Why is it forbidden for two photons to turn into one?
In the context of quantum field theory, why is it impossible for two photons (or other massless bosons like gluons) to collide and produce a single photon? This kind of a process is supposed to be forbidden by momentum conservation, but it was not immediately obvious to me why this is.
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Think about how that is supposed to work. It seems you want two photons to somehow combine into a single photon and nothing else. That means the output photon must have the combined energy of the two input photons because you've provided no other place for the energy to go. And of course momentum needs to be conserved too.
However, a photon's momentum is also proportional to its energy. But how is that supposed to work when the two photons aren't traveling in the same direction? Take the extreme case of the two photons having equal energy but colliding head-on. The net momentum is 0, but the net energy twice that of each photon. You can't conserve both energy and momentum if the photons were allowed to combine.
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Imagine two equivalent (e.g. same frequency) photons colliding with each other head-on. The linear momentum of the system is $0$ because each photon's momentum has the same magnitude but is pointing in opposite directions.
If these two photons collide and form a single photon (and nothing else), then conservation of linear momentum would mean that this photon has $0$ momentum. But a photon is, definitionally, always moving at the speed of light, and its energy is purely from its momentum (in special relativity, $E^2 = p^2c^2 + m^2c^4$ with $m=0$). For it to have zero momentum, it would have to also have zero energy which would violate conservation of energy and correspond to their not being anything.
Even for a more glancing collision, we'd either lose energy from the cancelled out parallel portions of the momentum, or we'd need to increase momentum in the perpendicular direction to make up for it which would violate conservation of linear momentum in that direction.
Thus, to satisfy conservation of energy and conservation of linear momentum simultaneously, we either need multiple particles moving at the speed of light afterwards, or the photons need to annihilate and produce a massive particle which can have zero momentum and absorb the energy as its mass (or some mixture of massive and massless particles whose net momentum is zero).
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