It seems to me that quantum mechanics can be formulated within the general mathematical framework of variational principles.
Derivation of the equations of nonrelativistic quantum mechanics based on the principle of minimum Fisher information.
https://journals.aps.org/pra/abstract/10.1103/PhysRevA.58.1775
On the other hand, it looks like that is not the case for general relativity.
The Einstein field equations cannot be derived, based on the minimum or maximum of the action integral.
https://link.springer.com/article/10.1007/BF00756194
As long ago as 1900, Hilbert formulated problem 23 (In his famous list of problems ).
https://en.m.wikipedia.org/wiki/Hilbert%27s_twenty-third_problem
You have to read the original formulation of this problem, not the Wikipedia version. The connection with this question will be clear.
Question 1. Is this the source of the difficulty in the attempt to unify quantum mechanics with general relativity?
Question 2. Is it possible to imagine some other measure of information, related to our knowledge of a physical system (analogy with Fisher information ), so that general relativity can be cast within the mathematical framework of variational principles?
