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I am reading a physics textbook which implicitly uses this assumption in dealing with photon electron scattering. In general, I don't see why this is true. I can imagine the first two particle momenta spanning the x-y plane and the second two spanning the x-z plane, for instance, as long as the z-components of the final momenta are equal and opposite.

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    $\begingroup$ They are probably using the center of mass frame, in which the scattering does occur in a plane. $\endgroup$ – Javier Oct 11 '17 at 20:59
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    $\begingroup$ You can always perform a Lorentz transformation to a frame where the scattering is in a plane. $\endgroup$ – Prahar Oct 11 '17 at 21:24
  • $\begingroup$ @Prahar I am looking specifically at Schwartz's QFT problem 9.1. where he considers $\gamma \phi \to \gamma \phi$ in scalar QED in the COM frame. Now it is clear to me that there exists some frame where scattering is in a plane. But I don't see why it holds for the COM frame. $\endgroup$ – Dwagg Oct 12 '17 at 19:50
  • $\begingroup$ @Javier (I tried to tag you as well) $\endgroup$ – Dwagg Oct 12 '17 at 19:50
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    $\begingroup$ It holds in the COM frame because the initial velocities are parallel and hence don't define a plane, only a line, and the same happens for the final velocities. Therefore, initial+final velocities define a plane. $\endgroup$ – Javier Oct 12 '17 at 19:52
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You are right about there being no reason for 2-particle collisions to be restricted to a plane. However, if one of the particles has mass, then there is always a frame of reference where the action happens in a plane. For the electron-photon collision, if you go to the rest frame of the electron, then there are three momentum vectors: the photon's pre-collision momentum, the photon's post-collision momentum, and the electron's post-collision momentum. To conserve momentum, these three vectors have to lie in the same plane.

If you are studying Compton scattering, then the electrons in experiments are bound to atoms, making them effectively motionless in the lab frame compared to the incoming photon. This makes it valid to study the problem in two dimensions.

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No, it is not necessarily in a plane. The textbook has simplified it for analysis. Firstly, if it were always in a plane then that would be major news. Secondly, the cross section (which is perpendicular to particle movement) has to have two dimensions for there to be any chance of collision. So there is at least some amount of a third dimension in their paths.

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