a phase for the QCD vacuum.

The number of 19 independent fundamental physical constants is subject to change under possible extensions of the Standard Model, notably by the introduction of neutrino mass (equivalent to seven additional constants, i.e. 3 Yukawa couplings and 4 lepton mixing parameters).^[3]

The discovery of variability in any of these constants would be equivalent to the discovery of "new physics".^[3]

The question as to which constants are "fundamental" is neither straightforward nor meaningless, but a question of interpretation of the physical theory regarded as fundamental; as pointed out by Lévy-Leblond 1977, not all physical constants are of the same importance, with some having a deeper role than others. Lévy-Leblond 1977 proposed a classification schemes of three types of constants:

• A: physical properties of particular objects
• B: characteristic of a class of physical phenomena
• C: universal constants

The same physical constant may move from one category to another as the understanding of its role deepens; this has notably happened to the speed of light, which was a class A constant (characteristic of light) when it was first measured, but became a class B constant (characteristic of electromagnetic phenomena) with the development of classical electromagnetism, and finally a class C constant with the discovery of special relativity.^[5]

Tests on time-independence

Main article: Time-variation of fundamental constants

By definition, fundamental physical constants are subject to measurement, so that their being constant (independent on both the time and position of the performance of the measurement) is necessarily an experimental result and subject to verification.

Paul Dirac in 1937 speculated that physical constants such as the gravitational constant or the fine-structure constant might be subject to change over time in proportion of the age of the universe. Experiments can in principle only put an upper bound on the relative change per year. For the fine-structure constant, this upper bound is comparatively low, at roughly 10⁻¹⁷ per year (as of 2008).^[6]

The gravitational constant is much more difficult to measure with precision, and conflicting measurements in the 2000s have inspired the controversial suggestions of a periodic variation of its value in a 2015 paper.^[7] However, while its value is not known to great precision, the possibility of observing type Ia supernovae which happened in the universe's remote past, paired with the assumption that the physics involved in these events is universal, allows for an upper bound of less than 10⁻¹⁰ per year for the gravitational constant over the last nine billion years.^[8]

Similarly, an upper bound of the change in the proton-to-electron mass ratio has been placed at 10⁻⁷ over a period of 7 billion years (or 10⁻¹⁶ per year) in a 2012 study based on the observation of methanol in a distant galaxy.^[9][10]

It is problematic to discuss the proposed rate of change (or lack thereof) of a single dimensional physical constant in isolation. The reason for this is that the choice of units is arbitrary, making the question of whether a constant is undergoing change an artefact of the choice (and definition) of the units.^[11][12][13]

For example, in SI units, the speed of light was given a defined value in 1983. Thus, it was meaningful to experimentally measure the speed of light in SI units prior to 1983, but it is not so now. Similarly, with effect from May 2019, the Planck constant has a defined value, such that all SI base units are now defined in terms of fundamental physical constants. With this change, the international prototype of the kilogram is being retired as the last physical object used in the definition of any SI unit.

Tests on the immutability of physical constants look at dimensionless quantities, i.e. ratios between quantities of like dimensions, in order to escape this problem. Changes in physical constants are not meaningful if they result in an observationally indistinguishable universe. For example, a "change" in the speed of light c would be meaningless if accompanied by a corresponding change in the elementary charge e so that the expression e²/(4πε₀ħc) (the fine-structure constant) remained unchanged.^[14]

Dimensionless physical constants

Any ratio between physical constants of the same dimensions results in a dimensionless physical constant, for example, the proton-to-electron mass ratio. The fine-structure constant α is the best known dimensionless fundamental physical constant. It is the value of the elementary charge squared expressed in Planck units. This value has become a standard example when discussing the derivability or non-derivability of physical constants. Introduced by Arnold Sommerfeld, its value and uncertainty as determined at the time was consistent with 1/137. This motivated Arthur Eddington (1929) to construct an argument why its value might be 1/137 precisely, which related to the Eddington number, his estimate of the number of protons in the Universe.^[15] By the 1940s, it became clear that the value of the fine-structure constant deviates significantly from the precise value of 1/137, refuting Eddington's argument.^[16]

