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In this Phys.SE answer Ron Maimon stats:

there is no relativistic particle formalism in which the particles have postive energies and casual propagation. You can either deal with fields in which case the particle notion is non local or you can deal with particles. But then they go back in time.

Is this true that there can be no particle interpretation where particles only travel forward in time? I'm not asking about path intergrals but the particle interpretation in general, I'm also aware the field interpretation is the more popular.

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Did you go to the previous linked answer to a similar question? He gives a clear enough exposition of the statement:

But in Feynman's particle path-integral picture, when you parametrize particles by their worldline proper time, and you renounce a global causal picture in favor of particles splitting and joining, the particle trajectories are consistent with relativity, but only if the trajectories include back-in-time trajectories, where coordinate time ticks in the opposite sense to proper time.

Looked at in the Hamiltonian formalism, the coordinate time is the only notion of time. So those paths where the proper time ticks in the reverse direction look like a different type of particle, and these are the antiparticles.

Bold mine.

Sometimes there is an identification, so that a particle is its own antiparticle.

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If you philosophically dislike acausal formalisms, you can say (in field theory) that the Hamiltonian formalism is fundamental, and that you believe in crossing and CPT, and then you don't have to talk about going back in time. Since crossing and CPT are the precise manifestations of the statement that antimatter is matter going back in time, you really aren't saying anything different, except philosophically. But the philosophy motivates crossing and CPT.

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