How is electric field arising from photons?

Question

This question has nothing in common with the question it is proposed to be duplicate e.g about virtual photons. The question is about a real radiation field where is known that no virtual photons exist.

The electric field E moving up and down (in its value) as a sinusoidal wave, which has been regularly shown in EM books should move a probe electron up and down (in space). How does a photon picture make this pattern, as the photon is moving forward (not up and down) its momentum is perpendicular to E? Of course there is also magnetic momentum but it is too faints. If one uses Feynman diagrams (as is proposed in the answer of Lubos in the proposed duplicate question) for electron photon scattering the momentum of the electron should also be in the photon's direction and not perpendicular as the EM picture suggests.

Answer 1

A photon is not a particle in the classical sense of the word.

A photon is a quanta of the radiation field.

Suppose you've got your sinusoidal Electric field, Suppose as you fix the energy say, hv, then the phase is maximally unknown.(Quantum Mechanics)

In a way a photon is much more similar to a field than a particle.

The emergence of the classical Electric field from photons can be understood in the "coherent state" representation of the Hilbert space.(The point is that, the field becomes classical in the limit where the average number of photons of a favored mode goes to infinity and the field becomes similar to a classical sinusoidal wave in the favored mode)

Answer 2

Photons in mainstream physics, are quantum mechanical entities which in great numbers build up the classical electromagnetic radiation, which is what you are describing with the changing fields.

This picture for circularly polarized light gives an intuition how the classical wave is built up from a quantum mechanical substrate of photons.

One needs the mathematics of quantum mechanics in order to really understand this.

Also this double slit experiment can give an intuition how photons, when detected one by one leave a footprint of a particle on the screen,

Figure 1. Single-photon camera recording of photons from a double slit illuminated by very weak laser light. Left to right: single frame, superposition of 200, 1’000, and 500’000 frames.

but in a large accumulation the classical interference pattern of the wave nature appears on the right, connected with the probabilistic nature of quantum mechanics.

One needs to study quantum mechanics to master this.