Fine-tuned universe

Main articles: Fine-tuned universe and Anthropic principle

Some physicists have explored the notion that if the dimensionless physical constants had sufficiently different values, our Universe would be so radically different that intelligent life would probably not have emerged, and that our Universe therefore seems to be fine-tuned for intelligent life.^[17] The anthropic principle states a logical truism: the fact of our existence as intelligent beings who can measure physical constants requires those constants to be such that beings like us can exist. There are a variety of interpretations of the constants' values, including that of a divine creator (the apparent fine-tuning is actual and intentional), or that the universe is one universe of many in a multiverse (e.g. the many-worlds interpretation of quantum mechanics), or even that, if information is an innate property of the universe and logically inseparable from consciousness, a universe without the capacity for conscious beings cannot exist.

Table of physical constants

Main article: List of physical constants

The table below lists some frequently used constants and their CODATA recommended values. For a more extended list, refer to List of physical constants.

Quantity	Symbol	Value^[18]	Relative standard uncertainty
elementary charge	${\displaystyle e}$	1.602176634×10^−19 C^[19]	0
Newtonian constant of gravitation	${\displaystyle G}$	6.67430(15)×10^−11 m^3⋅kg^−1⋅s^−2^[20]	2.2×10^−5
Planck constant	${\displaystyle h}$	6.62607015×10^−34 J⋅Hz^−1^[21]	0
speed of light in vacuum	${\displaystyle c}$	299792458 m⋅s^−1^[22]	0
vacuum electric permittivity	${\displaystyle \varepsilon _{0}}$	8.8541878128(13)×10^−12 F⋅m^−1^[23]	1.5×10^−10
vacuum magnetic permeability	${\displaystyle \mu _{0}}$	1.25663706212(19)×10^−6 N⋅A^−2^[24]	1.5×10^−10
electron mass	${\displaystyle m_{\mathrm {e} }}$	9.1093837015(28)×10^−31 kg^[25]	3.0×10^−10
fine-structure constant	${\displaystyle \alpha =e^{2}/2\varepsilon _{0}hc}$	7.2973525693(11)×10^−3^[26]	1.5×10^−10
Josephson constant	${\displaystyle K_{\mathrm {J} }=2e/h}$	483597.8484...×10^9 Hz⋅V^−1^[27]	0
Rydberg constant	${\displaystyle R_{\infty }=\alpha ^{2}m_{\mathrm {e} }c/2h}$	10973731.568160(21) m^−1^[28]	1.9×10^−12
von Klitzing constant	${\displaystyle R_{\mathrm {K} }=h/e^{2}}$	25812.80745... Ω^[29]	0

