Asymptotic in & out states in QFT

Question

For any free scalar field, $\Phi(x) $, the creation operator $a^{\dagger}(p)$ is written without any time dependence or time label on it. So when it acts on the ground state $|0\rangle $, it creates a momentum eigenstate. This state is just an honest vector in the Hilbert state, $|p\rangle$, which is an eigenstate of 4-momentum operator $P=(H,P_1,P_2,P_3)$. It doesn't have any time label on it. So it's not , for example, a state like $|p,t\rangle$ (which is an eigenstate of Heisenberg operator $P(t)= e^{iHt} P e^{-iHt}$).

Now, in textbooks, people point out that for an interacting theory, the field behaves like a free field at early and late times. They define those free fields $\Phi_{in}$ and $\Phi_{out}$ to have their own creation operators $a^\dagger_{in}$ and $a^\dagger_{out}$ (Maggiore QFT book, for example). Now, just like any other free field creation operator, I assume that this $a^\dagger_{in}$ could create states like $|p\rangle$ that are eigenstates of the 4-momentum operator $P$, where $P=(H_{in}, P_1,P_2,P_3$). But it seems to me from textbooks that this $a^\dagger_{in}$ can create momentum states with time label on them, like $|p, -\infty\rangle$ which are eigenstates of the Heisenberg operator $P(-\infty)= e^{iH_{in}(-\infty)} P e^{-iH_{in}(-\infty)}$.

Answer 1

In brief, the full Hilbert space of the theory $\mathscr{H}$ contains two subspaces $\mathscr{H}_\text{in}$ and $\mathscr{H}_\text{out}$ that represent the asymptotic in and out states. If $\mathscr{H} = \mathscr{H}_\text{in} = \mathscr{H}_\text{out}$, the theory is asymptotically complete. On both $\mathscr{H}_\text{in/out}$ we have free field operators $\phi_\text{in/out}$ of mass $m$ and it is their creation/annihilation operators that create the in/out states. One can construct those in/out spaces essentially as the spaces spanned by states created by the interacting field operators $\phi(\vec x,t)$ as $t\to\pm\infty$, modulo mathematical subtleties.

With the Møller operators $\Omega^\pm = \lim_{t\to\pm\infty} U(t)^\dagger J U_0(t)$ where $J$ is a map from the abstract free Hilbert space $\mathscr{H}_0$ into $\mathscr{H}$ that in some sense has the free states at time $t=0$ as its image, one can then show that the image of $\Omega^\pm$ is $\mathscr{H}_\text{in/out}$ and construct a map $(\Omega^+)^\dagger \Omega^- : \mathscr{H}_\text{in}\to\mathscr{H}_\text{out}$ - this is the S-matrix, or $U(-\infty,\infty)$. (At the physical level of rigor, we often omit the construction of the map $J$ and pretend both the free $H_0$ and interacting Hamiltonian $H$ can live on the same Hilbert space)

So there is an actual physical sense in which the states labeled by "in" are in the infinite past and "out" is in the infinite future - the limit of the time evolution operator $U(t_i,t_f)$ for $t_i\to -\infty, t_f\to\infty$ maps $\mathscr{H}_\text{in}$ to $\mathscr{H}_\text{out}$, i.e. the in-states evolve into the out-states in the limit of infinite time.

For a mathematically rigorous treatment of this setup for Haag-Ruelle scattering theory, see Reed and Simon's "Methods of Modern Mathematical Physics, Vol. III: Scattering Theory".