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## Main Question or Discussion Point

I'm currently reading Schwartz's QFT text "Quantum Field Theory and the Standard Model" (although this question does not specifically pertain to this text) and I've got two questions:

1. At the beginning of chapter 6 he talks about how we may consider the system to be non-interacting in asymptotic times, ostensibly because "the particles are far apart". I find this very hand-wavy so my question is can this notion be made rigorous?

Take a massless scalar field with a ## \phi^3## interaction, what's stopping a final momentum eigenstate from splitting into two particles each with half the momenta?I don't see how one could find free asymptotic states in a theory such as this.

2. My second question relates to the first question. Take a massive scalar field with a ## \phi^3## interaction for example. Now even though the momentum operator is unaltered by this term, and hence the ##a_p## at fixed times are still creating/annihilating momentum eigenstates, the Hamiltonian is altered with a ## - \frac g {3!} \phi^3## term which means that the ##a^{\dagger}| 0 \rangle## states are no longer energy eigenstates. However the asymptotic states are said to be on-shell, so does that mean that we are altering the Lagrangian/Hamiltonian at asymptotic times? I'm very confused.

1. At the beginning of chapter 6 he talks about how we may consider the system to be non-interacting in asymptotic times, ostensibly because "the particles are far apart". I find this very hand-wavy so my question is can this notion be made rigorous?

Take a massless scalar field with a ## \phi^3## interaction, what's stopping a final momentum eigenstate from splitting into two particles each with half the momenta?I don't see how one could find free asymptotic states in a theory such as this.

2. My second question relates to the first question. Take a massive scalar field with a ## \phi^3## interaction for example. Now even though the momentum operator is unaltered by this term, and hence the ##a_p## at fixed times are still creating/annihilating momentum eigenstates, the Hamiltonian is altered with a ## - \frac g {3!} \phi^3## term which means that the ##a^{\dagger}| 0 \rangle## states are no longer energy eigenstates. However the asymptotic states are said to be on-shell, so does that mean that we are altering the Lagrangian/Hamiltonian at asymptotic times? I'm very confused.

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