Distinction between real and virtual photon

[exponent137]

a.) One electron emits one real photon and it hits another electron.

b.) One electron emits one virtual photon and it hits another electron. (Repulsion because of charge repulsion)

What is gradual transition between cases a. and b. ? Or maybe, does it not exist?

 

[Bill_K]

The difference lies in the polarization vector. "Real" photons, as those found in electromagnetic waves, are transverse. "Virtual" photons, assuming you do mean those involved in Coulomb repulsion, are longitudinal.

 

[exponent137]

What is then polarisation difference between real and virtual gravitons? Does real gravitons have only two (transversal) polarisations, and virtual gravitons are only longitudinaly polarised?

 

[Bill_K]

Ok, without making it too complicated, a gravitational wave traveling along the x-axis is h_μν(x, t) = e_μν e^{i(kx - ωt)} where e_μν is the polarization tensor for the wave. e_μν must be symmetric.

Just as in electromagnetism, in gravity there are gauge transformations that can be used to eliminate some of the components, and in fact we can arrange it so that all of the components of e_μν vanish except for e_yy, e_yz and e_zz = - e_yy. This leaves just two independent components, and is what we mean when we say that the polarization tensor of a gravitational wave is transverse and traceless.

 

[exponent137]

I see now those pictures of polarisation in Carroll's book on arXiv.
Is it possible to compare light and gravity?
Is quadropole waving of light tensor?
Can be on any pseudo way possible to say, that light is traceless?

 

[Darwin123]

a.) One electron emits one real photon and it hits another electron.

b.) One electron emits one virtual photon and it hits another electron. (Repulsion because of charge repulsion)

What is gradual transition between cases a. and b. ? Or maybe, does it not exist?

Sometimes, I think of the virtual state as being a type of polarization. In my advanced QM courses, they talk about the polarization of a virtual photon. The virtual nature of the photon enters the mathematical description in the same variables that includes the polarization.
Here is how I visualize the four polarization states of an electromagnetic photon moving in a certain direction with a certain energy.
First, I imagine the photon as being composed of a wavevector and a polarization vector attached at the same point. The wave vector gives the direction and energy of the photon.
The polarization vector of an electromagnetic wave in a vacuum has to be perpendicular to the wave vector. So there are two transverse polarization states of a photon in optics. One can call these two states "horizontal" and "vertical". These two types of photons are "real" photons.
The polarization vector of a static electromagnetic field can be parallel to the wave vector. You could call this a longitudinal polarization state. However, there are two types of longitudinal states. One can have a longitudinal state describing a static electric field. One can have a longitudinal state describing a static magnetic field.
Hence there are two longitudinal polarization states of an electromagnetic field: electric and magnetic. Photons which are in one of these two longitudinal states are referred to as "virtual photons".
So in my view, "real photons" have "transverse polarization" and "virtual photons" have "longitudinal polarization". The "virtual photons" spontaneously disappear at short distances from an electric charge, although they can also disappear if they collide with an electric charge. The "real photons" persist until they collide with another electric charge.
This is just my visualization to help with the Feynman diagrams. In my view, the squiggle in the Feynman diagrams stands for longitudinal polarization. How seriously you want to take this is up to you.
String analogy time. I present a heuristic picture using an analogy between a vacuum and a string.
A lot of teachers use the waves on a string as an analogy to light waves. The waves on a taunt string that everyone visualizes are transverse. Just pluck a taunt string and you can see them. However, longitudinal waves can also carry down a taunt string. This is children can talk to each other using tin cans attached by a taunt string.
Consider the example of a taunt string. The transverse waves can carry down a string even if it isn't taunt. However, the longitudinal waves don't travel very far if the string isn't taunt. One can make an analogy between a string which isn't taunt and a vacuum. The vacuum state isn't taunt enough to carry a longitudinal wave very far down the string.
Call the quantized excitations of a string the "stringons". When children jump rope, they see the "real stringons" as transverse vibrations on the rope. When children talk to each other using a string connecting two tin cans, they are using the "virtual stringons".
The children don't care whether which "stringons" are "real" or "virtual". Each refers to a different type of energy transfer.
I don't have the software for posting the corresponding equations. However, I hope this analogy is useful.

 

[Darwin123]

a.) One electron emits one real photon and it hits another electron.

b.) One electron emits one virtual photon and it hits another electron. (Repulsion because of charge repulsion)

What is gradual transition between cases a. and b. ? Or maybe, does it not exist?

I stated in another post that the real/virtual dichotomy of photons is homologous to the transverse/longitudinal dichotomy in polarization. I said that I couldn’t write down the equations to support this heuristic. However, I have a reference that contains these equations.
Sorry, no links. I will quote a paper book <gasp> that I have in my personal library. It was very expensive, sorry.
From:
The Photon: A Virtual Reality” by David L. Andrews in The Nature of Light edited by C. Roychouder et al. (CRC, 2008) pp. 275-276.
“Despite the fact that the virtual photon formulation leading to equation 18.6 is cast in terms of electromagnetic fields that are purely transverse with respect to the photon propagation direction p, the field equation 18.8 contains elements that are manifestly non transverse against R. To exhibit this explicitly, the given expression can be decomposed into terms that are transverse and longitudinal with respect to R. <equations 18.9 and 18.10>
One immediate conclusion that can be drawn from the prominence of the longitudinal component of the short-range region is the fact that photons with p not parallel to R are involved in the energy transfer – which is consistent with the position-momentum uncertainty principle…Physically, this relates to the fact that with increasing distance the propagating field loses its near field character and is increasingly dominated by its transverse component, conforming ever more closely to what is expected of “real” photon transmission.”