The Cluster's Fine Structure Constant

A page about the fine structure constant — the dimensionless coupling constant of electromagnetism.

The fine structure constant

The fine structure constant is: alpha = e^2 / (4 pi epsilon_0 hbar c) = 1 / 137.035999... In Gaussian units: alpha = e^2 / (hbar c). It is dimensionless, so its value is independent of the unit system. It characterizes the strength of the electromagnetic interaction between charged particles.

In the cluster, the edit fine structure constant is an edit dimensionless coupling.

The physical meaning

Alpha governs:

• The ratio of the electron's velocity to the speed of light in the ground state of hydrogen: v / c = alpha
• The ratio of the Bohr radius to the Compton wavelength: a_0 / lambda_C = 1 / alpha
• The strength of any electromagnetic vertex in QED: each vertex contributes a factor of alpha^{1/2}
• The splitting of the hydrogen fine structure: Delta E_FS / E_n ~ alpha^2

In the cluster, the edit physical meaning governs an edit ratio.

The running of alpha

In QED, alpha is not a constant — it depends on the energy scale (it "runs"): alpha(mu) = alpha(0) / (1 - (alpha(0) / (3 pi)) ln(mu^2 / m_e^2)) At the Z boson mass (mu = m_Z = 91.2 GeV): alpha(m_Z) ~ 1 / 127.9. The increase is due to vacuum polarization by virtual electron-positron pairs screening the charge.

In the cluster, the edit running of alpha gives an edit running value.

The hierarchy

Alpha is small enough that perturbation theory works well in QED:

• Each loop in a Feynman diagram adds a factor of ~ alpha / pi ~ 10^{-3}
• The electron g-2 has been calculated to 5 loops: O(alpha^5) ~ 10^{-13}
• The theoretical prediction agrees with experiment at the 10^{-13} level — the most precise agreement between theory and experiment in all of science

In the cluster, the edit hierarchy gives an edit perturbation series.

The magic number

Feynman called it a "magic number": alpha = 1 / 137.035999... There are 157 digits, so if you count from the first digit (1), 137 is the 157th digit. Feynman reportedly said: "It's one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by us."

In the cluster, the edit magic number is an edit mystery.

This constant

This page is about the fine structure constant. alpha = 1/137.036. Dimensionless. Runs from 1/137 at low energy to 1/128 at the Z mass. The constant is real.