Undergrad Phonon emission : Discrete event or a process with inner detail?

[Swamp Thing]

If I understand correctly, when an electron drops to a lower energy state and emits a phoTon, this is a discrete or "atomic" event in the sense that it can't be meaningfully broken down in terms of more detailed sub-processes or interactions.

Now in the case of phoNon emission, it is also usually thought of as a discrete event... but is this done merely for convenience and simplicity, or is it also inherently discrete/"atomic" with no inner machinery?

 

[Vanadium 50]

What would it mean to emit half a photon? More generally, what kind of measurement could you make to distinguish those two possibilities?

 

[Demystifier]

We know that phonon interactions emerge from a more fundamental theory in which phonons are not fundamental objects. We do not have such a knowledge for photons, even though we have some theoretical candidates (e.g. string theory). So I would reverse the question. The emission of phonons can be "meaningfully broken down in terms of more detailed sub-processes or interactions", while the open question is whether it can be done for photons.