Once coupled, life becomes complicated
Once the electron field and the electromagnetic field are coupled together, the resulting path integral can no longer be evaluated. The action for quantum electrodynamics S QED (ψ, A) consists of two terms, S Maxwell (A) and S Dirac (ψ), both quadratic, plus the interaction term S coupling (ψ, A), which unfortunately, but by its very nature, is cubic in the fields. This cubic term renders the theory impossible to solve analytically, as discussed in chapter IV.2.The way forward after World War II was driven by Richard Feynman, Julian Schwinger, and Shin'ichiro Tomonaga, who later shared a Nobel prize for their work on quantum electrodynamics. It's worth noting that Feynman and Schwinger were both 21 in 1939 (Schwinger was older by three months, if you want to know), while Tomonaga was 12 years older. Each had his own approach to quantum field theory. Since I have never met Tomonaga and since I was educated by Schwinger and Feynman, directly and indirectly, I naturally tend to describe the approaches I know best.Undeniably, over the intervening decades, Feynman's diagrammatic method has become by far the most popular, because it is so easy to learn. The reader, however, should not get the impression, often propagated in popular media, that it is the only possible approach. How else could Tomonaga and Schwinger accomplish their great work?During the war, Feynman worked on the atomic bomb, while Schwinger worked on radar. They both lamented later that the war had robbed them of potentially the most creative periods of their lives.
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