In and Out States

Interacting states can be compared with free multiparticle states only when stable particle wave packets separate far enough in the remote past or future that their residual interaction becomes asymptotically negligible. When the corresponding strong limits exist, Møller operators map a free asymptotic Hilbert space into the interacting scattering subspace. They need not exist in every QFT, and their existence alone does not prove that every physical state is a scattering state.

Required background. One-particle states supplies the invariant shell measure and sharp-state normalization. Multiparticle states supplies the bosonic or fermionic Fock organization used for asymptotic labels.

Helpful background. Interacting Fields, Asymptotic Observables, and Effective Descriptions explains why interacting fields are not simply free fields at finite time. Resonances, Infraparticles, and Limits of Particle Language identifies important failures of the sharp-particle premise.

Wave packets, not particles at a clock time

Work in a Poincaré-invariant vacuum theory with self-adjoint Hamiltonian $H$ on the physical Hilbert space $\mathcal H$. Construct a self-adjoint auxiliary Hamiltonian $H_0$ on an asymptotic Hilbert space $\mathcal H_{\mathrm{as}}$ from the physical masses and stable species, including stable bound states when they occur. Its generalized multiparticle momentum kets are distributional, so the limiting comparison must be made on smooth wave packets

$$|f\rangle_0 =\sum_{\boldsymbol\sigma} \int \prod_{j=1}^n \mathrm d\Pi_j\, f_{\boldsymbol\sigma}(p_1,\ldots,p_n) |p_1,\sigma_1;\ldots;p_n,\sigma_n\rangle_0,$$

where

$$\mathrm d\Pi_j =\frac{\mathrm d^3\mathbf p_j}{(2\pi)^3 2E_j}.$$

The packet is chosen so that different outgoing velocity supports separate as $t\to+\infty$, and different incoming supports separate as $t\to-\infty$. Plane waves never become spatially separated; writing limits directly on sharp momentum kets hides the distributional step. Weinberg formulates the asymptotic condition on smooth energy superpositions for exactly this reason Weinberg 1995, § 3.1, pp. 107–113.

For a smooth packet of compact momentum support, stationary phase concentrates its large-$|t|$ profile near

$$\frac{\mathbf x}{t}=\nabla_{\mathbf p}E_{\mathbf p} =\frac{\mathbf p}{E_{\mathbf p}}.$$

Outside the associated velocity cone, repeated nonstationary-phase integration suppresses the wavefunction. This is concentration, not compact spatial support; locality and spectral estimates are still needed to turn separated cones into an interacting scattering-state theorem.

Møller operators and asymptotic labels

The two Hamiltonians need not act on the same Hilbert space. To give their limiting comparison a precise meaning, suppose the theory supplies a bounded comparison map $J:\mathcal H_{\mathrm{as}}\to\mathcal H$. This is additional, model-dependent data; an interacting QFT is not thereby identified with the free Fock representation. The two-space formulation explicitly retains such a map Güneysu 2024, § 2, p. 1369, PDF. That reference restricts to absolutely continuous spectral subspaces; here $\mathcal H_{\mathrm{as}}$ denotes the chosen particle channels, and existence is a separate assumption to be checked for those channels.

Write $W(t)=e^{iHt}Je^{-iH_0t}$ and, initially on a dense set of packets, define

$$\begin{aligned} \Omega_{\mathrm{in}} &=\underset{t\to-\infty}{\operatorname{s-lim}} e^{+iHt}Je^{-iH_0t},\\ \Omega_{\mathrm{out}} &=\underset{t\to+\infty}{\operatorname{s-lim}} e^{+iHt}Je^{-iH_0t}. \end{aligned}$$

The limit is strong: it asks for convergence of the vector obtained from each packet, not convergence of the operators in norm. Since $\lVert W(t)\rVert\leq\lVert J\rVert$, convergence on a dense packet set extends to all of $\mathcal H_{\mathrm{as}}$. The interacting states are

$$|f,\mathrm{in}\rangle=\Omega_{\mathrm{in}}|f\rangle_0, \qquad |g,\mathrm{out}\rangle=\Omega_{\mathrm{out}}|g\rangle_0.$$

Probability-preserving wave operators require the additional asymptotic normalization

$$\lim_{t\to\mp\infty} \left\lVert Je^{-iH_0t}f\right\rVert_{\mathcal H} =\lVert f\rVert_{\mathrm{as}}, \qquad f\in\mathcal H_{\mathrm{as}}.$$

It suffices to check this condition on the dense packet set. Strong convergence then gives $\lVert\Omega_{\mathrm{in/out}}f\rVert=\lVert f\rVert$; polarization gives preservation of inner products. These maps are isometries, not necessarily surjective. Existence alone does not imply this: for $H=H_0$ on one space and $J=2I$, the limit is $2I$, which doubles every norm.

