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 :)

  • $\begingroup$ Do you imagine that a light bulb conserves photon number? $\endgroup$
    – John Doty
    Commented Aug 17, 2023 at 23:59
  • $\begingroup$ Not really, but anyways in classical physics radiation can be absorbed and emitted by matter. What I don't see is what this type of process would look like in QFT $\endgroup$
    – Rond point
    Commented Aug 18, 2023 at 0:15
  • $\begingroup$ A cartoon might be an electron emitting a photon while exchanging a virtual photon with a nucleus, conserving momentum. The electron and nucleus survive, the virtual photon disappears, and you're left with an extra photon. You can wave your hands and claim that this "explains" a light bulb, but really, it's not much of a model. You need something less recherché than QFT to model a light bulb. $\endgroup$
    – John Doty
    Commented Aug 18, 2023 at 0:23
  • $\begingroup$ I see thank you ! $\endgroup$
    – Rond point
    Commented Aug 18, 2023 at 0:39

1 Answer 1


So I guess my main question is are there elementary particle processes where the overall particle number is not conserved ?

Yes, there are.

Consider, for example the very first figure shown in the Wikipedia article on Feynman diagrams.

This shows an electron and a positron annihilating to become a photon. So, if we were to cut the diagram at this point, already particle number is not conserved. (Two particles became one particle.) The photon is then shown as proceeding along until it becomes a quark, antiquark, and a gluon. Clearly the number of electrons wasn't conserved, the number of positrons wasn't conserved, the number of photons wasn't conserved, etc.

Further, the initial state had two particles and the final state has three particles. So, even considering just the initial and final states in the diagram there is no conservation of particle number.

Indeed, one of the main motivations for introducing Quantum Field Theory (QFT) is to account for particle number non-conservation.

  • $\begingroup$ Now I feel silly I passed right by that diagram on the wikipedia. Thank you for the quick answer anyways. $\endgroup$
    – Rond point
    Commented Aug 18, 2023 at 0:30

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