I'm not as familiar with relativistic QM as I would like to be, but it seems like the given formula actually does depend on the four momentum. The k in transform corresponds to the three spatial components of the four-momentum while the m corresponds to mass, which in turn depends on the time component of the four-momentum. Why the fourth component of the four-momentum is left out of the Fourier Transform, on the other hand, is unfamiliar to me.In the Wikipedia link:
http://en.wikipedia.org/wiki/Yukawa_potential
it says that the Fourier transformation of the Yukawa potential is the amplitude for two fermions to scatter. But the Fourier transform ignores 4-momentum and only has 3-momentum. The amplitude to scatter should depend on a 4-momentum squared, and not 3-momentum. So how is this reconciled?
It is known that, from the definition of V (in a static field) and the first of Maxwell's equations that:There is a problem in Jackson's E&M book which asks you to derive the charge distribution corresponding to this potential. I could never quite get it right.
The simplified version of the Yukawa derivation takes place in the barycentric system where the energy component of the 4-momentum transfer vanishes. Then the 3D Fourier T can be made.it says that the Fourier transformation of the Yukawa potential is the amplitude for two fermions to scatter. But the Fourier transform ignores 4-momentum and only has 3-momentum. The amplitude to scatter should depend on a 4-momentum squared, and not 3-momentum. So how is this reconciled?
Does barycentric system mean center of mass frame between the two fermions? Does this mean that the Yukawa potential is only valid in a center of mass frame, since you break Lorentz invariance by choosing a specific frame?The simplified version of the Yukawa derivation takes place in the barycentric system where the energy component of the 4-momentum transfer vanishes. Then the 3D Fourier T can be made.
May I come to know about the barycentric systemsThe simplified version of the Yukawa derivation takes place in the barycentric system where the energy component of the 4-momentum transfer vanishes. Then the 3D Fourier T can be made.