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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.

I want to know which theory can predict to the best possible extent whether a photon will go through the upper slit or the lower slit and where it will end up on the screen.

Out of all these theories (special and general relativity, quantum mechanics, quantum field theory, string theory, quantum gravity, M-theory etc) which can predict this to best possible extent?

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    $\begingroup$ Do you know what superposition is? $\endgroup$ – user36790 Jul 7 '15 at 8:59
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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.

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Photons are elementary particles and as such obey the laws of quantum mechanics. Quantum mechanics is the underlying framework of nature, at the microscopic level ; classical mechanics, classical electrodynamics are macroscopic theories that emerge from the underlying quantum mechanical frame.

In contrast to classical mechanics, where the trajectories of particles can be calculated as functions of (x,y,z,t) quantum mechanical solutions give only the probability of finding the particle at (x,y) in this case the slits. The behavior of photons impinging on two slits is described well by quantum mechanics .

The probability functions are the complex conjugate square, $\Psi\cdot \Psi^*$, of the quantum mechanical solution, $\Psi$, of the system "two slits and their fields a photon impinging". These solutions are sinusoidal, that is why $\Psi$ is called a wave function , because quantum mechanical equations are wave equations. Because of this functional behavior interference patterns appear even when the photon impinges one by one. This is very clear with the simple experiment of single electrons through double slit, where the slow build up of the interference in the accumulated data is seen (a measurement of the probability function).

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