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In physics, a dimensionless physical constant is a
physical constant A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant, ...
that is
dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1 ...
, i.e. a pure number having no units attached and having a numerical value that is independent of whatever
system of units A system of measurement is a collection of units of measurement and rules relating them to each other. Systems of measurement have historically been important, regulated and defined for the purposes of science and commerce. Systems of measurement i ...
may be used. For example, if one considers one particular
airfoil An airfoil ( American English) or aerofoil (British English) is the cross-sectional shape of an object whose motion through a gas is capable of generating significant lift, such as a wing, a sail, or the blades of propeller, rotor, or turbine. ...
, the
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dom ...
value of the laminar–turbulent transition is one relevant dimensionless physical constant of the problem. However, it is strictly related to the particular problem: for example, it is related to the airfoil being considered and also to the type of fluid in which it moves. On the other hand, the term fundamental physical constant is used to refer to some ''universal'' dimensionless constants. Perhaps the best-known example is the fine-structure constant, ''α'', which has an approximate value of . The correct use of the term ''fundamental physical constant'' should be restricted to the dimensionless universal physical constants that currently cannot be derived from any other source. This precise definition is the one that will be followed here. However, the term ''fundamental physical constant'' has been sometimes used to refer to certain universal dimensioned
physical constant A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant, ...
s, such as the speed of light ''c'',
vacuum permittivity Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric consta ...
''ε''0, Planck constant ''h'', and the gravitational constant ''G'', that appear in the most basic theories of physics.
NIST The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sci ...
and
CODATA The Committee on Data of the International Science Council (CODATA) was established in 1966 as the Committee on Data for Science and Technology, originally part of the International Council of Scientific Unions, now part of the International ...
sometimes used the term in this in the past.

# Characteristics

There is no exhaustive list of such constants but it does make sense to ask about the minimal number of fundamental constants necessary to determine a given physical theory. Thus, the
Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. ...
requires 25 physical constants, about half of them are the masses of fundamental particles (which become "dimensionless" when expressed relative to the Planck mass or, alternatively, as coupling strength with the Higgs field along with the gravitational constant). Fundamental physical constants cannot be derived and have to be measured. Developments in physics may lead to either a reduction or an extension of their number: discovery of new particles, or new relationships between physical phenomena, would introduce new constants, while the development of a more fundamental theory might allow the derivation of several constants from a more fundamental constant. A long-sought goal of theoretical physics is to find first principles ( theory of everything) from which all of the fundamental dimensionless constants can be calculated and compared to the measured values. The large number of fundamental constants required in the Standard Model has been regarded as unsatisfactory since the theory's formulation in the 1970s. The desire for a theory that would allow the calculation of particle masses is a core motivation for the search for " Physics beyond the Standard Model".

# History

In the 1920s and 1930s,
Arthur Eddington Sir Arthur Stanley Eddington (28 December 1882 – 22 November 1944) was an English astronomer, physicist, and mathematician. He was also a philosopher of science and a populariser of science. The Eddington limit, the natural limit to the lumin ...
embarked upon extensive mathematical investigation into the relations between the fundamental quantities in basic physical theories, later used as part of his effort to construct an overarching theory unifying quantum mechanics and cosmological physics. For example, he speculated on the potential consequences of the ratio of the electron radius to its mass. Most notably, in a 1929 paper he set out an argument based on the Pauli exclusion principle and the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
that fixed the value of the reciprocal of the fine-structure constant as 𝛼−1 = 16 + × 16 × (16 − 1) = 136. When its value was discovered to be closer to 137, he changed his argument to match that value. His ideas were not widely accepted, and subsequent experiments have shown that they were wrong (for example, none of the measurements of the fine-structure constant suggest an integer value; in 2018 it was measured at α = 1/137.035999046(27)). Though his derivations and equations were unfounded, Eddington was the first physicist to recognize the significance of universal dimensionless constants, now considered among the most critical components of major physical theories such as the
Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. ...
and ΛCDM cosmology. He was also the first to argue for the importance of the
cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field equ ...
Λ itself, considering it vital for explaining the
expansion of the universe The expansion of the universe is the increase in distance between any two given gravitationally unbound parts of the observable universe with time. It is an intrinsic expansion whereby the scale of space itself changes. The universe does not exp ...
, at a time when most physicists (including its discoverer,
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
) considered it an outright mistake or mathematical artifact and assumed a value of zero: this at least proved prescient, and a significant positive Λ features prominently in ΛCDM. Eddington may have been the first to attempt in vain to derive the basic dimensionless constants from fundamental theories and equations, but he was certainly not the last. Many others would subsequently undertake similar endeavors, and efforts occasionally continue even today. None have yet produced convincing results or gained wide acceptance among theoretical physicists. An empirical relation between the masses of the electron, muon and tau has been discovered by physicist Yoshio Koide, but this formula remains unexplained.

