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I have been investigating the scenario of two photons traveling in space (in a lab frame of reference) of different frequencies are about to collide with an angle $\theta$ between them. I set their 4-momentum vectors to be $(hν_1/c , hν_1/c \cos(\theta) ,hν_1/c \sin(\theta), 0)$ for a gamma photon and $(hν_2/c , hν_2/c , 0 , 0)$ for a CMB photon. I am having a hard time figuring out what the energy would be in a zero momentum frame due to the angle. The pair production along with a Lorentz transformation lead me to believe that $hν_1/c \sin(\theta)$ would always equal zero which is hard to believe as I can adjust the angle in other frames.

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  • $\begingroup$ Do photons interact like this? I’ve never heard of a photon-photon “collision”. $\endgroup$
    – Ben51
    Feb 7, 2021 at 17:01
  • $\begingroup$ It is used to create matter, making positron and electron similar to the Breit-Wheeler process. $\endgroup$
    – Oclock
    Feb 7, 2021 at 17:09
  • $\begingroup$ Thanks. Didn’t know that was possible. $\endgroup$
    – Ben51
    Feb 7, 2021 at 17:14
  • $\begingroup$ As far as i know this is not directly possible, only through some higher-order effects. But I guess this is valid as an approximation or an exercise. $\endgroup$
    – NDewolf
    Feb 7, 2021 at 18:36
  • $\begingroup$ I am doing it as an exercise. I am thinking of that, but there has to be a way to transition a system of photons to a center of momentum frame. It is the reverse of pair antihalation. $\endgroup$
    – Oclock
    Feb 7, 2021 at 21:09

1 Answer 1

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Set c=1, and, so $h\nu=E=p$, much less cluttered. Assuming you are serious about the signs you have chosen, the total energy, momentum four vector in the lab frame is $$ (p_1+p_2, p_2+ p_1 \cos\theta, p_1\sin\theta, 0). $$ In the center of momentum frame, $(E,0,0,0)$. But $s=E^2-\vec{p}^2$ is a relativistic invariant, and so the same in both frames, so $$ E^2= 2p_1p_2 (1-\cos\theta), $$ with a minimum at 0, and a maximum at 4$p_1p_2$. This is the smallest energy squared at which a pair of particles of mass m will be produced at rest, where $m^2=p_1p_2$.

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