About constants

2.1 About constants

Our physical theories introduce various structures to describe the phenomena of Nature. They involve various fields, symmetries and constants. These structures are postulated in order to construct a mathematically-consistent description of the known physical phenomena in the most unified and simple way.

We define the fundamental constants of a physical theory as any parameter that cannot be explained by this theory. Indeed, we are often dealing with other constants that in principle can be expressed in terms of these fundamental constants. The existence of these two sets of constants is important and arises from two different considerations. From a theoretical point of view we would like to extract the minimal set of fundamental constants, but often these constants are not measurable. From a more practical point of view, we need to measure constants, or combinations of constants, which allow us to reach the highest accuracy.

Therefore, these fundamental constants are contingent quantities that can only be measured. Such parameters have to be assumed constant in this theoretical framework for two reasons:

from a theoretical point of view: the considered framework does not provide any way to compute these parameters, i.e., it does not have any equation of evolution for them since otherwise it would be considered as a dynamical field,
from an experimental point of view: these parameters can only be measured. If the theories in which they appear have been validated experimentally, it means that, at the precisions of these experiments, these parameters have indeed been checked to be constant, as required by the necessity of the reproducibility of experimental results.

This means that testing for the constancy of these parameters is a test of the theories in which they appear and allow to extend our knowledge of their domain of validity. This also explains the definition chosen by Weinberg [526 ] who stated that they cannot be calculated in terms of other constants “…not just because the calculation is too complicated (as for the viscosity of water) but because we do not know of anything more fundamental”.

This has a series of implications. First, the list of fundamental constants to consider depends on our theories of physics and, thus, on time. Indeed, when introducing new, more unified or more fundamental, theories the number of constants may change so that this list reflects both our knowledge of physics and, more important, our ignorance. Second, it also implies that some of these fundamental constants can become dynamical quantities in a more general theoretical framework so that the tests of the constancy of the fundamental constants are tests of fundamental physics, which can reveal that what was thought to be a fundamental constant is actually a field whose dynamics cannot be neglected. If such fundamental constants are actually dynamical fields it also means that the equations we are using are only approximations of other and more fundamental equations, in an adiabatic limit, and that an equation for the evolution of this new field has to be obtained.

The reflections on the nature of the constants and their role in physics are numerous. We refer to the books [29 , 215 , 510 , 509 ] as well as [59 , 165 , 216 , 393 , 521 , 526 , 538 ] for various discussions of this issue that we cannot develop at length here. This paragraph summarizes some of the properties of the fundamental constants that have attracted some attention.

2.1.1 Characterizing the fundamental constants

Physical constants seem to play a central role in our physical theories since, in particular, they determined the magnitudes of the physical processes. Let us sketch briefly some of their properties.

How many fundamental constants should be considered?

The set of constants, which are conventionally considered as fundamental [213

] consists of the electron charge $e$ , the electron mass $me$ , the proton mass $mp$ , the reduced Planck constant $ℏ$ , the velocity of light in vacuum $c$ , the Avogadro constant $NA$ , the Boltzmann constant $kB$ , the Newton constant $G$ , the permeability and permittivity of space, $𝜀0$ and $μ0$ . The latter has a fixed value in the SI system of unit ( $μ = 4 π × 10−7 H m −1 0$ ), which is implicit in the definition of the Ampere; $𝜀 0$ is then fixed by the relation $−2 𝜀0μ0 = c$ . However, it is clear that this cannot correspond to the list of the fundamental constants, as defined earlier as the free parameters of the theoretical framework at hand. To define such a list we must specify this framework. Today, gravitation is described by general relativity, and the three other interactions and the matter fields are described by the standard model of particle physics. It follows that one has to consider 22 unknown constants (i.e., 19 unknown dimensionless parameters): the Newton constant $G$ , 6 Yukawa couplings for the quarks ( $hu,hd,hc,hs,ht,hb$ ) and 3 for the leptons ( $he,hμ,h τ$ ), 2 parameters of the Higgs field potential ( $ˆμ,λ$ ), 4 parameters for the Cabibbo–Kobayashi–Maskawa matrix (3 angles $𝜃 ij$ and a phase $δ CKM$ ), 3 coupling constants for the gauge groups $SU (3)c × SU (2)L × U(1)Y$ ( $g1,g2,g3$ or equivalently $g2,g3$ and the Weinberg angle $𝜃W$ ), and a phase for the QCD vacuum ( $𝜃QCD$ ), to which one must add the speed of light $c$ and the Planck constant $h$ . See Table 1 for a summary and their numerical values.