# Examples

Dimensionless fundamental physical constants include: * ''α'', the fine-structure constant, the
coupling constant In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two ...
for the
electromagnetic interaction In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
(≈ ). Also the square of the
electron charge The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundame ...
, expressed in Planck units, which defines the scale of charge of
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions (quarks, leptons, a ...
s with charge. * ''μ'' or ''β'', the
proton-to-electron mass ratio In physics, the proton-to-electron mass ratio, ''μ'' or ''β'', is the rest mass of the proton (a baryon found in atoms) divided by that of the electron (a lepton found in atoms), a dimensionless quantity, namely: :''μ'' = The number in parenthe ...
, the
rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
of the proton divided by that of the electron (≈1836). More generally, the ratio of the
rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
es of any pair of
elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions (quarks, leptons, a ...
s. * ''α''s, the
coupling constant In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two ...
for the
strong force The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the ...
(≈ 1)

## Fine-structure constant

One of the dimensionless fundamental constants is the fine-structure constant: :$\alpha = \frac= \frac \approx \frac,$ where ''e'' is the
elementary charge The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundame ...
, ''ħ'' is the reduced Planck constant, ''c'' is the speed of light in a vacuum, and ''ε''0 is the
permittivity of free space Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric consta ...
. The fine-structure constant is fixed to the strength of the
electromagnetic force In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
. At low energies, ''α'' ≈ , whereas at the scale of the
Z boson In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , an ...
, about 90 GeV, one measures ''α'' ≈ . There is no accepted theory explaining the value of ''α'';
Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superflu ...
elaborates:

## Standard model

The original
standard model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. ...
of particle physics from the
1970s File:1970s decade montage.jpg, Clockwise from top left: U.S. President Richard Nixon doing the V for Victory sign after his resignation from office following the Watergate scandal in 1974; The United States was still involved in the Vietnam ...
contained 19 fundamental dimensionless constants describing the masses of the particles and the strengths of the
electroweak In particle physics, the electroweak interaction or electroweak force is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very differe ...
and
strong force The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the ...
s. 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 The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. ...
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: * the fine structure constant; * the strong coupling constant; * fifteen masses of the fundamental particles (relative to the Planck mass ''m''P = ), namely: ** six
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...
s ** six
lepton In particle physics, a lepton is an elementary particle of half-integer spin (spin ) that does not undergo strong interactions. Two main classes of leptons exist: charged leptons (also known as the electron-like leptons or muons), and neut ...
s ** the
Higgs boson The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field, one of the fields in particle physics theory. In the Sta ...
** the
W boson In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , an ...
** the
Z boson In particle physics, the W and Z bosons are vector bosons that are together known as the weak bosons or more generally as the intermediate vector bosons. These elementary particles mediate the weak interaction; the respective symbols are , , an ...
* four parameters of the CKM matrix, describing how
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...
s oscillate between different forms; * four parameters of the Pontecorvo–Maki–Nakagawa–Sakata matrix, which does the same thing for neutrinos. * Note: whenever we are talking about angle we are talking about radians which are equal to 1. *

## Cosmological constants

The
cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field equ ...
, which can be thought of as the density of
dark energy In physical cosmology and astronomy, dark energy is an unknown form of energy that affects the universe on the largest scales. The first observational evidence for its existence came from measurements of supernovas, which showed that the univer ...
in the universe, is a fundamental constant in
physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of ...
that has a dimensionless value of approximately 10−122. Other dimensionless constants are the measure of homogeneity in the universe, denoted by ''Q'', which is explained below by Martin Rees, the baryon mass per photon, the cold dark matter mass per photon and the neutrino mass per photon.

