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Your question touches upon the characteristic features and controversies of quantum mechanics. You want to know whether any theory can predict or explain which slit a photon passed through in a double-slit experiment.

With a few caveats, the answer is that there is no such theory. Relativity, quantum field theory, string theory etc say nothing about the puzzles in quantum mechanics. In quantum mechanics, it makes no sense to speak of the behaviour of a system in between your observations. In that time, it won't have definite values for observable quantities and photons etc won't follow definite paths, but superposition of all possible paths.

In other words, when you write

In Young's double-slit experiment, we know that a photon goes through either one of the slits but we don't know which one, and it ends up on a screen.

you need to be careful. If we didn't perform the measurement, all we know is that the photon was in a superposition of all possible paths, some going through the first slit and some going through the second slit. The classical intuition that the photon must have gone through one of the two slits and not the other is incorrect.

Now, of course, many over the years objected to this situation, and attempted to construct so-called "hidden-variable" theories, in which a system had predictable behaviour, including which path in a double-slit experiment. As it turned out, though, there are strong constraints on such a theory (e.g. Bell's inequalities ) - the fact is that experiments demonstrate quantum mechanical rather than classical behaviour.

It seems quite unlikely that any theory in the future could be constructed that agrees with our observations and predicts/explains which path a photon travelled through in a double-slit experiment. The interference fringes on the screen result from the fact that the particles don't travel through a definite slit.