Table 1: List of the fundamental constants of our standard model. See Ref. [379

] for further details on the measurements.

Constant	Symbol	Value
Speed of light	$c$	299 792 458 m s^–1
Planck constant (reduced)	$ℏ$	1.054 571 628(53) × 10^–34 J s
Newton constant	$G$	6.674 28(67) × 10^–11 m² kg^–1 s^–2
Weak coupling constant (at $mZ$ )	$g2(mZ )$	0.6520 ± 0.0001
Strong coupling constant (at $mZ$ )	$g3(mZ )$	1.221 ± 0.022
Weinberg angle	$2 sin 𝜃W$ (91.2 GeV) $MS-$	0.23120 ± 0.00015
Electron Yukawa coupling	$he$	2.94 × 10^–6
Muon Yukawa coupling	$hμ$	0.000607
Tauon Yukawa coupling	$hτ$	0.0102156
Up Yukawa coupling	$hu$	0.000016 ± 0.000007
Down Yukawa coupling	$hd$	0.00003 ± 0.00002
Charm Yukawa coupling	$hc$	0.0072 ± 0.0006
Strange Yukawa coupling	$hs$	0.0006 ± 0.0002
Top Yukawa coupling	$ht$	1.002 ± 0.029
Bottom Yukawa coupling	$hb$	0.026 ± 0.003
Quark CKM matrix angle	$sin 𝜃12$	0.2243 ± 0.0016
	$sin 𝜃23$	0.0413 ± 0.0015
	$sin 𝜃13$	0.0037 ± 0.0005
Quark CKM matrix phase	$δ CKM$	1.05 ± 0.24
Higgs potential quadratic coefficient	$2 ˆμ$	?
Higgs potential quartic coefficient	$λ$	?
QCD vacuum phase	$𝜃 QCD$	< 10^–9

Again, this list of fundamental constants relies on what we accept as a fundamental theory. Today we have many hints that the standard model of particle physics has to be extended, in particular to include the existence of massive neutrinos. Such an extension will introduce at least seven new constants (3 Yukawa couplings and 4 Maki–Nakagawa–Sakata (MNS) parameters, similar to the CKM parameters). On the other hand, the number of constants can decrease if some unifications between various interaction exist (see Section 5.3.1 for more details) since the various coupling constants may be related to a unique coupling constant $αU$ and an energy scale of unification $MU$ through

−1 −1 bi MU α i (E ) = α U + ---ln ---, 2π E

where the $bi$ are numbers, which depend on the explicit model of unification. Note that this would also imply that the variations, if any, of various constants shall be correlated.

Relation to other usual constants.

These parameters of the standard model are related to various constants that will appear in this review (see Table 2). First, the quartic and quadratic coefficients of the Higgs field potential are related to the Higgs mass and vev, $∘ ------- mH = −μˆ2 ∕2$ and $∘ ------- v = −μˆ2 ∕λ$ . The latter is related to the Fermi constant $G = (v2√2-)−1 F$ , which imposes that $v = (246.7 ± 0.2) GeV$ while the Higgs mass is badly constrained. The masses of the quarks and leptons are related to their Yukawa coupling and the Higgs vev by $√ -- m = hv∕ 2$ . The values of the gauge couplings depend on energy via the renormalization group so that they are given at a chosen energy scale, here the mass of the $Z$ -boson, $mZ$ . $g1$ and $g2$ are related by the Weinberg angle as $g1 = g2 tan 𝜃W$ . The electromagnetic coupling constant is not $g1$ since $SU (2)L × U (1)Y$ is broken to $U (1)elec$ so that it is given by

