Proving Isolated Photon Doesn't Split into Two Photons

[Xamfy19]

Here is the question I had trouble with it:

How to prove that an isolated photon cannot split into two photons going in direction other than the original direction.

I thought photon doesn't have mass, then

energy = momentum.
from momentum conservation law. If photon split into two part and travel in different direction, then it must have different mass (or at opposite direction).

That's all I have for the time being.
Thanks for the help.

 

[dextercioby]

$$p^{\mu}=p_{1}^{\mu}+p_{2}^{\mu}$$ (1)

Workin' in Heaviside-Lorentz units,

$$\omega=\omega_{1}+\omega_{2}$$ (2)

$$\vec{p}=\vec{p}_{1}+\vec{p}_{2}$$ (3)

Square (2) & (3) and use $\omega=|\vec{p}|$ (which is valid for all 3 photons).

Denote the angle between the momenta of outgoing photons by $\theta$

Daniel.

 

[Xamfy19]

Great, but what are the p^u, w and p?

I guess
w = w1 + w2 is the energy conservation
p = p1 + p2 is the momentum conservation, and what is

the first equation ?

thanks

 

[dextercioby]

Conservation of 4-momentum,what else...?That's where the other 2 eqns come from.

Daniel.

 

[Xamfy19]

thanks, Daniel