What does it mean for a theory to be right? In science, we measure rightness by confrontation with experiment. A theory predicts a number, experimentalists measure that number, and we compare. The closer the match, the more confidence we place in our theoretical framework.

By this standard, quantum electrodynamics—QED—stands alone in human intellectual history. Its predictions agree with measurements to more than ten significant figures. To appreciate this precision: if you measured the distance from New York to Los Angeles, QED-level accuracy would pin it down to within the width of a human hair.

This extraordinary success didn't come easily. It required decades of theoretical innovation, experimental ingenuity, and the development of entirely new mathematical techniques. Understanding why QED achieves such precision reveals deep truths about the structure of quantum field theory itself.

Every quantum field theory calculation involves an infinite number of processes. When an electron scatters off another electron, it can exchange one photon, or two, or three, or infinitely many. It can create virtual particle-antiparticle pairs that flicker in and out of existence. How do we possibly sum all these contributions?

The answer lies in a remarkable gift from nature: the fine-structure constant α ≈ 1/137. This dimensionless number measures the strength of electromagnetic interactions. Because it's small—much less than one—each additional photon exchange contributes roughly α times less than the previous one.

This hierarchy transforms an impossible infinite sum into a rapidly converging series. The leading term captures most of the physics. The next correction is roughly 1% of that. The next is roughly 1% of the correction. Each order of perturbation theory adds decimal places of precision rather than fundamentally changing the answer.

Not all quantum field theories enjoy this luxury. The strong force, described by quantum chromodynamics, has a coupling constant near unity at low energies. Perturbation theory fails catastrophically there. QED's weak coupling isn't a feature we engineered—it's a fortunate accident of our universe that makes precision calculations possible.

Takeaway

A small coupling constant transforms intractable infinities into manageable approximations, enabling theoretical predictions of unprecedented accuracy.

Feynman diagrams provide the bookkeeping system for perturbation theory. Each diagram represents a particular way particles can interact, and the rules for translating diagrams into numbers are precise and mechanical. The challenge is that the number of diagrams explodes at each order.

At first order in α, the electron magnetic moment involves one simple diagram. At second order, there are seven diagrams. At third order, seventy-two. At fourth order, the number exceeds nine hundred. At fifth order—the current frontier—tens of thousands of diagrams contribute, each requiring sophisticated computer algebra and numerical integration.

These calculations push the boundaries of human ingenuity and computational power. Some diagrams involve integrals over hundreds of dimensions. New mathematical techniques had to be invented to evaluate them. The fifth-order QED contribution to the electron magnetic moment required years of effort by multiple research groups.

Yet the investment pays off spectacularly. Each completed order of calculation extends our theoretical precision by roughly two decimal places. The current theoretical prediction includes contributions through fifth order in α, plus leading corrections from muons, tau leptons, and even hadronic physics. Every piece matters when you're hunting the eleventh significant figure.

Takeaway

Systematic improvement requires exponentially more effort at each stage, yet the payoff—additional decimal places of certainty—justifies the investment.

The electron's magnetic moment provides the ultimate testing ground for QED. Dirac's equation, written in 1928, predicted that the electron's g-factor—the ratio of its magnetic moment to its angular momentum in natural units—should be exactly 2. But Dirac's equation ignores quantum fluctuations.

In the full quantum field theory, the electron constantly emits and reabsorbs virtual photons. These quantum fluctuations slightly modify its interaction with magnetic fields, shifting g away from 2. The anomalous magnetic moment a = (g-2)/2 captures this purely quantum effect.

Schwinger calculated the leading correction in 1948: a = α/(2π) ≈ 0.00116. This was revolutionary—a precise, unambiguous prediction from quantum field theory. Modern calculations extend to fifth order and beyond, yielding a theoretical value known to twelve significant figures.

The experimental side matches this ambition. Using Penning traps, physicists confine single electrons and measure their precession frequencies with extraordinary precision. The current experimental value agrees with theory to better than one part in a trillion. This agreement represents the most precise confirmation of any theory in science. When theory and experiment match to ten decimal places, something profound is being revealed about nature's deep structure.

Takeaway

The electron g-2 measurement stands as humanity's most precise test of physical theory—a triumph demonstrating that quantum field theory genuinely captures how nature works at its foundations.

QED's success is not merely technical—it's philosophically significant. A framework invented to reconcile quantum mechanics with special relativity, refined through decades of struggle with infinities and renormalization, ultimately produces predictions accurate to one part in a trillion.

This precision suggests we've touched something true about reality's fabric. The mathematical structures of gauge symmetry and quantum fields aren't just convenient fictions. They correspond to genuine features of the physical world.

Yet QED is also incomplete. It describes electromagnetism beautifully but says nothing about gravity. Its success sets the standard that any deeper theory must meet. Whatever replaces the Standard Model must reproduce QED's extraordinary accuracy while explaining what QED cannot.