Fundamental Constants

• Intuitive
• Concrete
• Abstract

Intuitive

The laws of physics contain a handful of numbers that the laws themselves do not explain: the speed of light, the strength of gravity, the charge of the electron, the masses of the elementary particles. We measure them and put them into the equations. These are the fundamental constants.

Some of them carry units, like the speed of light in metres per second. Their numerical values only tell us about our choice of units. A metre and a second were defined by human convention, so the number 299 792 458 says more about the metre and the second than about light. The speed of light itself acts as a conversion factor between space and time, and Planck's constant acts as a conversion factor between energy and frequency.

Other constants are pure numbers without units. The best known is the fine-structure constant, about one divided by 137, which sets the strength of electromagnetic interactions. Another is the ratio of the proton mass to the electron mass, about 1836. Every observer using any system of units finds the same values for these. With different values, atoms and stars would be built differently. Why they take the values they do is an open question.

Concrete

Defined constants in the SI. Since 2019 the SI fixes the units by assigning exact numerical values to several constants:

• speed of light: $c = 299\,792\,458\ \mathrm{m\,s^{-1}}$ (exact since 1983, when the metre was defined through it),
• Planck constant: $h = 6.626\,070\,15\times 10^{-34}\ \mathrm{J\,s}$, with $\hbar = h/2\pi$,
• elementary charge: $e = 1.602\,176\,634\times 10^{-19}\ \mathrm{C}$,
• Boltzmann constant: $k_B = 1.380\,649\times 10^{-23}\ \mathrm{J\,K^{-1}}$,
• Avogadro constant: $N_A = 6.022\,140\,76\times 10^{23}\ \mathrm{mol^{-1}}$.

These values carry no uncertainty because they define the metre, kilogram, ampere, kelvin and mole. Other constants are measured. Newton's constant $G \approx 6.674\times 10^{-11}\ \mathrm{m^3\,kg^{-1}\,s^{-2}}$ is the least precisely known of the commonly used ones, with a relative uncertainty of order $10^{-5}$.

Dimensionless constants. Combining constants so that the units cancel gives numbers that do not depend on any unit convention. The fine-structure constant is

$$\alpha = \frac{e^2}{4\pi\varepsilon_0\hbar c} \approx \frac{1}{137.036} .$$

It sets the scale of atomic physics. For hydrogen, the Bohr radius and the binding energy of the ground state are

$$a_0 = \frac{\hbar}{m_e c\,\alpha} \approx 5.29\times 10^{-11}\ \mathrm{m}, \qquad E_1 = -\frac{1}{2}\alpha^2 m_e c^2 \approx -13.6\ \mathrm{eV},$$

so atoms are larger than the electron's Compton wavelength $\hbar/(m_e c)$ by a factor $1/\alpha$, and atomic binding energies are smaller than $m_e c^2 \approx 511\ \mathrm{keV}$ by a factor of order $\alpha^2$.

Planck units. From $\hbar$, $c$ and $G$ one can form a length, a time and a mass:

$$\ell_P = \sqrt{\frac{\hbar G}{c^3}} \approx 1.6\times 10^{-35}\ \mathrm{m}, \qquad t_P = \frac{\ell_P}{c} \approx 5.4\times 10^{-44}\ \mathrm{s}, \qquad m_P = \sqrt{\frac{\hbar c}{G}} \approx 2.2\times 10^{-8}\ \mathrm{kg}.$$

They mark the scale at which quantum effects of gravity are expected to become important. Measured in Planck units, the masses of elementary particles become dimensionless numbers, for example $m_e/m_P \approx 4\times 10^{-23}$.

Parameters of the Standard Model. In the Standard Model the free parameters include the three gauge couplings, the quark and charged lepton masses, the three quark mixing angles and the CP phase, the Higgs mass and vacuum expectation value, and the QCD vacuum angle $\theta$. Without neutrino masses the usual count is 19. Neutrino masses and mixing add further parameters.

* Constants of Nature: The Royal Road to Fundamental Physics by Dr. Alexander Unzicker, PhD

Abstract

Trialogue on the number of fundamental constants by M. J. Duff, L. B. Okun, G. Veneziano:

This paper consists of three separate articles on the number of fundamental dimensionful constants in physics. We started our debate in summer 1992 on the terrace of the famous CERN cafeteria. In the summer of 2001 we returned to the subject to find that our views still diverged and decided to explain our current positions. LBO develops the traditional approach with three constants, GV argues in favor of at most two (within superstring theory), while MJD advocates zero.

Leads to: Planck Length.