Haag–Ruelle Scattering-State Construction

Haag–Ruelle theory constructs incoming and outgoing multiparticle states directly from local observables and an isolated massive one-particle shell. Approximants whose wave packets have disjoint velocity supports separate linearly in time; locality makes their commutators decay, and a Cook-type estimate gives strong limits. The theorem proves existence and Fock structure of scattering states, not that these states exhaust the physical Hilbert space.

Required background. Jost points, edge-of-the-wedge, and locality supplies spacelike commutation; clustering, vacuum uniqueness, and mass-gap implications supplies decay and spectral separation; and particles and one-particle subspaces supplies the stable shell.

Helpful background. LSZ reduction: poles, residues, and stable external states gives the physical scattering bridge.

Haag–Ruelle approximants

Work in a local massive vacuum theory with unique vacuum, positive energy, and a stable isolated mass hyperboloid $H_m^+$. Choose a local operator $A$ for which $P_1A\Omega\neq0$, where $P_1=E(H_m^+)$. After spacetime smearing, its energy–momentum transfer can be confined to a small neighborhood of the shell; the resulting operator is almost local.

For a smooth compact momentum packet $\widetilde f$ on $H_m^+$, let

$$f_t(\mathbf x) =\int\frac{\mathrm d^3\mathbf p}{(2\pi)^3\,2\omega_{\mathbf p}} \widetilde f(\mathbf p)e^{-i\omega_{\mathbf p}t+i\mathbf p\cdot\mathbf x}, \qquad \omega_{\mathbf p}=\sqrt{\mathbf p^2+m^2},$$

and form

$$A_t(f)=\int\mathrm d^3\mathbf x\, f_t(\mathbf x)\,A(t,\mathbf x),$$

with the conventional energy-smearing or Klein–Gordon derivative factor chosen so that $A_t(f)\Omega=P_1A(f)\Omega$ is independent of $t$. The precise normalization is secondary to the spectral projection and domain statement.

For operators $A_k$ and packets $f_k$ with pairwise disjoint velocity supports,

$$\Gamma(f_k)= \left\{\frac{\mathbf p}{\omega_{\mathbf p}}: \mathbf p\in\operatorname{supp}\widetilde f_k\right\},$$

the outgoing state is

$$\Psi^{\mathrm{out}} =\lim_{t\to+\infty}A_{1,t}(f_1)\cdots A_{n,t}(f_n)\Omega,$$

and the incoming state uses $t\to-\infty$.

Why the strong limit exists

Stationary-phase estimates concentrate $f_{k,t}$ near $\mathbf x=t\mathbf v$ with $\mathbf v\in\Gamma(f_k)$. Disjoint velocity supports therefore make the localization regions spacelike separated by a distance proportional to $|t|$. Exact locality for local approximants and rapid tails for almost-local operators imply commutators that decrease faster than any inverse power:

$$\|[A_{j,t}(f_j),A_{k,t}(f_k)]\|=O(|t|^{-N}) \quad\text{for every }N.$$

Differentiate the product approximant. The one-particle parts are time-independent, so after commuting derivatives through the other factors, the derivative is bounded by integrable commutator tails. Cook’s criterion makes the family Cauchy in norm and gives the strong limit. This is the central argument of Ruelle 1962, pp. 147–163, extending Haag’s local-observable asymptotic construction Haag 1958, pp. 669–673.

Clustering and locality also give the Fock inner-product formula: scalar products of scattering states are sums over pairings of their one-particle scalar products. The limit depends only on $P_1A_k(f_k)\Omega$, not on the chosen interpolating operators. Bose or Fermi statistics enter through the corresponding permutation signs.

The convergence statement is first a statement about vectors on a controlled finite-energy domain, not norm convergence of the operators $A_t(f)$ on all of $\mathcal H$. Energy bounds keep every product approximant in a common domain, and the inner-product formula extends the resulting multilinear map by continuity to the finite-particle Fock space. Independence of the interpolating operator is checked by replacing one $A_k$ with a difference whose one-particle projection vanishes: spectral separation and the same commutator estimates make that difference disappear in the scattering limit. Thus the limit is attached to the one-particle wave packet, not to a chosen field coordinate.

Two particles in P(φ)₂

In a massive constructed $P(\phi)_2$ model, choose local interpolating operators $A_1,A_2$ coupling to the stable scalar and packets supported in disjoint velocity intervals. The derivative estimate above is integrable in $1+1$ dimensions, so

$$\Psi^{\mathrm{out}}_{12} =\lim_{t\to+\infty}A_{1,t}(f_1)A_{2,t}(f_2)\Omega$$

exists. Its norm is

$$\|\Psi^{\mathrm{out}}_{12}\|^2 =\|\psi_1\|^2\|\psi_2\|^2 +|\langle\psi_1,\psi_2\rangle|^2, \qquad \psi_k=P_1A_k(f_k)\Omega,$$

for identical bosons. With disjoint momentum supports the cross term vanishes. This gives the rigorous construction underlying in and out states without calculating a cross section.

An independent check is geometric: if the closed velocity supports have separation $\delta>0$, the packet centers differ by at least $\delta|t|$, whereas each packet’s nonstationary tail decreases rapidly. The commutator region is therefore asymptotically spacelike.

Failure test: where the proof stops

If velocity supports overlap, the localization centers need not separate and the displayed commutator estimate is unavailable. The limit might exist by model-specific methods, but this proof cannot be cited. If the mass shell is not isolated or regular, the approximant cannot keep a fixed one-particle component while suppressing continuum contamination. Charged infraparticles fail precisely here.

Existence of every finite Haag–Ruelle product only defines a scattering subspace. Equality of that subspace with $\mathcal H$ is asymptotic completeness, a separate theorem addressed later.

Exercises

Why is disjoint momentum support stronger than needed, while disjoint velocity support is the geometric condition used in the proof?

Solution

Large-time propagation follows group velocity $\mathbf v=\nabla_{\mathbf p}\omega_{\mathbf p}$. Two disjoint momentum sets can, for more general dispersion relations or species, have overlapping velocity images; conversely distinct momentum points can be organized directly by separated velocity sets. It is separation of the spacetime rays $\mathbf x\sim t\mathbf v$ that makes locality useful.

References

• Haag, Rudolf. 1958. “Quantum Field Theories with Composite Particles and Asymptotic Conditions.” Physical Review 112: 669–673. DOI.
• Ruelle, David. 1962. “On the Asymptotic Condition in Quantum Field Theory.” Helvetica Physica Acta 35: 147–163. Digitized article.