What is really going on when two natural magnets repel each other? I was listening to Frank Wilczek talking about how the electromagnetic force is actually a field, which is mediated by photons. What is our current/deepest understanding of what is happening when two natural magnets repel or attract each other?
Are they exchanging photons? Is the photon field "bending", similar to how the gravitational field bends? If photons are exchanged, can a detector see these photons?
Also what field of physics is the "real" theory of this kind of magnetism covered in? For instance, what textbook would I pick up to read about our current level of understanding of how magnets work, not the emergent explanation in electrodynamics?
 A: No, there are no photons being exchanged this way in such a case. What you are describing is photon exchange in the quantum electromagnetic force, which is not similar to how a large macroscopic magnet works. Instead, that is a purely quantum mechanical process that is described by the framework of quantum field theory. This can be explained in any introductory book on quantum field theory.
To understand the mechanism for such large magnets, and why they attract or repel each other, it is better to try to understand it from classical electromagnetic theory.
In this case, we imagine magnets to be composed of many tiny current loops made from electrons circling$^1$ in atoms. These loops are magnetic dipoles that each produce their own magnetic field.
The currents loops (magnetic moments) each add up to produce a total magnetic moment which results in the permanent magnet
(this is explained thoroughly in any text on classical electrodynamics).
The end result is the classical magnetic force you observe when magnets are brought close to each other.
$^1$ This is a simplification, and how electrons "circle" is more complicated, and in fact electrons form probability clouds in atoms.
A: I have been asking similar questions here for a while, relating to both electrostatic and magnetostatic fields. The short answer appears to be that there is a gap in the Standard Model. Quantum field theory postulates a zero-point magnetic field and simply plugs in Maxwell's classical equations to describe an actual magnetic field as a disturbance of that zero-point.
For electromagnetic radiation, an electron emits a real photon which carries away energy and momentum. But for magnetostatic and electrostatic fields which essentially just sit there exerting forces on things, there is only an appeal to the classical laws of Farady and co. (while quite possibly accompanied by a lecture that those same "forces" are classical things which do not exist at the quantum scale and are therefore a snare and a delusion!). Some people do talk of "virtual photons", but others say that makes no sense - without offering anything more quantised than the good Mr. Faraday.
Devices such as the superconducting quantum interference device (SQUID) are sensitive enough to pick up the stepwise changes in magnetic field strength as individual field quanta come and go, so it seems there is a quantum model to be had.
Experimentally there are no issues, but in terms of understanding there is just a big gap between those last two paragraphs.
A: We have that electromagnetic field is thought of as acting as mediator of the electromagnetic interaction.
To that end the electromagnetic field is thought of as capable of carrying momentum.
In terms of any theory that describes interaction in terms of that interaction being mediated by a field:
Particles are not interacting directly with each other, they are all interacting with the local field.
In that way it is possible to account for the fact that when there is a change of the source of the field there is no immediate consequence for distant particles. The change propagates away from the source at a finite velocity.
Now let's take the case of an antenna that is used to broadcast a radio signal. The electromagnetic field around the antenna is changed continuously, and that change profile propagates away from the antenna.
As we know, electromagnetic waves are not dependent on the source, in the sense that once emitted electromagnetic waves continue to propagate after the source has stopped emitting.

Photons
Reception of electromagnetic energy can be set up in such a way that the energy is absorbed in discrete units. One example of a device that is designed to be triggered one discrete unit at a time is a photomultiplier
Of course, in the case of electromagnetic waves these discrete units are referred to as 'photons'.

The thing is: a measured photon is a vanished photon. When a photomultiplier is triggered the unit of energy is absorbed.
Now we go back to the electromagetic field as mediator of electromagenetic interaction.
It is possible to describe the interaction in terms of interactions with virtual photons. It is referred to as a virtual photon because any photon that is measured is not available to act as mediator of electromagnetic interaction.

In my opinion it is a dodgy thing to say: "So-and-so is happening, but it is inherently impossible to measure it." If it is inherently outside the scope of measurement then it is better to not include it in the theory. In general, include in the theory only the things that are necessary in order to make the theory work.
In my opinion the concept of 'photons' becomes necessary when you get ever closer to definite detection, such as with a photomultiplier.
The context of the electromagnetic interaction is very field-like; nothing is definite yet.
So my recommendation is to not think in terms of interactions with virtual photons, and to concentrate on thinking in terms of interaction with a field.

Finally, you may be interested in the note titled Magnetism, Radiation, and Relativity by Daniel Schroeder. Daniel Schroeder points out that it is possible to describe magnetism as a relativistic manifestation of the Coulomb force (the electrostatic force).
