How do you connect classical electromag, the photon, the wavefunction, and QFT to form a picture of light?

Question

Classically, if you wiggle an electron in a sinusoidal pattern up and down, you get a smooth electromagnetic wave that propagates, kinda like when you wiggle a jump rope. Do this fast enough and you can get visible light.

However, then there's the Nobel-prize-winning idea that light comes in packets called photons. But how "big" is a photon? Wavelengths span over space. radio waves can have wavelengths a meter long; is a radio photon a meter large? Classical electromagnetics will tell you that resolving power depends on wavelength because larger wavenlengths diffract more. However, if photons are point like, why does the wavelength matter? If I send a photon towards an electron, it should hit it no matter the wavelength right? All that should change is how much momentum it imparts via $E=hf$

Furthermore, if we go back to our wiggling electron, there is no longer an electric and magnetic field, no "jump rope" that spans space. Instead it's all the exchange of "virtual" photons. But what determines where these photons are at any given point? What determines the strength and shape of the electric and magnetic fields? Is the amplitude of the fields equal to the number of photons at that point?

Moving on we get the wavefunction. So light is a photon, and any waviness comes from the fact that the probability distribution of a photon is a wave equation? I.e. is this the reason why photons interfere or diffract?

Finally there's QED/QFT which says there's the electromagnetic field and photons are just quantized oscillations. What causes the quantization? If I continuously wiggle that electron from before, shouldn't the field oscillate continuously? Where did the individual photons go? And what has become of the wavefunction? Is the photon no longer probabilistic? Is the waviness of light now due to the fact that it's a field?

There's so many different phenomena and each one is explained by whatever the most convenient model is, but I have no idea how these models connect together. I know QFT is the most accurate and that the other models are just approximations, but I'm very unclear how one model turns into another.

Contrast that with General Relativity vs Newtonian gravity. I know that gravity is really caused by acceleration via curved spacetime; we go through spacetime at the speed of light but the curvature causes us to accelerate. When you switch between non-inertial reference frames, accelerations look like forces, so gravity is just the force we in our reference frame call this acceleration. I don't know the math of GR in the slightest, but I buy that it's possible to do the calculations and see that in very simple cases you get that the magnitude this force is approximately $Gm_1m_2(1/r^2)$. I understand how the true underlying theory and the approximation fit together. I'm curious how QED, the wavefunction, the photon, and classical electromagnetics fit together in such a neat picture.

Answer 1

If you're dealing with phenomenon that are large and slow, classical E&M fields describe light. If you're dealing with phenomenon that are very small (atom-sized) but slow, Quantum Mechanics (wavefunctions) describe light. If you're dealing with phenomenon that are very small and very fast (near light-speed), QFT describes light.

QFT is the most general description. But if you use its equations with the understanding that the speeds involved are slow, the equations "simplify" into QM wavefunctions. If you use those equations with the understanding that the sizes involved are large, the equations "simplify" into classical E&M fields. Those are the "limiting cases".