Dimensionless Physical Constant - Introduction

Introduction

The numerical values of dimensional physical constants are dependent on the units used to express these physical constants. As such it is possible to define a basis set of units so that selected dimensional physical constants are normalized to 1 solely because of the choice of units. The basis set may consist of time, length, mass, charge, and temperature, or an equivalent set. A choice of units is called a system of units.

For example, the SI, the international system of units, is such a system of units solely defined as convenient to human use and the numerical values of dimensional physical constants have no natural significance, only in a manner that relates to the human experience. As another example, a system of natural units called Planck units are defined so that the numerical values of the speed of light (in a vacuum), the universal gravitational constant, and the constants of Planck, Coulomb, and Boltzmann, are all set to 1. Because, merely from the choice of units, these five dimensional physical constants disappear from equations of physical law, they are considered not fundamental in an operationally distinguishable sense.

In contrast, the numerical values of dimensionless physical constants are independent of the units used. These constants cannot be eliminated by any choice of a system of units. Such constants include:

• α, the fine structure constant, the coupling constant for the electromagnetic interaction (≈1/137.036). Also the square of the electron charge, expressed in Planck units. This defines the scale of charge of elementary particles with charge.
• μ or β, the proton-to-electron mass ratio, the rest mass of the proton divided by that of the electron (≈1836.15). More generally, the rest masses of all elementary particles relative to that of the electron.
• α_s, the coupling constant for the strong force (≈1)
• α_G, the gravitational coupling constant (≈10−38) which is the square of the electron mass, expressed in Planck units. This defines the scale of the mass of elementary particles.

At the present time, the values of the dimensionless physical constants cannot be calculated; they are determined only by physical measurement. This is one of the unsolved problems of physics.

The best known of the dimensionless constants is the fine structure constant:

where e is the elementary charge, ħ is the reduced Planck's constant, c is the speed of light in a vacuum, and ε₀ is the permittivity of free space. The fine structure constant is fixed to the strength of the electromagnetic force. Note that at low energies, α ≈ 1/137, whereas at the scale of the Z boson, about 90 GeV, one measures α ≈ 1/127. There is no accepted theory explaining the value of α.

The analog of the fine structure constant for gravitation is the gravitational coupling constant. This constant requires the arbitrary choice of a pair of objects having mass. The electron and proton are natural choices because they are stable, and their properties are well measured and well understood. If α_G is calculated from two protons, its value is ≈10−38.

The list of dimensionless physical constants increases in length whenever experiments measure new relationships between physical phenomena. The list of fundamental dimensionless constants, however, decreases when advances in physics show how some previously known constant can be computed in terms of others. A long-sought goal of theoretical physics is to find first principles from which all of the fundamental dimensionless constants can be calculated and compared to the measured values. A successful "Theory of Everything" would allow such a calculation, but so far, this goal has remained elusive.