There are already similar topics with interesting answers such as When particle number can change in quantum physics? but I still don't understand much.
I often read about the non-conservation of particle number in (Relativistic) Quantum Field Theory but it remains quite obscure to me. First of all, isn't particle number trivially conserved in any free field theory such as the Klein Gordon field ? In quantizing the theory we construct a Hilbert space that is the direct sum of n-particles Hilbert spaces (Fock Space) but to the best of my knowledge states of different particle numbers are orthogonal when there are no interactions.
Once you include interactions such as in QED, you can of course have processes such as $ \gamma + \gamma \longrightarrow particle + antiparticle $. Then I see that Hilbert states of different particle numbers (I use particle as a general term for particles and antiparticles alike) for e.g the Dirac field will be connected through a "flow" of photons into $ e^{+} + e^{-} $. So the number of particles of a given type is not conserved but particle number is still conserved overall in such a process, there is simply a transfer from one field to another.
So I guess my main question is are there elementary particle processes where the overall particle number is not conserved ? I can only think of Feynman diagrams that have the same number of ingoing and outgoing particles and it seems to me that momentum conservation etc would not be respected otherwise. But then the overall number of particles in the Universe (as in excitations of any field) would be conserved. That would also imply that when we say e.g. an atom absorbed a photon and emitted two lower frequency ones it would only be an approximation of some kind.
I would appreciate it if someone could shed some light on this :)
