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I've seen and read that calculations before Feynman diagrams involved solving differential equations, but I can't find sample calculations like these. I've read that they would take months and pages to do, so I'm curious what the equations were that were being solved. Does anyone know an old textbook or papers where these are shown?

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Original renormalization papers that came out in 1947-48 by Bethe, Lewis, Schwinger and Tomonaga did not use Feynman diagrams, not even Feynman's own 1948 paper did. Here is Cao-Schweber's description and comparison of their approaches:

"Neither Lewis nor Schwinger nor Tomonaga made explicit use of a cutoff. They directly identified the divergent terms with corrections to the mass and the charge, and removed them from the expressions for real processes by redefining the masses and charges. By contrast, Feynman's efficient calculational algorithm... is based on the explicit use of a relativistic cutoff. The latter consists of a set of rules for regularization, which makes it possible to calculate physical quantities in a relativistically and gauge invariant manner, but still results in divergent expressions in the limit as the cutoff mass goes to infinity. With a finite cutoff, this artifice transforms essentially purely formal manipulations of divergent quantities, i.e., the redefinition of parameters, into quasi respectable mathematical operations. If, after the redefinition of mass and charge, other processes are insensitive to the value of the cutoff, then a renormalized theory can be defined by letting the cutoff go to infinity."

Some of these papers are available online, Bethe's is pretty short.

Bethe, H. A.: 1947, The Electromagnetic Shift of Energy Levels, Physical Review 72, 339-341.

Feynman, R. P.: 1948, Relativistic Cut-off for Quantum Electrodynamics, Physical Review 74, 1430-38.

Schwinger, J.: 1948 Quantum Electrodynamics. I. A Covariant Formulation', Physical Review 74, 1439-61.

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