Defining the fine-structure constant as $αEM = g2EM∕ ℏc$ , the (usual) zero energy electromagnetic fine structure constant is $αEM = 1∕137.03599911 (46)$ is related to $αEM (mZ ) = 1∕(127.918 ± 0.018)$ by the renormalization group equations. In particular, it implies that $2- (--------m2Z0--------) αEM ∼ α (mZ ) + 9π ln m4um4cmdmsmbm3em3μm3τ$ . We define the QCD energy scale, $ΛQCD$ , as the energy at which the strong coupling constant diverges. Note that it implies that $Λ QCD$ also depends on the Higgs and fermion masses through threshold effects. More familiar constants, such as the masses of the proton and the neutron are, as we shall discuss in more detail below (see Section 5.3.2), more difficult to relate to the fundamental parameters because they depend not only on the masses of the quarks but also on the electromagnetic and strong binding energies.

Table 2: List of some related constants that appear in our discussions. See Ref. [379

Constant	Symbol	Value
Electromagnetic coupling constant	$gEM = e = g2sin𝜃W$	0.313429 ± 0.000022
Higgs mass	$mH$	> 100 GeV
Higgs vev	$v$	(246.7 ± 0.2) GeV
Fermi constant	$√--2 GF = 1∕ 2v$	1.166 37(1) × 10^–5 GeV^–2
Mass of the $W ±$	$m W$	80.398 ± 0.025 GeV
Mass of the $Z$	$mZ$	91.1876 ± 0.0021 GeV
Fine structure constant	$αEM$	1/137.035 999 679(94)
Fine structure constant at $mZ$	$αEM (mZ )$	1/(127.918 ± 0.018)
Weak structure constant at $mZ$	$αW (mZ )$	0.03383 ± 0.00001
Strong structure constant at $mZ$	$αS(mZ )$	0.1184 ± 0.0007
Gravitational structure constant	$2 αG = Gm p∕ℏc$	$∼$ 5.905 × 10^–39
Electron mass	$√ -- me = hev∕ 2$	510.998910 ± 0.000013 keV
Mu mass	$√ -- m μ = hμv∕ 2$	105.658367 ± 0.000004 MeV
Tau mass	$√ -- m τ = h τv∕ 2$	1776.84 ± 0.17 MeV
Up quark mass	$√ -- mu = huv ∕ 2$	(1.5 – 3.3) MeV
Down quark mass	$√ -- md = hdv ∕ 2$	(3.5 – 6.0) MeV
Strange quark mass	$ms = hsv∕√2--$	$105+25 − 35$ MeV
Charm quark mass	$m = h v∕√2-- c c$	$1.27+0.07 −0.11$ GeV
Bottom quark mass	$√ -- mb = hbv ∕ 2$	$+0.17 4.20−0.07$ GeV
Top quark mass	$√ -- mt = htv∕ 2$	171.3 ± 2.3 GeV
QCD energy scale	$ΛQCD$	(190 – 240) MeV
Mass of the proton	$mp$	938.272013 ± 0.000023 MeV
Mass of the neutron	$m n$	939.565346 ± 0.000023 MeV
proton-neutron mass difference	$Qnp$	1.2933321 ± 0.0000004 MeV
proton-to-electron mass ratio	$μ = mp ∕me$	1836.15
electron-to-proton mass ratio	$¯μ = me ∕mp$	1/1836.15
$d − u$ quark mean mass	$mq = (mu + md)∕2$	(2.5 – 5.0) MeV
$d − u$ quark mass difference	$δmq = md − mu$	(0.2 – 4.5) MeV
proton gyromagnetic factor	$gp$	5.586
neutron gyromagnetic factor	$gn$	–3.826
Rydberg constant	$R∞$	10 973 731.568 527(73) m^–1

Are some constants more fundamental?

