Historical Context of the Coupling Constant - Time Travel, Quantum Entanglement and Quantum Computing

Origin of the QED Coupling Constant

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).

4. Historical Attempts to Explain $1/137$

The unusual appearance of the fine-structure constant sparked fascination and speculation among some of the
greatest physicists of the 20th century:

Arthur Eddington: Proposed a numerological approach, claiming that $\alpha^{-1}$ was exactly 137.
He sought a purely mathematical derivation of this number, linking it to cosmology and fundamental constants.
His attempts, though elegant, are not considered physically correct.
Paul Dirac: Dirac was deeply intrigued by the number 137, suspecting it pointed to a deep connection
between quantum mechanics and cosmology. He noted that certain large-number coincidences (ratios of cosmic
to microscopic quantities) involved numbers near powers of 137. He hoped future theory would derive $\alpha$
rather than taking it as input.
Richard Feynman: Famously described $\alpha$ as “one of the greatest damn mysteries of physics: a magic number
that comes to us with no understanding by man.” He emphasized that while QED uses $\alpha$ to extraordinary precision,
the theory itself does not explain why it has this value.
Modern Views: In grand unified theories (GUTs), $\alpha$ is not fundamental but emerges from a common high-energy
coupling that splits into the three Standard Model couplings as the universe cools. In string theory, coupling constants
may arise from geometric features of extra dimensions or vacuum expectation values of scalar fields (“moduli”).
Still, no unique derivation of 1/137 has been achieved.

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.
The fascination with 137 reflects a century-long quest for a deeper understanding of nature’s most mysterious number.