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I was thinking about how to reconcile the picture of the eletromagnetic wave as a changing electromagnetic field and as a collection of photons.

If i have a charged particle, that is free in space and then an electromagnetic wave passes through it. In the field picture, the particle will move by the eletromagnetic force acting on it in a direction specified by the wave direction and polatization, lets suppose the wave makes the particle go downward, then upwards and etc.

Now in the picture of photons, there is a shower of ziolion of photons coming, they interact with the charged particle by Compton scattering or something. In this picture I cannot see how a particular direction can be being pursued.

Actually, i can't either understand how the photon picture connects with the wave frequency and polarization.
But i know that eletromagnetic events are due to the QED, this is, virtual photons interactions with fermions and etc.

Someone knows how to properly connect eletromagnetic classical waves to QED photons, or just, classical quantum mechanics photons?

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    $\begingroup$ I don't think forums are a good places to ask questions like that. The response will necessarily be too short. You need a book. I would suggest going for something on quantum optics rather than full-blown QED. Loudon's Quantum Theory of Light is a good start. There you will learn that quantum state of the electomagnetic field is one thing and the specific electric and magnetic fields, which are operators that operate on that state, are something else. $\endgroup$ – Cryo Dec 12 '20 at 23:10
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Claudio Saspinski's answer answers the special case of the radio wave. Your question is much broader. Therefore I will also answer.

... there is a shower of ziolion of photons coming, they interact with the charged particle by Compton scattering or something. In this picture I cannot see how a particular direction can be being pursued.

Correct. All these photons are from thermic sources, with different energy content (frequencies), with different spatial positions of their crests and with different directions - over all 360°- to the direction of propagation. Interacting with a body, these photons gave their energy to the atoms of the body, rising the temperature above the surrounding temperature. The atoms of the body for their part emit photons (mostly with lower frequencies) and a the thermic equilibration. =>
The atoms are doing chaotic displacement by the absorption and emission of the photons.

If i have a charged particle, that is free in space and then an electromagnetic wave passes through it.

That is the special case mentioned by Claudio Saspinski. The photons from the first paragraph are not measurable as a wave. The are electromagnetic radiation. A measurable EM wave are radio waves. Electrons on the surface of a conductor (antenna rod) get synchronously accelerated forth and back and during each acceleration they emit photons, all with the same direction of their electric field component (parallel to the rod). The EM radiation is a polarized one.
Furthermore the number of emitted photons and the direction of their electric field changes periodically. From zero to a maximum number and back to zero - all with the direction of their electric field to the top -, and than again to the maximum number - this time with the direction of the electric field to the bottom.

On the receiving conductive rod the electrons, absorbing these polarized photons, get accelerated all in the same direction. Of course only as long, as the incoming wave contents photons. On the next half wave the incoming photons accelerate the electrons on the rod to the opposite direction.

enter image description here

In short, you are right to be concerned about the difference between electromagnetic waves and photons.

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A typical example is a LC circuit, where a EM radio wave with the ressonant frequency of the device generates an alternating current.

The LC circuit is an harmonic oscillator, and everything can be understood without thinking of photons.

But if the power of the incoming wave becoming too weak, things change. There is a minimum of energy for each frequency: $E = \hbar \nu$. So, the behaviour of the circuit must be understood as a quantum harmonic oscillator.

Only when the energy level of the QHO is high enough, it converges to the behaviour of a classical harmonic oscillator. But $\hbar$ is a very small number, so that most of what we call weak signals are stronger enough to behave classically.

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Are you asking what will happen if charged particle meets a beam of non perfectly coherent and polarized light?

You may see such examples in plasma being opaque, if dense enough and also early universe being opaque due to distances and amount of charged particles.

Particle will interact with incoming photons and randomly change direction, with average force being close to zero. Each time charged particle changes direction it emits a photon, that weakens the original one and amplifies the one that goes sideways, where charged particles was moving.

If you ask how specific direction can be selected as preferred one for emitting photons, you may want to check this device:

https://en.wikipedia.org/wiki/Klystron

If you ask how specific direction can be selected as preferred one for accelerating electrons, you may want to check this device, the one with RF:

https://en.m.wikipedia.org/wiki/Linear_particle_accelerator

If you ask about reference frame of photon, then its not meaningful. It has no time, from it prespective it did not react with anything other than the target electron, immidiately after being emitted by the source.

If you ask about popular images of photon that shows spin or polarization when it hits the target, its not useful in your question. Electron doesnt see this complex pattern, for it photon is a simple object that makes one force, depending on polarization and phase on impact.

If you ask if photons can interact with each other, answer is usually no. Only through gravity or matter creation for extremely high energies.

If you ask how to go in calculations from one photon to many photon beam: consider that only polarised and coherent part of the beam pushes the electron in the same direction. So, if beam is 23% vertical polarisation and 77% horizontal polarization, consider that 46% (smallest, 23% multiplied by 2) of the beam is scattered and 54% is useful in polarization. If 12% of the beam has phase of 0 deg and 88% has phase of 180deg, that is unlikely, but easier to calculate, then part that is scattered is 24% (smallest part, 12% multiplied by 2) and 76% is useful in phase. In total 76% of the beam being useful in phase and 54% is useful in polarization. 76% multiplied by 54% is 41%, and this part of the beam acts as one strong photon, pushing the electron in the same direction. The rest of the beam is wasted, pushing the electron in random directions.

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