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Picture a wire with current directed out of the page. The electrons in the wire emit virtual photons in all directions which mediate the produced magnetic field. What is the orientation of the momenta of the emitted virtual photons, or are they randomly oriented?

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I disagree with the other answer. If the electrons are moving with a constant velocity (i.e. not accelerating), then there are no real photons emitted.

In the rest frame of the moving electrons, the electrons will set up an electric field (in this case a field that will point inwards and drop like a log of the distance from the line of charge). The Fourier transform of this field will give the direction of the momenta of the virtual photons.