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:
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.
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.
Constant |
Symbol | Value |
Speed of light |
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299 792 458 m s–1 |
Planck constant (reduced) |
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1.054 571 628(53) × 10–34 J s |
Newton constant |
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6.674 28(67) × 10–11 m2 kg–1 s–2 |
Weak coupling constant (at |
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0.6520 ± 0.0001 |
Strong coupling constant (at |
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1.221 ± 0.022 |
Weinberg angle |
![]() ![]() |
0.23120 ± 0.00015 |
Electron Yukawa coupling |
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2.94 × 10–6 |
Muon Yukawa coupling |
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0.000607 |
Tauon Yukawa coupling |
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0.0102156 |
Up Yukawa coupling |
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0.000016 ± 0.000007 |
Down Yukawa coupling |
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0.00003 ± 0.00002 |
Charm Yukawa coupling |
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0.0072 ± 0.0006 |
Strange Yukawa coupling |
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0.0006 ± 0.0002 |
Top Yukawa coupling |
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1.002 ± 0.029 |
Bottom Yukawa coupling |
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0.026 ± 0.003 |
Quark CKM matrix angle |
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0.2243 ± 0.0016 |
|
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0.0413 ± 0.0015 |
|
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0.0037 ± 0.0005 |
Quark CKM matrix phase |
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1.05 ± 0.24 |
Higgs potential quadratic coefficient |
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? |
Higgs potential quartic coefficient |
![]() |
? |
QCD vacuum phase |
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< 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 and an energy scale of unification
through
Constant |
Symbol | Value |
Electromagnetic coupling constant |
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0.313429 ± 0.000022 |
Higgs mass |
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> 100 GeV |
Higgs vev |
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(246.7 ± 0.2) GeV |
Fermi constant |
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1.166 37(1) × 10–5 GeV–2 |
Mass of the |
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80.398 ± 0.025 GeV |
Mass of the |
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91.1876 ± 0.0021 GeV |
Fine structure constant |
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1/137.035 999 679(94) |
Fine structure constant at |
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1/(127.918 ± 0.018) |
Weak structure constant at |
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0.03383 ± 0.00001 |
Strong structure constant at |
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0.1184 ± 0.0007 |
Gravitational structure constant |
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![]() |
Electron mass |
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510.998910 ± 0.000013 keV |
Mu mass |
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105.658367 ± 0.000004 MeV |
Tau mass |
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1776.84 ± 0.17 MeV |
Up quark mass |
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(1.5 – 3.3) MeV |
Down quark mass |
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(3.5 – 6.0) MeV |
Strange quark mass |
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![]() |
Charm quark mass |
![]() |
![]() |
Bottom quark mass |
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Top quark mass |
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171.3 ± 2.3 GeV |
QCD energy scale |
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(190 – 240) MeV |
Mass of the proton |
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938.272013 ± 0.000023 MeV |
Mass of the neutron |
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939.565346 ± 0.000023 MeV |
proton-neutron mass difference |
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1.2933321 ± 0.0000004 MeV |
proton-to-electron mass ratio |
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1836.15 |
electron-to-proton mass ratio |
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1/1836.15 |
|
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(2.5 – 5.0) MeV |
|
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(0.2 – 4.5) MeV |
proton gyromagnetic factor |
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5.586 |
neutron gyromagnetic factor |
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–3.826 |
Rydberg constant |
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10 973 731.568 527(73) m–1 |
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.
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 . It can be measured
from, e.g., the triplet
transition in hydrogen, the frequency of which is related to the Rydberg
constant and other constants by assuming QED so that the accuracy of
is much lower than that of
the measurement of the transition. This could be solved by defining
as
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.
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 and
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.
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 (, 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
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.
http://www.livingreviews.org/lrr-2011-2 |
Living Rev. Relativity 14, (2011), 2
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