Is the speed of light constant?

The theory of relativity predicts that the speed of light in empty space is constant. Physical experiments, such as the famous Michelson–Morley experiment, confirm this. However, consider a theoretical experiment such as the double-slit experiment, built in such a way that there is a difference in distance between the two possible paths. According to Richard Feynman’s path integral formulation, or sum-over-paths, a quantum of light (a photon) can’t be seen as passing through one of the slits or the other, but both of them at once. Therefore, it will be considered as passing through two different distances at the same time, inevitably leading to a conclusion that the speed of light is not constant. This is what I call the paradox of the speed of light.

Why haven’t we seen this in experiments? I think because of the nature of light and the way we perceive it, its speed is always perceived as very close to a constant value, due to the probabilistic nature of waves in general, and the electromagnetic wave in particular. One of the reasons is because we always measure the speed of light for long distances, while what I describe is relevant to the tiny distances related to quantum mechanics. At the macroscopic level, indeed light seems to travel at a constant speed and also in more or less straight lines – according to light’s wave length and general relativity. But at the microscopic level (the quantum level) there is no constant speed. The speed of light, in this case, varies according to probability.

This means that spacetime itself might not be a constant thing at the microscopic level. I compare it to the surface of water in the sea. If electromagnetic waves can be compared to waves of sea water, then the spacetime itself can be compared to the water itself. The surface of water is never flat, when looking from close enough. But from a distance it looks flat enough (or actually a sphere around earth).

Does it mean a material object can reach or pass beyond the limit of speed of light? I think it does. If a particle can reach a speed close to the speed of light, if given enough energy, and when considering the wave features of particles and light – it appears that a particle with high energy is able to reach or pass the speed of light, under some circumstances. This might mean a particle can go back in time. For a massive body with many particles I think the chances are very low – almost zero – to do that. But for a tiny sub-atom particle I think it is possible, and the probability of it reaching the speed of light is high enough to actually allow this to happen once in a while.