This question is based on my rather shaky grasp of QFT, so if I'm missing a key concept then just let me know!
If you're deriving the Schrodinger equation from the path integral as Feynman did, then the obvious choice for the factor to associate with each path is a complex phase (as Dirac originally said) -- otherwise, different paths can't interfere in a meaningful way to explain e.g. the double-slit experiment.
But what about when you formulate QFT as a path integral? Now, my knowledge here is pretty vague, but this is how I understand it: in the case of QED for example, the paths themselves are Dirac spinor fields, and the Lagrangian ensures that the path of stationary action is something like the solution to the Dirac equation (depending on boundary conditions and accounting for the Maxwell term) -- which is a solution that already exhibits interference within itself. So as long as QED ensures that the system statistically tends toward the path of stationary action, then, at least qualitatively, interference will be present.
So then we have the statistical law; once again, this says that you integrate all paths with the phase factor $e^{iS/\hbar}$ for each. This introduces an additional layer of interference -- interference between the various field configurations. And it is this interference that makes QFT seem so hopelessly distant from classical comprehension.
However, it looks tantalizingly close: it's only a Wick rotation away from being a classical statistical theory. The problem is that absorbing the $i$ into time doesn't get rid of the mathematical interference, so there's still no recourse for the realist but to say the system adopts all configurations simultaneously.
But the fact that the system ends up in only one configuration makes it very tempting to believe that it really only adopted one along the way! So the question is, can the $e^{iS/\hbar}$ factor be replaced by a positive real number, representing the true, independent probability of that path alone...and still come up with the right overall transition probabilities?
I guess you could just throw in some random function of action and tweak it until you get the numbers to match; the real question is whether you can find a replacement function with a simple form and from which we could understand why some paths are more likely than others, via a deeper underlying principle.
With $e^{iS/\hbar}$, the main contribution is from the neighborhood of the stationary action, and it "decays" into rapid oscillation further away. So the real function would likewise be a decaying one, such as $e^{-S/\hbar}$ to give a simple example.
And if it had a form like that, perhaps the underlying theory would be a true stationary action principle after all, and the reason that apparently non-stationary paths of the system have non-zero probabilities is that they can actually be stationary depending on the sensitive microscopic details of the boundary conditions, which vary from one experiment to the next.
So I'm just wondering to what extent this idea has been investigated and whether any progress has been made.
