The Coupling Constant in QED

The QED coupling constant is one of the most famous “mystery numbers” in physics, better known as the
fine-structure constant, usually denoted:

$\alpha = \frac{e^2}{4 \pi \epsilon_0 \hbar c} \approx \frac{1}{137.035999...}.$

1. What is the coupling constant in QED?

In quantum electrodynamics (QED), the coupling constant is the strength with which charged particles
(like electrons) interact with the electromagnetic field (photons).
Mathematically, the electron charge $e$ enters the QED Lagrangian as the coefficient in the interaction term:

$\mathcal{L}_{\text{int}} = -e \, \bar{\psi}\gamma^\mu A_\mu \psi,$

where $\psi$ is the electron field and $A_\mu$ is the photon field.
In natural units ( $\hbar = c = 1$ ), the dimensionless form of the coupling is exactly
$\alpha = \frac{e^2}{4\pi}$ .

So, the origin of the QED coupling constant is: it’s the coefficient that sets the interaction strength
between the fundamental electron field and the photon field in the theory.

2. Why is it ~1/137?

This is the deeper mystery:

Experimental input: QED does not predict the value of $\alpha$ . Instead, it must be measured in experiments (atomic spectroscopy, electron g-2, quantum Hall effect).
Running with energy: $\alpha$ is not really constant. It “runs” with energy scale due to vacuum polarization. At low energies, $\alpha \approx 1/137$ . At the Z boson scale ( $\sim 90 \,\text{GeV}$ ), it increases to about $1/128$ .
Attempts at explanation: Many physicists (Dirac, Eddington) wondered whether $1/137$ has a deeper mathematical or cosmological origin. The Standard Model does not explain it; in GUT or string theory, coupling constants may emerge from vacuum expectation values of fields or geometry of extra dimensions.
Anthropic speculation: If $\alpha$ were very different, chemistry and stable matter might not exist. Its value could be constrained by conditions necessary for life.

3. Why that number matters

$\alpha$ controls atomic structure (fine splitting in hydrogen, hence the name).
It governs scattering probabilities in particle physics.
Its smallness explains why QED perturbation theory converges so well (each higher order suppressed by ~1/137).

Summary

The QED coupling constant originates from the coefficient of the electron–photon interaction in the QED Lagrangian.
Its numerical value ( $\alpha \approx 1/137$ ) is not derived from deeper principles in the Standard Model;
it is a fundamental constant determined by experiment. Why it has this particular value is one of the biggest unsolved
questions in physics — possibly to be explained only by a deeper unification theory or anthropic reasoning.