As pointed-out by Lévy-Leblond [328

], all constants of physics do not play the same role, and some have a much deeper role than others. Following [328

], we can define three classes of fundamental constants, class A being the class of the constants characteristic of a particular system, class B being the class of constants characteristic of a class of physical phenomena, and class C being the class of universal constants. Indeed, the status of a constant can change with time. For instance, the velocity of light was initially a class A constant (describing a property of light), which then became a class B constant when it was realized that it was related to electromagnetic phenomena and, to finish, it ended as a type C constant (it enters special relativity and is related to the notion of causality, whatever the physical phenomena). It has even become a much more fundamental constant since it now enters in the definition of the meter [413 ] (see Ref. [510

] for a more detailed discussion). This has to be contrasted with the proposition of Ref. [538

] to distinguish the standard model free parameters as the gauge and gravitational couplings (which are associated to internal and spacetime curvatures) and the other parameters entering the accommodation of inertia in the Higgs sector.

Relation with physical laws.

Lévy-Leblond [328 ] proposed to rank the constants in terms of their universality and he proposed that only three constants be considered to be of class C, namely $G$ , $ℏ$ and $c$ . He pointed out two important roles of these constants in the laws of physics. First, they act as concept synthesizer during the process of our understanding of the laws of nature: contradictions between existing theories have often been resolved by introducing new concepts that are more general or more synthetic than older ones. Constants build bridges between quantities that were thought to be incommensurable and thus allow new concepts to emerge. For example $c$ underpins the synthesis of space and time while the Planck constant allowed to related the concept of energy and frequency and the gravitational constant creates a link between matter and space-time. Second, it follows that these constants are related to the domains of validity of these theories. For instance, as soon as velocity approaches $c$ , relativistic effects become important, relativistic effects cannot be negligible. On the other hand, for speed much below $c$ , Galilean kinematics is sufficient. Planck constant also acts as a referent, since if the action of a system greatly exceeds the value of that constant, classical mechanics will be appropriate to describe this system. While the place of $c$ (related to the notion of causality) and $ℏ$ (related to the quantum) in this list are well argued, the place of $G$ remains debated since it is thought that it will have to be replaced by some mass scale.

Evolution.

There are many ways the list of constants can change with our understanding of physics. First, new constants may appear when new systems or new physical laws are discovered; this is, for instance, the case of the charge of the electron or more recently the gauge couplings of the nuclear interactions. A constant can also move from one class to a more universal class. An example is that of the electric charge, initially of class A (characteristic of the electron), which then became class B when it was understood that it characterizes the strength of the electromagnetic interaction. A constant can also disappear from the list, because it is either replaced by more fundamental constants (e.g., the Earth acceleration due to gravity and the proportionality constant entering Kepler law both disappeared because they were “explained” in terms of the Newton constant and the mass of the Earth or the Sun) or because it can happen that a better understanding of physics teaches us that two hitherto distinct quantities have to be considered as a single phenomenon (e.g., the understanding by Joule that heat and work were two forms of energy led to the fact that the Joule constant, expressing the proportionality between work and heat, lost any physical meaning and became a simple conversion factor between units used in the measurement of heat (calories) and work (Joule)). Nowadays the calorie has fallen in disuse. Indeed demonstrating that a constant is varying will have direct implications on our list of constants. In conclusion, the evolution of the number, status of the constants can teach us a lot about the evolution of the ideas and theories in physics since it reflects the birth of new concepts, their evolution and unification with other ones.

2.1.2 Constants and metrology

Since we cannot compute them in the theoretical framework in which they appear, it is a crucial property of the fundamental constants (but in fact of all the constants) that their value can be measured. The relation between constants and metrology is a huge subject to which we just draw the attention on some selected aspects. For more discussions, see [56 , 280 , 278 ].

The introduction of constants in physical laws is also closely related to the existence of systems of units. For instance, Newton’s law states that the gravitational force between two masses is proportional to each mass and inversely proportional to the square of their separation. To transform the proportionality to an equality one requires the use of a quantity with dimension of m³ kg^–1 s^–2 independent of the separation between the two bodies, of their mass, of their composition (equivalence principle) and on the position (local position invariance). With an other system of units the numerical value of this constant could have simply been anything. Indeed, the numerical value of any constant crucially depends on the definition of the system of units.

Measuring constants.

