Dear Arjen,
Your quote '.. impossible, to explain in any classical way (..) quantum mechanics' is exactly to the point. Quantummechanics indeed is impossible for classical physics to explain as it is a causal, religious kind of physics, the kind which creates artificial problems which never can be solved as they are brewed out of unfounded assumptions and an outdated notion of reality. As long as we cling to the idea of a particle as some tiny pellet acting as the source, the cause of its wavelike behaviour, we will never comprehend quantummechanics. If particles create and preserve each other by alternately borrowing and lending each other the energy to exist and by this continuous exchange at the same time express their properties, then they are as much the product as the source of their interactions, so there is no need for a pebble at the mass center of the particle to act as a source of gravity. (For the mechanism how particles keep existing even outside the particle cluster they forged each other in, see my reply dd 17 oct to your comment on my essay). To me there is no pebble at its center which can act as a source, no interaction-independent something. Though the De Broglie waves then must refer to the energy exchange between particles and their environment and can guide them, these waves have no pellet-like source at their center. If what we call a particle is an oscillating piece of spacetime, a pure wave phenomenon, it can, of course, behave like a tangible, pellet-like thing: as time passes slower where the field is stronger, nearer its mass center, it offers an increasing opposition to a probe trying to penetrate the field. So if with very tiny fingers we would try to pinch the particle/field, we would feel spacetime become more viscous, more frozen in time, offering a stronger opposition as we increase the pressure, so it would feel like a tiny pea. The harder we pinch, the nearer to its mass center we get, the more violent its energy exchange is expressed as a counterforce to our pinching.
As there's nothing special to find at its mass center and these pilot waves are involved in the continuous energy exchange with its environment to keep existing, and even are part of what we call a particle, this two-way traffic is subject to manipulation. As the setup of the screen with the two slits and the machine gun in the experiment affects the directions of the energy exchange of the bullets (electrons) with their environment, interference effects are to be expected as this exchange is instantaneous, so in this view there's nothing in quantummechanics to get excited about. If the indefiniteness in the position of an electron depends on the distance it is observed from, by an observing particle exchanging energy with the electron, then to a distant observer the indefiniteness in the position of the electron is so large that it contains the experimental setup, so the electron is already partly behind the screen with the slits, the position of the particles of the screen and the slit edges being similarly indefinite. Hence Schrodingers cat, hence the entanglement of particles.
What is surprising is that so long after the discovery of the quantum nature of our world we still cannot let go of the antique particle idea which implicitly refers to objects having an absolute kind of reality, an existence outside interactions, as if they are created by some creator outside the universe and thus exist even without interacting to their creator. By clinging to classical, causal physics, by thinking about particles as pellets, we require something of them which they simply don't posess: an autonomous, interaction-independent existence, an individual nature of their own. This is why particles of the same kind are identical, so they the nature they have they owe to, maintain by mutually exchanging energy, their exchange with other kinds of particles proceeding in a different way -the how being the main question in particle physics. The pellet idea is related to the similarly naïve idea of spacetime as something which comes for free, as if, though deformable, it doesn't have properties of its own and doesn't need any work to keep existing. Though we're aware that spacetime is filled with virtual particles, spacetime to us is still too obvious a quantity to be able to concede that it is the product as well as the source of all particles that inhabit it, so we still tend think about spacetime as something which exists only to accommodate particles.