See also

• List of common physics notations

References

1. "Fundamental Physical Constants from NIST". from the original on 2016-01-13. Retrieved 2016-01-14. NIST
2. "ISO 80000-1:2022 Quantities and units — Part 1: General". iso.org. Retrieved 2023-08-31.
3. Uzan, Jean-Philippe (2011). "Varying Constants, Gravitation and Cosmology". Living Reviews in Relativity. 14 (1): 2. arXiv:1009.5514. Bibcode:2011LRR....14....2U. doi:10.12942/lrr-2011-2. PMC 5256069. PMID 28179829.
4. Le Système international d’unités [The International System of Units] (PDF) (in French and English) (9th ed.), International Bureau of Weights and Measures, 2019, ISBN 978-92-822-2272-0
5. Lévy-Leblond, J. (1977). "On the conceptual nature of the physical constants". La Rivista del Nuovo Cimento. Series 2. 7 (2): 187–214. Bibcode:1977NCimR...7..187L. doi:10.1007/bf02748049. S2CID 121022139.Lévy-Leblond, J.-M. (1979). "The importance of being (a) Constant". In Toraldo di Francia, G. (ed.). Problems in the Foundations of Physics, Proceedings of the International School of Physics 'Enrico Fermi' Course LXXII, Varenna, Italy, July 25 – August 6, 1977. New York: NorthHolland. pp. 237–263.
6. T. Rosenband; et al. (2008). "Frequency Ratio of Al⁺ and Hg⁺ Single-Ion Optical Clocks; Metrology at the 17th Decimal Place". Science. 319 (5871): 1808–12. Bibcode:2008Sci...319.1808R. doi:10.1126/science.1154622. PMID 18323415. S2CID 206511320.
7. J.D. Anderson; G. Schubert; V. Trimble; M.R. Feldman (April 2015), "Measurements of Newton's gravitational constant and the length of day", EPL, 110 (1): 10002, arXiv:1504.06604, Bibcode:2015EL....11010002A, doi:10.1209/0295-5075/110/10002, S2CID 119293843
8. J. Mould; S. A. Uddin (2014-04-10), "Constraining a Possible Variation of G with Type Ia Supernovae", Publications of the Astronomical Society of Australia, 31: e015, arXiv:1402.1534, Bibcode:2014PASA...31...15M, doi:10.1017/pasa.2014.9, S2CID 119292899
9. Bagdonaite, Julija; Jansen, Paul; Henkel, Christian; Bethlem, Hendrick L.; Menten, Karl M.; Ubachs, Wim (December 13, 2012). "A Stringent Limit on a Drifting Proton-to-Electron Mass Ratio from Alcohol in the Early Universe" (PDF). Science. 339 (6115): 46–48. Bibcode:2013Sci...339...46B. doi:10.1126/science.1224898. hdl:1871/39591. PMID 23239626. S2CID 716087.
10. Moskowitz, Clara (December 13, 2012). "Phew! Universe's Constant Has Stayed Constant". Space.com. from the original on December 14, 2012. Retrieved December 14, 2012.
11. Duff, Michael (2015). "How fundamental are fundamental constants?". Contemporary Physics. 56 (1): 35–47. arXiv:1412.2040. Bibcode:2015ConPh..56...35D. doi:10.1080/00107514.2014.980093. hdl:10044/1/68485. S2CID 118347723.
12. Duff, M. J. (13 August 2002). "Comment on time-variation of fundamental constants". arXiv:hep-th/0208093.
13. Duff, M. J.; Okun, L. B.; Veneziano, G. (2002). "Trialogue on the number of fundamental constants". Journal of High Energy Physics. 2002 (3): 023. arXiv:physics/0110060. Bibcode:2002JHEP...03..023D. doi:10.1088/1126-6708/2002/03/023. S2CID 15806354.
14. Barrow, John D. (2002), The Constants of Nature; From Alpha to Omega – The Numbers that Encode the Deepest Secrets of the Universe, Pantheon Books, ISBN 978-0-375-42221-8
15. A.S Eddington (1956). "The Constants of Nature". In J.R. Newman (ed.). The World of Mathematics. Vol. 2. Simon & Schuster. pp. 1074–1093.
16. H. Kragh (2003). "Magic Number: A Partial History of the Fine-Structure Constant". Archive for History of Exact Sciences. 57 (5): 395–431. doi:10.1007/s00407-002-0065-7. S2CID 118031104.
17. Leslie, John (1998). Modern Cosmology & Philosophy. University of Michigan: Prometheus Books. ISBN 1573922501.
18. The values are given in the so-called concise form, where the number in parentheses indicates the standard uncertainty referred to the least significant digits of the value.
19. "2018 CODATA Value: elementary charge". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
20. "2018 CODATA Value: Newtonian constant of gravitation". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
21. "2018 CODATA Value: Planck constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2021-04-28.
22. "2018 CODATA Value: speed of light in vacuum". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
23. "2018 CODATA Value: vacuum electric permittivity". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
24. "2018 CODATA Value: vacuum magnetic permeability". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
25. "2018 CODATA Value: electron mass". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
26. "2018 CODATA Value: fine-structure constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
27. "2018 CODATA Value: Josephson constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
28. "2018 CODATA Value: Rydberg constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
29. "2018 CODATA Value: von Klitzing constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.

• Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2008). (PDF). Reviews of Modern Physics. 80 (2): 633–730. arXiv:0801.0028. Bibcode:2008RvMP...80..633M. doi:10.1103/RevModPhys.80.633. Archived from the original (PDF) on 2017-10-01.

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