https://www.physicsforums.com/threads/validity-of-theoretical-arguments-for-unruh-and-hawking-radiation.978501/page-3#post-6243209
https://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/
There might exist a classical field problem which resembles the renormalization problem. In the Navier-Stokes equation, a Millennium problem is to prove that solutions do not "blow up" because of turbulence.
In a realistic fluid there is a natural scale, the scale of molecules, at which the Navier-Stokes equation stops working. The blowup cannot happen. This sounds like an energy cutoff which is used to eliminate the divergence in renormalization.
The concept of an "effective theory" contains the idea that at very short distances there is new physics which provides the necessary cutoff.
If we try to model an electromagnetic field in a gravitational field, and consider the backreaction of the two fields when they interact, the renormalization problem may appear in the classical fields as a blowup problem. For example, the solution might not be stable under small perturbations.
After all, Feynman diagrams are perturbation calculations. If the perturbations diverge, then a classical solution might be unstable.
Which brings us to the old topic if general relativity has any solutions at all under realistic matter fields.
Christodoulou and Klainerman (1990) proved the "nonlinear stability" of the Minkowski metric under general relativity.
This is a very interesting question: if Feynman diagrams with gravitons diverge, how could Christodoulou and Klainerman show the stability in the (very restricted) case of the Minkowski metric?
In physics, if we are calculating with two fields, we usually ignore the backreaction. If we calculate the behavior of a laser beam which climbs out of a gravitational field, we assume that the backreaction on the gravitational field is negligible. But it might be that a precise calculation shows the the solution is not stable.
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