I am trying to find some names or models of a particle interpretation of quantum field theory which isn't a literal path integral approach? Are there any particle interpretations of quantum field theory which don't use path integrals?
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Quantum mechanics has a path integral interpretation, but it also has a description in terms of operators acting on a Hilbert space.
QFT is pretty much (special)relativistic quantum mechanics, so it turns out that particle number is not conserved and you can create/annihilate particles. So it's not enough to have a Hilbert space (fixed number of particles) but a Fock space (space of states of the theory) where you can have an arbitrary number of particles. $\mathcal{F} = \mathcal{H}_{1-particle} \oplus \mathcal{H}_{2-particles} \oplus \mathcal{H}_{3-particles} \oplus \ldots$
QFT can be described as a bunch of operators acting on this Fock space. Just like interacting QM, we can go to the interaction picture (where the Fock space corresponds to the space of states of the free theory, since we don't know the states of the full interacting theory). Here, the time evolution operator can be written in terms of the Dyson series. Each term in the Dyson series is the time-ordered product of a bunch of operators corresponding to the interacting part of the Hamiltonian (since the free part acts trivially on the Fock space). So, you can see that the Dyson series is inherently perturbative as each term corresponds to a higher order in the interaction coupling. One can then define (say) scattering amplitudes as the the Dyson series (representing time evolution) sandwiched between states of the Fock space. That's how one does QFT in the operator picture, without any mention of path integrals.
Let's now go over to the path integral description. In the path integrals picture, when we try do do the integral perturbatively, note that the perturbation series we get is similar to the Dyson series. That's because we've solved the free part (which is an easy Gaussian integral) and trying to solve for the interaction (by expanding out the exponential). That formulation is very similar to what we do in the interaction picture! Essentially, paths which interfere constructively and contribute to the amplitude correspond to events (alternatively) modeled as a bunch of interaction operators acting on a state in the Fock space. So I hope I've also managed to motivate how the two descriptions are connected.
Update: I recently came across this relevant and useful blog post by Lubos.
Siva
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In some sense 2nd quantization is equivalent to path integral approach when introduce quantized fields. But the point is, even if you use 2nd quantization you still need to calculate things like cross sections, which is more related to generating functional of path integral. Also, many things of more advanced QFT are based on path integral. I think you would like Ron's comment on path integral which is more professional: What is the fundamental probabilistic interpretation of Quantum Fields?.