The determination of the laboratory value of constants relies mainly on the measurements of lengths, frequencies, times, …(see [414 ] for a treatise on the measurement of constants and [213 ] for a recent review). Hence, any question on the variation of constants is linked to the definition of the system of units and to the theory of measurement. The behavior of atomic matter is determined by the value of many constants. As a consequence, if, e.g., the fine-structure constant is spacetime dependent, the comparison between several devices such as clocks and rulers will also be spacetime dependent. This dependence will also differ from one clock to another so that metrology becomes both device and spacetime dependent, a property that will actually be used to construct tests of the constancy of the constants. Indeed a measurement is always a comparison between two physical systems of the same dimensions. This is thus a relative measurement, which will give as result a pure number. This trivial statement is oversimplifying since in order to compare two similar quantities measured separately, one needs to perform a number of comparisons. In order to reduce the number of comparisons (and in particular to avoid creating every time a chain of comparisons), a certain set of them has been included in the definitions of units. Each units can then be seen as an abstract physical system, which has to be realized effectively in the laboratory, and to which another physical system is compared. A measurement in terms of these units is usually called an absolute measurement. Most fundamental constants are related to microscopic physics and their numerical values can be obtained either from a pure microscopic comparison (as is, e.g., the case for $me∕mp$ ) or from a comparison between microscopic and macroscopic values (for instance to deduce the value of the mass of the electron in kilogram). This shows that the choice of the units has an impact on the accuracy of the measurement since the pure microscopic comparisons are in general more accurate than those involving macroscopic physics. This implies that only the variation of dimensionless constants can be measured and in case such a variation is detected, it is impossible to determine, which dimensional constant is varying [183

It is also important to stress that in order to deduce the value of constants from an experiment, one usually needs to use theories and models. An example [278 ] is provided by the Rydberg constant. It can easily be expressed in terms of some fundamental constants as $R ∞ = α2 mec ∕2h EM$ . It can be measured from, e.g., the triplet $1s − 2s$ transition in hydrogen, the frequency of which is related to the Rydberg constant and other constants by assuming QED so that the accuracy of $R ∞$ is much lower than that of the measurement of the transition. This could be solved by defining $R ∞$ as $4νH (1s − 2s)∕3c$ but then the relation with more fundamental constants would be more complicated and actually not exactly known. This illustrates the relation between a practical and a fundamental approach and the limitation arising from the fact that we often cannot both exactly calculate and directly measure some quantity. Note also that some theoretical properties are plugged in the determination of the constants.

As a conclusion, let us recall that (i) in general, the values of the constants are not determined by a direct measurement but by a chain involving both theoretical and experimental steps, (ii) they depend on our theoretical understanding, (iii) the determination of a self-consistent set of values of the fundamental constants results from an adjustment to achieve the best match between theory and a defined set of experiments (which is important because we actually know that the theories are only good approximation and have a domain of validity) (iv) that the system of units plays a crucial role in the measurement chain, since for instance in atomic units, the mass of the electron could have been obtained directly from a mass ratio measurement (even more precise!) and (v) fortunately the test of the variability of the constants does not require a priori to have a high-precision value of the considered constants.

System of units.

Thus, one needs to define a coherent system of units. This has a long, complex and interesting history that was driven by simplicity and universality but also by increasing stability and accuracy [29 , 509

]. Originally, the sizes of the human body were mostly used to measure the length of objects (e.g., the foot and the thumb gave feet and inches) and some of these units can seem surprising to us nowadays (e.g., the span was the measure of hand with fingers fully splayed, from the tip of the thumb to the tip of the little finger!). Similarly weights were related to what could be carried in the hand: the pound, the ounce, the dram…. Needless to say, this system had a few disadvantages since each country, region has its own system (for instance in France there was more than 800 different units in use in 1789). The need to define a system of units based on natural standard led to several propositions to define a standard of length (e.g., the mille by Gabriel Mouton in 1670 defined as the length of one angular minute of a great circle on the Earth or the length of the pendulum that oscillates once a second by Jean Picard and Christiaan Huygens). The real change happened during the French Revolution during which the idea of a universal and non anthropocentric system of units arose. In particular, the Assemblée adopted the principle of a uniform system of weights and measures on 8 May 1790 and, in March 1791 a decree (these texts are reprinted in [510 ]) was voted, stating that a quarter of the terrestrial meridian would be the basis of the definition of the meter (from the Greek metron, as proposed by Borda): a meter would henceforth be one ten millionth part of a quarter of the terrestrial meridian. Similarly the gram was defined as the mass of one cubic centimeter of distilled water (at a precise temperature and pressure) and the second was defined from the property that a mean solar day must last 24 hours.

