Fine-structure Constant - Physical Interpretations

Physical Interpretations

The fine-structure constant α has several physical interpretations. α is:

• The square of the ratio of the elementary charge to the Planck charge

• The ratio of two energies: (i) the energy needed to overcome the electrostatic repulsion between two electrons a distance of d apart, and (ii) the energy of a single photon of wavelength (from a modern perspective, of angular wavelength r = d ; see Planck relation):

• The ratio of the velocity of the electron in the Bohr model of the atom to the speed of light. Hence the square of α is the ratio between the Hartree energy (27.2 eV = twice the Rydberg energy) and the electron rest mass (511 keV).
• The ratio of three characteristic lengths: the classical electron radius, the Bohr radius and the Compton wavelength of the electron :

• In quantum electrodynamics, α is the coupling constant determining the strength of the interaction between electrons and photons. The theory does not predict its value. Therefore α must be determined experimentally. In fact, α is one of the about 20 empirical parameters in the Standard Model of particle physics, whose value is not determined within the Standard Model.
• In the electroweak theory unifying the weak interaction with electromagnetism, α is absorbed into two other coupling constants associated with the electroweak gauge fields. In this theory, the electromagnetic interaction is treated as a mixture of interactions associated with the electroweak fields. The strength of the electromagnetic interaction varies with the strength of the energy field.
• Given two hypothetical point particles each of Planck mass and elementary charge, separated by any length, α is the ratio of their electrostatic repulsive force to their gravitational attractive force.
• In the fields of electrical engineering and solid-state physics, the fine-structure constant is one fourth the product of the characteristic impedance of free space, Z₀ = 1/(c ε₀), and the conductance quantum, G₀ = 2e2/h:

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When perturbation theory is applied to quantum electrodynamics, the resulting perturbative expansions for physical results are expressed as sets of power series in α. Because α is much less than one, higher powers of α are soon unimportant, making the perturbation theory extremely practical in this case. On the other hand, the large value of the corresponding factors in quantum chromodynamics makes calculations involving the strong nuclear force extremely difficult.

According to the theory of the renormalization group, the value of the fine-structure constant (the strength of the electromagnetic interaction) grows logarithmically as the energy scale is increased. The observed value of α is associated with the energy scale of the electron mass; the electron is a lower bound for this energy scale because it (and the positron) is the lightest charged object whose quantum loops can contribute to the running. Therefore 1/137.036 is the value of the fine-structure constant at zero energy. Moreover, as the energy scale increases, the strength of the electromagnetic interaction approaches that of the other two fundamental interactions, a fact important for grand unification theories. If quantum electrodynamics were an exact theory, the fine-structure constant would actually diverge at an energy known as the Landau pole. This fact makes quantum electrodynamics inconsistent beyond the perturbative expansions.