Quantum Field Theory

Quantum electrodynamics, the quantum field theory of electrons and light, was built in the late 1920s and at once went wrong. The electron constantly emits and reabsorbs virtual photons, and adding up their effects gave infinity for almost any correction beyond the simplest approximation. Through the 1930s many physicists concluded that the theory was fundamentally flawed.

After the war, experiments forced the issue. In 1947 Willis Lamb and Robert Retherford, using radar-era microwave techniques, found a tiny shift in hydrogen's energy levels that Dirac's theory said should not exist. The effect was real, and it came from the corrections that gave infinities. Hans Bethe estimated it on the train home from the conference where it was announced, by subtracting one infinity from another.

Between 1947 and 1949, Julian Schwinger, Richard Feynman and, independently in war-ravaged Tokyo, Sin-Itiro Tomonaga found systematic ways to do it. The infinities could all be absorbed into the electron's measured mass and charge, leaving finite corrections that could be calculated to any accuracy. Feynman's diagrams made the calculations visual and fast. Freeman Dyson showed that the three methods were the same theory. Many physicists, Dirac among them, regarded renormalisation as sweeping infinities under the rug. Phase transitions later showed what it really means: physics at one scale is insensitive to the details at much smaller scales.

In 1954 Chen-Ning Yang and Robert Mills generalised the symmetry behind electromagnetism. Their theory seemed to predict massless particles no one had seen, and it was set aside for years. It later became the framework of particle physics.

An electron is a tiny magnet. Its strength is measured by the g-factor. Dirac's equation predicts $g = 2$ exactly. Quantum field theory says the electron's cloud of virtual photons changes that slightly. The deviation, $a_e = (g - 2)/2$, is called the anomalous magnetic moment.

In 1948 Schwinger calculated the first correction. It depends only on the fine-structure constant $\alpha \approx 1/137.036$:

The measured value, as of 2023, is

Schwinger's single term is already right to about 0.15%. Physicists have since computed further terms, with more and more virtual particles, up to diagrams with five loops, more than 12,000 of them. With those included, theory and experiment agree to about one part in a trillion in $g$. The comparison is limited mainly by how precisely $\alpha$ itself is known. In fact, one of the two most precise determinations of $\alpha$ uses this very calculation. The other comes from measuring how atoms recoil when they absorb light.

It is often called the most accurate prediction in science. Schwinger had the formula $\alpha/2\pi$ engraved on his tombstone.

After Yang–Mills theory was shown in 1971 to be renormalisable, quantum field theory became the language of all particle physics, and much of condensed-matter physics too. It has one glaring gap: nobody has constructed a realistic interacting quantum field theory in four dimensions with full mathematical rigour, and the Clay Institute offers a million dollars for the first step. Gravity has resisted it entirely. General relativity cannot be renormalised in the same way, and a quantum theory of gravity is often called the deepest open problem in physics.