To make a long story short, this led to the creation of the metric system and then of the signature of La convention du mètre in 1875. Since then, the definition of the units have evolved significantly. First, the definition of the meter was related to more immutable systems than our planet, which, as pointed out by Maxwell in 1870, was an arbitrary and inconstant reference. He then suggested that atoms may be such a universal reference. In 1960, the International Bureau of Weights and Measures (BIPM) established a new definition of the meter as the length equal to 1650763 wavelengths, in a vacuum, of the transition line between the levels $2p10$ and $5d5$ of krypton-86. Similarly the rotation of the Earth was not so stable and it was proposed in 1927 by André Danjon to use the tropical year as a reference, as adopted in 1952. In 1967, the second was also related to an atomic transition, defined as the duration of 9 162 631 770 periods of the transition between the two hyperfine levels of the ground state of caesium-133. To finish, it was decided in 1983, that the meter shall be defined by fixing the value of the speed of light to c = 299 792 458 m s^–1 and we refer to [55 ] for an up to date description of the SI system. Today, the possibility to redefine the kilogram in terms of a fixed value of the Planck constant is under investigation [279 ].

This summary illustrates that the system of units is a human product and all SI definitions are historically based on non-relativistic classical physics. The changes in the definition were driven by the will to use more stable and more fundamental quantities so that they closely follow the progress of physics. This system has been created for legal use and indeed the choice of units is not restricted to SI.

SI systems and the number of basic units.

The International System of Units defines seven basic units: the meter (m), second (s) and kilogram (kg), the Ampere (A), Kelvin (K), mole (mol) and candela (cd), from which one defines secondary units. While needed for pragmatic reasons, this system of units is unnecessarily complicated from the point of view of theoretical physics. In particular, the Kelvin, mole and candela are derived from the four other units since temperature is actually a measure of energy, the candela is expressed in terms of energy flux so that both can be expressed in mechanical units of length [L], mass [M] and time [T]. The mole is merely a unit denoting numbers of particles and has no dimension. The status of the Ampere is interesting in itself. The discovery of the electric charge [Q] led to the introduction of a new units, the Coulomb. The Coulomb law describes the force between two charges as being proportional to the product of the two charges and to the inverse of the distance squared. The dimension of the force being known as [MLT^–2], this requires the introduction of a new constant $𝜀0$ (which is only a conversion factor), with dimensions [Q²M^–1L^–3T²] in the Coulomb law, and that needs to be measured. Another route could have been followed since the Coulomb law tells us that no new constant is actually needed if one uses [M^1/2L^3/2T^–1] as the dimension of the charge. In this system of units, known as Gaussian units, the numerical value of $𝜀0$ is 1. Hence the Coulomb can be expressed in terms of the mechanical units [L], [M] and [T], and so will the Ampere. This reduces the number of conversion factors, that need to be experimentally determined, but note that both choices of units assume the validity of the Coulomb law.

Natural units.

The previous discussion tends to show that all units can be expressed in terms of the three mechanical units. It follows, as realized by Johnstone Stoney in 1874² , that these three basic units can be defined in terms of 3 independent constants. He proposed [27 , 267 ] to use three constants: the Newton constant, the velocity of light and the basic units of electricity, i.e., the electron charge, in order to define, from dimensional analysis a “natural series of physical units” defined as

∘ ------- 2 tS = --Ge---∼ 4.59 × 10−45 s, 4π 𝜀0c6 ∘ ------- --Ge2-- −36 ℓS = 4π 𝜀0c4 ∼ 1.37 × 10 m, ∘ ------- e2 mS = -------∼ 1.85 × 10−9 kg, 4π 𝜀0G

where the $𝜀 0$ factor has been included because we are using the SI definition of the electric charge. In such a system of units, by construction, the numerical value of $G$ , $e$ and $c$ is 1, i.e., $−1 c = 1 × ℓS ⋅ tS$ etc. A similar approach to the definition of the units was independently proposed by Planck [418 ] on the basis of the two constants $a$ and $b$ entering the Wien law and $G$ , which he reformulated later [419 ] in terms of $c$ , $G$ and $ℏ$ as

∘ G-ℏ- tP = ----∼ 5.39056 × 10−44 s, ∘ -c5- G ℏ ℓP = --3-∼ 1.61605 × 10−35 m, ∘ -c- ℏc- −8 mP = G ∼ 2.17671 × 10 kg.

