https://inspirehep.net/files/20b84db59eace6a7f90fc38516f530e...
using integration over phase space instead of position or momentum space. Most people think you need an orthogonal basis set to do quantum mechanical calculation but it turns that "resolution of unity is all you need", that is, if you integrate |x><x| over all x you get 1. If you believe resolution of unity applies in quantum gravity, then Hawking was wrong about black hole information. In my case we were hoping we could apply the trace formula and make similar derivations to systems with unusual coordinates, such as spin systems.
There are quite a few calculations in physics that involve perturbation theory, for instance, people used to try to calculate the motion of the moon by expanding out thousands of terms that look like (112345/552) sin(32 θ-75 ϕ) and still not getting terribly good results. It turns out classic perturbation theory is pathological around popular cases such as the harmonic oscillator (frequency doesn't vary with amplitude) and celestial mechanics (the frequency to go around the sun, to get closer or further from sun, or to go above or below the plane of the plane of the ecliptic are all the same.) In quantum mechanic these are not pathological, notably perturbation theory works great for an electron going around an atom which is basically the same problem as the Earth going around the Sun.
I have a lot of skepticism about things like
https://en.wikipedia.org/wiki/Anomalous_magnetic_dipole_mome...
in high energy physics because frequently they're comparing a difficult experiment to an expansion of thousands of Feynman diagrams and between computational errors and the fact that perturbation theory often doesn't converge very well I don't get excited when they don't agree.
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Note that I used numerical calculations for "unit and integration testing", so if I derived an identity I could test that the identity was true for different inputs. As for formal systems, they only go so far. See
https://en.wikipedia.org/wiki/Principia_Mathematica#Consiste...
Interesting, and I used to think that math and sciences were invented by humans to model the world in a manner to avoid errors due to chains of fuzzy thinking. Also, formal languages allowed large buildings to be constructued on strong foundations.
From your anecdote it appears that the calculations in the paper were numerical ? but I suppose a similar argument applies to symbolic calculations.