Is the quantum concept of a photon still poorly understood?

[PeterDonis]

I don't see the difference in these statements

Here it is:

"Photons cannot be described by a wave function, they don't even have a position operator, you cannot apply the 1st-quantization formalism to photons."

These are all true statements about the mathematical formulation of quantum theory for photons, which is a quantum field theory. None of the statements you referenced in the textbook contradict any of these statements.

"Quantum fields can describe matter particles, but it can't describe a photon."

This is a claim that there is no quantum field theory for photons, which is manifestly false; see above. None of the statements you referenced in the textbook are making this claim.

 

[PeterDonis]

I don't see the difference in these statements

"Photons cannot be described by a wave function, they don't even have a position operator, you cannot apply the 1st-quantization formalism to photons."

"Quantum fields can describe matter particles, but it can't describe a photon."

If you think a quantum field theory means that there would have to be a wave function or a position operator or a 1st quantization formalism for photons, you need to go spend some time learning quantum field theory. None of those things are required for a quantum field theory.

 

[vanhees71]

No, it actually describes light as a probability wave, that's why I said it. Just 8 pages past the one with the quote from OP. Then describes experiments that explain this wave-particle view.

While describing all the different versions of experiments they do give the caveat:
"Bear in mind that the only thing we can know about photons is when light
interacts with matter—we have no way of detecting them without an interaction
with matter, such as with a detector or a screen."

This is of course correct, and it's the modern view (by the way that was also Planck's view in contradistinction to Einstein, but of course before 1926 there was no idea of field quantization in the modern sense).

Which is still true to this day. So quantum fields can describe matter particles, but it can't describe a photon.
I've described photons as a whole photon or no photon... to me that is also quantization and it doesn't really matter if you think of it as a field, particle, or wave. Mapping continuous infinite values to a smaller set of discrete finite values. Is this wrong, do I have to unlearn this?

Quantum fields perfectly describe both "matter particles" and "photons". QED, together with the rest of the Standard Model, is among the best theories ever discovered. It makes predictions for some quantities (like the anomalous magnetic moment of electrons and most probably also muons or the Lamb shift of the hydrogen-atom energy levels, etc.), which are accurate to more than 12 significant digits.