Yang-Mills Exists: What Does That Mean, and What Is It Good For?

Last week, OpenAI announced they tackled a longstanding problem in mathematical physics: the existence of quantum Yang-Mills theory. Since one of this blog’s first posts was an explainer about what Yang-Mills theory is, I figured I should probably comment. In 1954, Yang and Mills wrote down a type of theory that would turn out to describe three of the four fundamental force of nature. Electromagnetism uses the simplest version of the theory, while the two nuclear forces are more complicated. Gravity is the only force that isn’t described by Yang-Mills theory. But while Yang and Mills wrote down a theory that any physicist would be happy with, that’s not enough for mathematicians. Physicists take shortcuts and make assumptions in order to get a theory they can readily use, and for quantum theories those shortcuts and assumptions can paper over a lot of subtleties. Normally, that doesn’t matter much: when physicists want to use Yang-Mills theories to understand the real world, they use…

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