Thread by @mcnees

Freeman Dyson submitted a lovely little two-page paper to Physical Review #OTD in 1951, demonstrating that perturbation theory in quantum electrodynamics produces a divergent series. It's one of my favorites, an absolute classic of the field.
journals.aps.org/pr/abstract/10…

In QED we calculate physical quantities perturbatively, giving a series with increasing powers of a small number α ~ 1/137. So if we calculate the anomalous magnetic moment of the electron (classically it should be g=2) we get a series like:
g = 2 + (1/π) α + (0.656/π²) α² + …

This tells us that the actual magnetic moment of the electron is a little different than what we'd expect from classical considerations. The series above starts with the classical bit (g=2) and then all the subsequent terms represent various quantum mechanical effects.

Dyson pointed out a basic mathematical problem with this set up.

The problem is that if these series expansions for physical quantities converged for some small value of α (like 1/137), then they would have to be analytic functions of α around α = 0. That means these series would also have to converge for small *negative* values of α.

But physically, a theory with negative α would be pathological. You would have like charges (both positive or both negative) attracting instead of repelling, and all sorts of weirdness. Dyson describes "an explosive disintegration of the vacuum by spontaneous polarization."

Such a theory would not give a well-defined analytic function of α for physical quantities. Therefore, the radius of convergence for the perturbative series obtained from QED must be zero. For any non-zero value of α we must be calculating a _divergent_ series.

The first several terms in a divergent series may appear to converge towards a particular result. In QED, they agree with experiment to amazing precision: state-of-the-art calculation reproduce experimental results to 12 or 13 decimal places.

But eventually, the coefficients in front of subsequent powers of α in the series will start to grow. Slowly at first, but then fast enough to overwhelm even very large powers of (1/137). At that point the series veers off and runs away toward infinity.

Luckily, this isn't a practical concern. Dyson estimated that it happens somewhere around the term in the series proportional to α^{137}. That's so far beyond anything we could ever calculate that it has no impact on any prediction made by QED.

But why does it happen? In Feynman's approach to QED, the number of diagrams needed to calculate the α^n term in the series grows rapidly with n. Their sum eventually overwhelms even large powers of α!
Image: K.K. Gan, 3rd order contributions to electron anomalous magnetic moment

As @rerutled points out, this two-page paper submitted to PRD *70 years ago* is still behind a paywall. A google search turns up other sources, but it boggles the mind that this isn't freely available.