The limits also intertwine time evolution:

$$e^{-iHs}\Omega_{\mathrm{in/out}} =\Omega_{\mathrm{in/out}}e^{-iH_0s}.$$

Indeed, $e^{-iHs}W(t)=W(t-s)e^{-iH_0s}$, and a finite shift of $t$ leaves either limit unchanged. Differentiating this group identity gives $H\Omega f=\Omega H_0f$ for $f\in D(H_0)$; it also ensures $\Omega f\in D(H)$.

Thus an in or out label says how the same interacting history looks when resolved into separated particles at one temporal end. The two sets are not separate physical universes. Their overlap defines the scattering operator on the auxiliary asymptotic Hilbert space,

$$S=\Omega_{\mathrm{out}}^\dagger\Omega_{\mathrm{in}}.$$

For these isometric wave operators, if their ranges coincide, $S$ is unitary on the asymptotic Hilbert space, and the common range is the interacting scattering subspace. If, more strongly, that range exhausts the relevant physical Hilbert space, the theory is asymptotically complete. Neither conclusion follows merely from having written the symbols $\Omega_{\mathrm{in/out}}$; Weinberg’s physical construction and normalization argument appear in Weinberg 1995, §§ 3.1–3.2, pp. 107–116.

A sufficient Cook estimate

Here is a convenient sufficient domain assumption for a Cook estimate: $J D(H_0)\subset D(H)$, with $H,H_0$ self-adjoint and $J$ bounded as above. Choose a dense packet set $\mathcal D\subset D(H_0)$. For $f\in\mathcal D$, the product rule is then valid in norm and gives

$$\frac{\mathrm d}{\mathrm dt}W(t)f =i e^{iHt}(HJ-JH_0)e^{-iH_0t}f.$$

The domain condition matters because the generators are unbounded. The standard product rule and common-space Cook criterion are stated in Richard 2016, Lemma 5.1.7, p. 65, and Proposition 5.2.4, p. 71, PDF; inserting $J$ gives the following two-space argument.

If every $f\in\mathcal D$ satisfies

$$\int_0^\infty\mathrm dt\, \left\lVert(HJ-JH_0)e^{-iH_0t}f\right\rVert_{\mathcal H}<\infty,$$

then, for $t>s\geq0$,

$$\lVert W(t)f-W(s)f\rVert_{\mathcal H} \leq\int_s^t\mathrm d\tau\, \left\lVert(HJ-JH_0)e^{-iH_0\tau}f\right\rVert_{\mathcal H} \longrightarrow0.$$

Thus the outgoing limit exists on $\mathcal D$ and extends by the uniform bound to $\mathcal H_{\mathrm{as}}$. The incoming argument uses integrability over $(-\infty,0]$. This supplies existence, while the asymptotic norm condition above still supplies isometry. In the special common-space case $J=I$, the defect reduces to $H-H_0$.

Cook’s method originates in Cook 1957, pp. 82–87. This sufficient estimate is not a general QFT existence theorem: the comparison map, its domains, and the needed decay remain physical and mathematical work. In particular, the Haag–Ruelle construction linked below proves scattering limits by local operators and separated velocity supports, rather than assuming that every QFT admits this bounded-$J$ description.

A two-packet separation check

Let two massive scalar packets have momentum supports concentrated near $\mathbf p_1$ and $\mathbf p_2$, with group velocities

$$\mathbf v_j=\frac{\mathbf p_j}{E_j}, \qquad \mathbf v_1\neq\mathbf v_2.$$

Under free evolution their centers separate as

$$\Delta\mathbf x(t) \simeq (\mathbf v_1-\mathbf v_2)t.$$

For a massive theory with sufficiently short-range effective interactions, matrix elements of the interaction between these separated packets can decay as $|t|$ grows. This is the physical mechanism behind the free comparison. Equal velocity support is a warning: the packets need not separate, and bound-state or threshold channels may require a different asymptotic description.

