How does quantum mechanics affect the classical electrostatic force?

[exponent137]

I read Feynman's quantum electrodynamics, but I do not clearly understand, where quantum mechanics enter.
So if clasicaly force in electrostatic field equals
F = e^2 /4 pi epsilon 0 r^2. What happens, when quanum mechanics is included.
I think that for elementary principle we do need renormalization and screening of charge and so one. We need only F = function of h?

Jump from classical gravitational field to quantum gravitational field is unknown, but jump from classical electrostatic field to quantum electrostatic field is known.

It is possible to say that electrostatic force is consequence of virtual photons, but some say that virtual photons are only mathematical help in approximative area and they are not real.

 

[dextercioby]

Well, there's a quantum correction to the electrostatic potential. It partly accounts for the Lamb shift in Hydrogen.

Daniel.

 

[exponent137]

I know for these correction. But I do not understand where is different formalism and no matter if it is the same result as classically?
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For instance: Clasically electromagnetic field pushes "space sail"
Quanum view: Photons pushes "space sail"

Result of calculations in both regimes is the same, but formalism is not.

But what is difference in formalism of force between two charges or electrons classically and in quantum regime?

It is expected that this are virtual photons. But, some say that virtual photons are used only as approximation for convergence and that they are not real.

 

[exponent137]

I will ask differently:I have one question about quantum fields.

As first, i will try to simplify question about quantum fields.
We have one big light ray – coherent.
We can describe it with Maxwell equations or with many of photons. For instance energy, which is produced by this light-ray, can be described in both way.

But we have one standstill charge e_1 with infinite mass. It influence with force
e_1 e_2/(4 \pi \epsilon_0 r^2 on another charge, which is standstill. How this force or momentum or energy can be described with quantum field.

This can be simpified that all spins equals zero. Maybe we can see this problem only in one dimension (two plates) or so on.

I am not interested in screening of charge in lamb shift, only in principle, how Planck constant enter here.

How these Feynman's Fourier formulae show r^-2 dependence.

I suppose that virtual photons can describe this example, but I read that virtual photons do not exists, they are used only as mathematics in theory of perturbations.

 

[exponent137]

Still another form of question: We describe real photons with creation and anihilation operators a+, a. Do these operators are used in any form for virtual photons.

Let us simplify as much as possible to describe virtual photons, so:
unrelativistic Quantum field theory, no spins, one particle have infinite mass, maybe the other electron is also described in without quantum mechanics, scalar field. And what is still possible to simplify virtual photon? Maybe two or one dimension instead of three dimensions.