Many problems in physics are perturbative. Then, approximate methods suffice to get reliable answers. In Asimov’s “relativity of wrong” it’s the perturbative

Many problems in physics are perturbative. Then, approximate methods suffice to get reliable answers. In Asimov’s “relativity of wrong” it’s the perturbative nature of gravity—-gravity is very weak—-that makes “the Earth is a sphere” less wrong than “the Earth is flat”. So, if you need an answer to only such-and-such precision, the approximation scheme lets me know how hard I must work to achieve reliability at that scale. Once I get the precision I need I can stop. Electrodynamics is like that. There is a number, the fine structure constant, about 1/137, that gives the natural scale for how big the next step in the approximation is, compared to the size of the current step. So, if I need to know the answer to 1 part in 10^9, I’m going to need to do 4 or 5 steps of fixing up the approximation (each fix, of course, being a great deal more arduous). QCD, and other “strongly coupled” or “non-perturbative” problems are not like that. If you make the dumbest approximation (flat-earth) and then fix it up a bit (sphere earth), answers don’t change just a little. They change completely. In QCD the number that characterizes the “obvious” approximation (the Feynman diagram approach)—-the number that’s 1/137 for electrodynamics—-is about 1.5. That’s a disaster! The approximation scheme is obviously no good—-you learn that you can never stop improving your approximation, because if you only worked “a little harder” your answer could change completely. Other approaches are required.