Asymptotic Completeness: Definitions and Known Models

Asymptotic completeness is surjectivity, not merely existence: the ranges of the appropriate incoming or outgoing channel wave operators must exhaust the physical sector being claimed. In a massive vacuum sector this often means Fock scattering states span the Hilbert space. With bound states, superselection sectors, massless radiation, or infraparticles, the asymptotic space and the word “complete” must be reformulated before a theorem is stated. A factorizing two-dimensional model provides the chapter’s exact elasticity and factorization application.

Required background. Existence, construction, reconstruction, and continuum claims fixes the claim strength; Haag–Ruelle construction gives state existence; wave operators and asymptotic fields gives the range question; and Araki–Haag detectors and particle weights supplies a weaker asymptotic observable concept.

Helpful background. Integrable-QFT casebook, exact data, and limits gives the model class in which a full theorem is available.

Completeness is relative to channels and sectors

For one massive neutral species, let

$$\Omega^{\mathrm{in/out}}:\mathcal F(\mathcal H_1)\to\mathcal H$$

be the Haag–Ruelle wave operators. Standard vacuum-sector asymptotic completeness is

$$\operatorname{Ran}\Omega^{\mathrm{in}} =\operatorname{Ran}\Omega^{\mathrm{out}} =\mathcal H.$$

If stable bound states form additional particle species, their one-particle spaces must be included in the asymptotic Fock space. If the theory has superselection sectors, one states completeness sector by sector or sums specified charge channels. For several cluster decompositions, channel wave operators replace a single Fock map. A theorem that omits a possible bound or topological sector has not proved completeness.

Massless neutral radiation may require Fock spaces over null one-particle subspaces and different propagation estimates. Charged infraparticles may not admit a sharp-mass Fock space at all; completeness can instead concern asymptotic electromagnetic fields, inclusive detectors, or particle-weight decompositions. These are inequivalent definitions, not weaker spellings of one universal equation.

Existence, range equality, and completeness

Three claims must remain separate:

1. Existence: each finite collection of one-particle packets has an in/out limit.
2. Scattering-space equality: incoming and outgoing ranges coincide, making the scattering operator unitary on that space.
3. Asymptotic completeness: the common scattering range equals the specified physical Hilbert sector.

The first does not imply the second or third. Detector completeness is different again: a family of asymptotic observables may separate or generate a prescribed scattering subspace without proving that all vectors are conventional multiparticle states.

An independent check uses projections $P^{\mathrm{in/out}}=\Omega^{\mathrm{in/out}}(\Omega^{\mathrm{in/out}})^*$. Completeness is exactly $P^{\mathrm{in}}=P^{\mathrm{out}}=1$ on the claimed sector. Norm preservation gives only $(\Omega^\#)^*\Omega^\#=1$ on the asymptotic space.

A proven factorizing-model theorem

Lechner’s two-dimensional construction begins with a regular factorizing two-particle scattering function $S_2$, builds wedge-local fields and nontrivial local algebras, and identifies ordered rapidity scattering states. For $n$ particles, incoming and outgoing states are ordered products of Zamolodchikov–Faddeev creation operators. Varying ordered smooth rapidity packets gives a total set in each $n$-particle subspace.

Consequently the direct sum of these scattering states is dense in the constructed Hilbert space. Lechner 2008, § 6, Proposition 6.2, pp. 33–34 of the open manuscript proves asymptotic completeness for the models with regular scattering functions; Theorem 6.3, pp. 34–35 computes the resulting factorized $S$-matrix.

The theorem is model-class specific. It depends on the constructed Hilbert-space decomposition, regularity of $S_2$, locality obtained through modular nuclearity, and totality of ordered rapidity products. It does not prove that every integrable Lagrangian has been constructed or that general four-dimensional QFT is asymptotically complete.

Known scope and open boundaries

Completeness has been proved for important free theories and selected interacting low-dimensional constructive or factorizing models. There is no theorem deriving ordinary massive Fock completeness for every local QFT from the Haag–Kastler or Wightman axioms alone. Confinement, topological charges, unstable excitations, and long-range gauge forces each change or obstruct the conventional statement.

Araki–Haag detectors can establish precise range results in restricted energy windows—for example, identifying two-particle scattering states selected by two counters—without yielding global completeness. Such partial theorems should be reported as energy- and channel-local results.

Adversarial test: the omitted sector

Suppose every finite-particle incoming state has been constructed, but the theory also has a stable topological soliton sector not included in $\mathcal H_1$. The wave operator is still an isometry and its range contains all states it was designed to create. Its orthogonal complement can contain the soliton sector. Without proving that complement zero, completeness remains unproved.

The same logic applies to bound states: either add each stable bound species to the channel space and prove totality, or state completeness only for the restricted scattering sector.

Exercises

Let $V:\mathcal K\to\mathcal H$ be an isometry. Which operator equality is equivalent to surjectivity?

Solution

Isometry gives $V^*V=1_{\mathcal K}$. Surjectivity is equivalent to $VV^*=1_{\mathcal H}$, because $VV^*$ is the orthogonal projection onto the closed range of $V$. Confusing these two identities is exactly the isometry/completeness error.

References

• Araki, Huzihiro, and Rudolf Haag. 1967. “Collision Cross Sections in Terms of Local Observables.” Communications in Mathematical Physics 4: 77–91. DOI.
• Lechner, Gandalf. 2008. “Construction of Quantum Field Theories with Factorizing S-Matrices.” Communications in Mathematical Physics 277: 821–860. DOI. Open manuscript.