The check also explains why the masses in $H_0$ must be the observed pole masses. If a bare mass were used instead, the phase difference between $e^{-iHt}$ and $e^{-iH_0t}$ would grow linearly with time, preventing the limit even for a stable particle.

When the construction stops

Ordinary Fock in/out states require more than a field appearing in a Lagrangian.

Situation	What fails	Correct direction
unstable resonance	no normalizable eigenstate with a real isolated mass shell	use resonance poles and process-dependent observables
charged particle with unscreened massless radiation	the sharp mass pole and finite-photon Fock asymptotics can fail	specify inclusive observables or a controlled dressed-state construction
confined colored field	the field does not create a physical isolated colored state	use color-singlet asymptotic states and the model-specific hadronic description
time-dependent or curved background without common asymptotic regions	no single preferred free particle comparison at both ends	use the appropriate in-in or curved-spacetime particle framework
incomplete scattering theory	wave-operator ranges do not exhaust the physical space	keep $S$ restricted to the constructed scattering subspace

In Srednicki’s one-loop scalar φ³ example, the massless limit brings the one-particle pole to the multiparticle threshold and makes the usual on-shell residue condition ill defined Srednicki 2006 draft, § 27, p. 172, PDF. This illustrates a failure of the ordinary isolated-pole premise.

The theorem-level construction of scattering states under mass-gap and locality hypotheses belongs to Haag–Ruelle Scattering-State Construction. Wave Operators and Asymptotic Fields treats the operator limits and completeness questions more precisely.

Common pitfalls

“Interacting particles become literally free at a finite time.” The statement is asymptotic and packet-dependent. It compares dynamics in a limit; it does not switch the interaction Hamiltonian off at some universal clock time.

“A stable pole is enough to prove an S-matrix exists.” A pole identifies a candidate one-particle sector. Multi-particle wave operators, separation estimates, and the relevant completeness statement are additional requirements.

“In and out states require two separate physical Hilbert spaces.” They are two maps from the same asymptotic channel space into the physical space. The S-matrix compares their images; the auxiliary channel space itself need not be identified with the physical space.

“Every state must be asymptotic for $S$ to be useful.” A unitary S-matrix can be defined on a scattering subspace without proving full asymptotic completeness, provided its domain and range are stated.

Check your understanding

1. Why is $\Omega_{\mathrm{in}}$ defined with $t\to-\infty$ while $\Omega_{\mathrm{out}}$ uses $t\to+\infty$?

   Answer

   Incoming packets are required to resemble freely evolving separated particles before the collision; outgoing packets are required to do so after it. The opposite limits encode those two boundary conditions on the same interacting dynamics.

2. Suppose $H_0$ uses mass $m_0$ while the stable particle has physical energy $E_{\mathbf p}=\sqrt{\mathbf p^2+m^2}$. What obstructs the Møller limit when $m_0\neq m$?

   Answer

   Even on a narrow packet, the two evolutions accumulate a relative phase approximately $[E_{\mathbf p}(m)-E_{\mathbf p}(m_0)]t$. It does not approach a time-independent vector as $|t|\to\infty$. The physical mass must therefore be part of the asymptotic Hamiltonian.

Continue

S-Matrix and T-Matrix Normalization turns the overlap of the two asymptotic bases into a convention-explicit amplitude. If the obstruction is infrared rather than short-range, continue instead to Dressed States and Infrared-Finite Scattering. If it is instability, use Resonance Poles, Riemann Sheets, and Unstable States.

References

• Cook, James M. “Convergence to the Møller Wave-Matrix.” Journal of Mathematics and Physics 36, no. 1–4 (1957): 82–87. DOI.
• Güneysu, Batu. “Asymptotic Equivalence of Identification Operators in Geometric Scattering Theory.” Documenta Mathematica 29 (2024): 1367–1379. DOI. Open PDF.
• Richard, Serge. Hilbert Space Methods for Quantum Mechanics. Nagoya University lecture notes, spring 2016, Chapter 5, “Scattering Theory.” Course page. Open PDF.
• Srednicki, Mark. Quantum Field Theory. Author’s prepublication draft, 2006, § 27, p. 172; pagination of this draft. Author’s draft page and published-edition errata. Open PDF.
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI.