[FairWitness]

One of the things I was taught in Chemistry is that there is no such thing as "action at a distance". In other words, chemistry happens only when atoms (or more accurately, their electron "clouds") touch each other. On the other hand, at least some of what I have heard of quantum physics (which is admittedly not a lot) indicates that that is not necessarily true for elementary particle physics. Fascinating (in my best Spock imitation).

[TopQuark]

there is no such thing as "action at a distance". This was the focus of extended philosophical and scientific debate in the second half of the XIX century(Mach and Sommerfeld were central figures, if I remember correctly). There was a notion of ether introduced --- a medium for the alleged transport of action (force) at a distance.

It has been concluded then and believed since that, as you pointed out, there is no such thing indeed: all interactions of a particle happen at the point where that particle is located. Thus, two electrons repel each other not "at a distance," but by means of an electomagnetic field: each electron creates a field which influences the other at the point where that other is located.

Quantum mechanics does not actually add anything to that picture.

[lepton]

On the other hand, at least some of what I have heard of quantum physics (which is admittedly not a lot) indicates that that is not necessarily true for elementary particle physics

One of the precepts of quantum mechanics is that "objects" don't really exist at a specific place, but could more reasonably be represented as probabilities centered around a certain area. If one cancels a probability in one place, that naturally increases the probability that the energy will exhibit itself somewhere else. This is kind of what the photon teleportation experiment appears to have done, by creating a barrier, sufficiently narrow that some small portion of the photon beam would mute itself to the other side of the barrier, and continue on its path, thereby arriving before it was expected.