The two systems are clearly related by the fine-structure constant since $e2∕4π 𝜀0 = αEMhc$ .

Indeed, we can construct many such systems since the choice of the 3 constants is arbitrary. For instance, we can construct a system based on ( $e,me, h)$ , that we can call the Bohr units, which will be suited to the study of the atom. The choice may be dictated by the system, which is studied (it is indeed far fetched to introduce $G$ in the construction of the units when studying atomic physics) so that the system is well adjusted in the sense that the numerical values of the computations are expected to be of order unity in these units.

Such constructions are very useful for theoretical computations but not adapted to measurement so that one needs to switch back to SI units. More important, this shows that, from a theoretical point of view, one can define the system of units from the laws of nature, which are supposed to be universal and immutable.

Do we actually need 3 natural units?

is an issue debated at length. For instance, Duff, Okun and Veneziano [165 ] respectively argue for none, three and two (see also [535 ]). Arguing for no fundamental constant leads to consider them simply as conversion parameters. Some of them are, like the Boltzmann constant, but some others play a deeper role in the sense that when a physical quantity becomes of the same order as this constant, new phenomena appear; this is the case, e.g., of $ℏ$ and $c$ , which are associated respectively to quantum and relativistic effects. Okun [392 ] considered that only three fundamental constants are necessary, as indicated by the International System of Units. In the framework of quantum field theory + general relativity, it seems that this set of three constants has to be considered and it allows to classify the physical theories (with the famous cube of physical theories). However, Veneziano [514 ] argued that in the framework of string theory one requires only two dimensionful fundamental constants, $c$ and the string length $λs$ . The use of $ℏ$ seems unnecessary since it combines with the string tension to give $λs$ . In the case of the Nambu–Goto action $∫ ∫ S ∕ℏ = (T ∕ℏ) d(Area ) ≡ λ−s 2 d(Area )$ and the Planck constant is just given by $λ− 2 s$ . In this view, $ℏ$ has not disappeared but has been promoted to the role of a UV cut-off that removes both the infinities of quantum field theory and singularities of general relativity. This situation is analogous to pure quantum gravity [388 ] where $ℏ$ and $G$ never appear separately but only in the combination $∘ ------ ℓPl = G ℏ∕c3$ so that only $c$ and $ℓPl$ are needed. Volovik [520 ] made an analogy with quantum liquids to clarify this. There an observer knows both the effective and microscopic physics so that he can judge whether the fundamental constants of the effective theory remain fundamental constants of the microscopic theory. The status of a constant depends on the considered theory (effective or microscopic) and, more interestingly, on the observer measuring them, i.e., on whether this observer belongs to the world of low-energy quasi-particles or to the microscopic world.

Fundamental parameters.

Once a set of three independent constants has been chosen as natural units, then all other constants are dimensionless quantities. The values of these combinations of constants does not depend on the way they are measured, [110 , 164 , 437 ], on the definition of the units etc.…. It follows that any variation of constants that will leave these numbers unaffected is actually just a redefinition of units. These dimensionless numbers represent, e.g., the mass ratio, relative magnitude of strength etc.…. Changing their values will indeed have an impact on the intensity of various physical phenomena, so that they encode some properties of our world. They have specific values (e.g., $αEM ∼ 1 ∕137$ , $mp ∕me ∼ 1836$ , etc.) that we may hope to understand. Are all these numbers completely contingent, or are some (why not all?) of them related by relations arising from some yet unknown and more fundamental theories. In such theories, some of these parameters may actually be dynamical quantities and, thus, vary in space and time. These are our potential varying constants.