Four momentum of Photon in Center of Mass frame?

[infiniteen]

Hi,

I have just been pondering the problem of electron - positron annihilation into a single photon in the CM frame.

I was stuck at a discrepancy - that in the center of mass frame, the total momentum of the particles was zero, but the energy is the sum of the energies of the original particles.

Say in an annihilation process, with electrons 1 and 2 annihilating into a photon,

1+2 -> gamma

$$P^{\mu}_{1}+P^{\mu}_{2}= P^{\mu}_{\gamma}$$

in CM frame,
$$P^{\mu}_{1}=(E*_{1}, p_{x}, p_{y}, p_{z})$$
$$P^{\mu}_{2}=(E*_{2}, -p_{x}, -p_{y}, -p_{z})$$
hence
$$P^{\mu}_{\gamma}=(E*_{1}+E*_{2}, 0, 0, 0)$$
but for photon, isn't
$$P=(hf, hf, 0, 0)$$?

How do you reconcile these facts?

Thanks in advance for the help.

 

[atyy]

Maybe it's not possible to create only one photon.
http://en.wikipedia.org/wiki/Electron-positron_annihilation

 

[Meir Achuz]

You have proved that particles cannot annihilatr il\nto a single photon.