## Barrow and Tipler

Barrow and Tipler (1986) anchor their broad-ranging discussion of
astrophysics Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline said, Astrophysics "seeks to ascertain the nature of the hea ...
, cosmology,
quantum physics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...
, teleology, and the
anthropic principle The anthropic principle, also known as the "observation selection effect", is the hypothesis, first proposed in 1957 by Robert Dicke, that there is a restrictive lower bound on how statistically probable our observations of the universe are, beca ...
in the fine-structure constant, the
proton-to-electron mass ratio In physics, the proton-to-electron mass ratio, ''μ'' or ''β'', is the rest mass of the proton (a baryon found in atoms) divided by that of the electron (a lepton found in atoms), a dimensionless quantity, namely: :''μ'' = The number in parenthe ...
(which they, along with Barrow (2002), call β), and the
coupling constant In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two ...
s for the
strong force The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the ...
and gravitation.

## Martin Rees's Six Numbers

Martin Rees Martin John Rees, Baron Rees of Ludlow One or more of the preceding sentences incorporates text from the royalsociety.org website where: (born 23 June 1942) is a British cosmologist and astrophysicist. He is the fifteenth Astronomer Royal, ...
, 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 electrostatic and the gravitational forces between two protons. This ratio is denoted α/αG in Barrow and Tipler (1986). ''N'' governs the relative importance of gravity and electrostatic attraction/repulsion in explaining the properties of
baryonic matter In particle physics, a baryon is a type of composite subatomic particle which contains an odd number of valence quarks (at least 3). Baryons belong to the hadron family of particles; hadrons are composed of quarks. Baryons are also classif ...
;Rees, M. (2000), p. . * ''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 In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between two ...
for the
strong force The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the ...
; * Ω ≈ 0.3: the ratio of the actual density of the universe to the critical (minimum) density required for the universe to eventually collapse under its gravity. Ω determines the
ultimate fate of the universe The ultimate fate of the universe is a topic in physical cosmology, whose theoretical restrictions allow possible scenarios for the evolution and ultimate fate of the universe to be described and evaluated. Based on available observational ev ...
. If Ω ≥ 1, the universe may experience a
Big Crunch The Big Crunch is a hypothetical scenario for the ultimate fate of the universe, in which the expansion of the universe eventually reverses and the universe recollapses, ultimately causing the cosmic scale factor to reach zero, an event potentia ...
. If Ω < 1, the universe may expand forever; * ''λ'' ≈ 0.7: The ratio of the energy density of the universe, due to the
cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field equ ...
, to the critical density of the universe. Others denote this ratio by $\Omega_$; * ''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 A supercluster is a large group of smaller galaxy clusters or galaxy groups; they are among the largest known structures in the universe. The Milky Way is part of the Local Group galaxy group (which contains more than 54 galaxies), which in ...
, expressed as a fraction of the energy equivalent to the
rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
''m'' of that structure, namely ''mc''2;Rees, M. (2000), p. 118. * ''D'' = 3: the number of macroscopic spatial
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
s. ''N'' and ''ε'' govern the fundamental interactions of physics. The other constants (''D'' excepted) govern the size,
age Age or AGE may refer to: Time and its effects * Age, the amount of time someone or something has been alive or has existed ** East Asian age reckoning, an Asian system of marking age starting at 1 * Ageing or aging, the process of becoming older ...
, and expansion of the universe. These five constants must be estimated empirically. ''D'', on the other hand, is necessarily a nonzero natural number and does not have an uncertainty. 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.

# Use in SI

In 2019, fundamental physical constants have been introduced for the definition of all
SI unit The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. E ...
s and derived units.

