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TL;DR: How can states of an interacting QFT asymptotically follow the trajectories governed by the free Hamiltonian, when, say, the free and interacting groundstates are different, and the states look like localized excitations on the groundstate?

I'm confused about Weinberg's discussion of the S-matrix, encapsulated in Eq. 3.1.12. (There have been other posts here about this same equation, I know!) His argument is on page 110 of his first QFT volume, and it's largely repeated on Wiki: https://en.wikipedia.org/wiki/S-matrix#From_free_particle_states.

He posits a split of the full Hamiltonian into $$H=H_0+V, $$ where $H_0$ is a free Hamiltonian defined to have the correct physical ("renormalized") mass. Ignoring issues about bound states, $H_0$ and $H$ will then have the same spectrum. Let $\Phi_\alpha$ be eigenstates of $H_0$ with energy $E_\alpha$, with corresponding "in" states $\Psi_\alpha$ that are eigenstates of $H$ with energy $E_\alpha$. Then Weinberg writes that as $\tau \to -\infty$, $$ \int d\alpha e^{-i E_\alpha \tau} g(\alpha) \Psi_\alpha \to \int d\alpha e^{-i E_\alpha \tau}g(\alpha) \Phi_\alpha. \tag{3.1.12}$$

I understand one must choose $g$ to be a sufficiently well-behaved function of the energy and momentum associated to $\alpha$. (Choosing $g(\alpha)$ as a $\delta$-function to select a single energy eigenstate would yield problems.) I also understand why 3.1.12 holds in quantum-mechanical (first-quantized) scattering problems.

Meanwhile, in QFT, we have (if I understand Weinberg correctly) something like $$H_0=\pi^2+(\nabla \phi)^2 + m_{phys}^2 \phi^2 \tag{1}$$ and $$V=(m_{bare}^2-m_{phys}^2)\phi^2 + \lambda \phi^4. \tag{2}$$ I understand why $H_0$ and $H$ have the same spectrum, and I understand a certain sense in which the LHS of eq. 3.1.12 approaches a free trajectory.

Still, I'd think our equation 3.1.12 can't hold as posited. Say $g(\alpha)$ is a smooth function whose support only intersects the spectrum for $\alpha$ corresponding to the groundstate. (This is possible because the ground state is gapped.) Then 3.1.12 can't hold, because $H$ and $H_0$ have very different groundstates. Or choose $g(\alpha)$ to have support that only intersects the spectrum on the single-particle mass shell. (This is again possible because the shell is isolated.) Then eq. 3.1.12 still seems wrong, because although both sides describe single-particle localized excitations on top of the ground state, the excitation on each side is on top of very different ground states (and the excitations look different, too).

Again, in a first-quantized quantum-mechanical problem, you don't have this issue, because the groundstate and asymptotic excitations of the free and interacting theory look the same.

By the way, I think I can define an $H_0$ such that 3.1.12 holds, but it's not local in the field $\phi$, whereas I suspect Weinberg had in mind something more like eq. 1. Also, you can avoid these problems in the Haag-Ruelle approach to scattering formalism.

I don't think the answer is that Weinberg intends eq. 3.1.12 to be true only when the interactions of $H$ are adiabatically switched off at large times. He probably would have said so, and it would make the equation more trivial.

Maybe you can point out a simple misunderstanding.

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