Disclaimer: this is a homework question, so I am happy with just a hint or the expressions needed to proceed with my understanding.
I am working on the momentum conservation of a particle/anti-particle annihilation process, and I have been asked to show that the annihilation of a particle with a finite mass and its anti-particle cannot lead to the emission of only one photon.
I understand why this happens: the conservation of momentum. However, I would like show this in a more sophisticated 4-momentum proof...how would I go about showing that momentum is conserved for two photons but it is not conserved when the annihilation process creates just one photon...?
This may be a duplicate of: Proving the conservation of 4-momentum for a particle collision $A+B\to C+D$
Answer
There are many possible proofs. Here is one that involves some practice with four-vectors. I write with mostly-minus metric st $p^2=m^2$.
You can write four-momentum conservation as $$ p_1 + p_2 = a $$ Now Minkowski-square, finding $$ 2m^2 + 2 p_1 \cdot p_2 = 0 \Rightarrow p_1 \cdot p_2 < 0$$ Try to show that the latter inequality is impossible. Hint: evaluate the Minkowski product in the rest frame of $p_1$ or $p_2$.
Alternatively, as @Danu suggests, think about the centre of momentum frame, in which $\vec p_1 + \vec p_2 =0$. Can a photon have zero momentum but non-zero energy?
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