Every QFT textbook starts by basically postulating that we have discrete states connected by creation and annihilation operators. In Quantum Mechanics, we started from a differential equation and found that the eigenstates of the Hamiltonian can be considered to be discrete states. So in QM, quanta are an emergent phenomenon from our postulates. Can we get discrete particles out of a QFT with the path integral formulation? I want to focus on the path integral because it's covariant, and because canonical quantization seems to have these assumptions about discrete particles baked into it.
I'm thinking that maybe if we do an example of a Klein-Gordon field with periodic boundary conditions then we should be able to see how the quanta appear. That is, a $1+1$ dimensional QFT with Lagrangian $$L=\frac12 \partial^\mu \phi \partial_\mu \phi - \frac12 m^2 \phi^2$$ and $x \sim x + l$. I would do the calculation myself, but I need help with where to start in terms of carefully defining what space of states we're working over and getting an equation of motion (or something equivalent) out of the path integral. Can anyone help?