*
Cabibbo–Kobayashi–Maskawa matrix In the Standard Model of particle physics, the Cabibbo–Kobayashi–Maskawa matrix, CKM matrix, quark mixing matrix, or KM matrix is a unitary matrix which contains information on the strength of the flavour-changing weak interaction. Technical ...
( Cabibbo angle) *
Dimensionless numbers in fluid mechanics Dimensionless numbers in fluid mechanics are a set of dimensionless quantities that have an important role in analyzing the behavior of fluids. Common examples include the Reynolds or the Mach numbers, which describe as ratios the relative magnitu ...
*
Dirac large numbers hypothesis The Dirac large numbers hypothesis (LNH) is an observation made by Paul Dirac in 1937 relating ratios of size scales in the Universe to that of force scales. The ratios constitute very large, dimensionless numbers: some 40 orders of magnitude i ...
* Neutrino oscillation *
Physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of ...
*
Standard Model The Standard Model of particle physics is the theory describing three of the four known fundamental forces (electromagnetic, weak and strong interactions - excluding gravity) in the universe and classifying all known elementary particles. ...
*
Weinberg angle The weak mixing angle or Weinberg angle is a parameter in the Weinberg– Salam theory of the electroweak interaction, part of the Standard Model of particle physics, and is usually denoted as . It is the angle by which spontaneous symmetry bre ...
*
Fine-tuned universe The characterization of the universe as finely tuned suggests that the occurrence of life in the universe is very sensitive to the values of certain fundamental physical constants and that the observed values are, for some reason, improbable. If ...
*
Koide formula The Koide formula is an unexplained empirical equation discovered by Yoshio Koide in 1981. In its original form, it relates the masses of the three charged leptons; later authors have extended the relation to neutrinos, quarks, and other famil ...

# Bibliography

*
Martin Rees Martin John Rees, Baron Rees of Ludlow One or more of the preceding sentences incorporates text from the royalsociety.org website where: (born 23 June 1942) is a British cosmologist and astrophysicist. He is the fifteenth Astronomer Royal, ...
, 1999. ''Just Six Numbers: The Deep Forces that Shape the Universe''. London: Weidenfeld & Nicolson. * Josef Kuneš, 2012
''Dimensionless Physical Quantities in Science and Engineering''
Amsterdam: Elsevier.

# External articles

;General *
John D. Barrow John David Barrow (29 November 1952 – 26 September 2020) was an English cosmologist, theoretical physicist, and mathematician. He served as Gresham Professor of Geometry at Gresham College from 2008 to 2011. Barrow was also a writer of popul ...
, 2002. ''The Constants of Nature; From Alpha to Omega The Numbers that Encode the Deepest Secrets of the Universe''. Pantheon Books. . * *
Michio Kaku Michio Kaku (, ; born January 24, 1947) is an American theoretical physicist, futurist, and popularizer of science ( science communicator). He is a professor of theoretical physics in the City College of New York and CUNY Graduate Center. Kaku ...
, 1994. '' Hyperspace: A Scientific Odyssey Through Parallel Universes, Time Warps, and the Tenth Dimension''. Oxford University Press.
Fundamental Physical Constants from NIST

Values of fundamental constants.
CODATA The Committee on Data of the International Science Council (CODATA) was established in 1966 as the Committee on Data for Science and Technology, originally part of the International Council of Scientific Unions, now part of the International ...
, 2002. *
John Baez John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, appl ...
, 2002,
How Many Fundamental Constants Are There?
* Simon Plouffe, 2004,

;Articles on variance of the fundamental constants * *
John D. Barrow John David Barrow (29 November 1952 – 26 September 2020) was an English cosmologist, theoretical physicist, and mathematician. He served as Gresham Professor of Geometry at Gresham College from 2008 to 2011. Barrow was also a writer of popul ...
and Webb, J. K.,
Inconstant Constants – Do the inner workings of nature change with time?
''Scientific American'' (June 2005). * Michael Duff, 2002
Comment on time-variation of fundamental constants.
* * * * * {{cite journal , last1=Webb , first1=J. K. , last2=Murphy , first2=M. T. , last3=Flambaum , first3=V. V. , last4=Dzuba , first4=V. A. , last5=Barrow , first5=J. D. , last6=Churchill , first6=C. W. , last7=Prochaska , first7=J. X. , last8=Wolfe , first8=A. M. , authorlink8=Arthur M. Wolfe, title=Further Evidence for Cosmological Evolution of the Fine Structure Constant , journal=Physical Review Letters , volume=87 , issue=9 , date=2001-08-09 , issn=0031-9007 , doi=10.1103/physrevlett.87.091301 , pmid=11531558 , page=091301, arxiv=astro-ph/0012539, bibcode=2001PhRvL..87i1301W , s2cid=40461557