Particles, Scattering Theory, and Infrared Structure

A relativistic theory need not admit an ordinary particle S-matrix. This chapter derives scattering states from spectral and locality assumptions, then identifies what changes for massless radiation and charged infrared sectors.

Enter this chapter

Start with the one-particle subspace and Haag–Ruelle construction before approaching LSZ or completeness. Helpful background. LSZ reduction, poles, residues, and stable states supplies the perturbative amplitude language compared with the operator limits here. Resonances, infraparticles, and the limits of particle language distinguishes spectral eigenstates from poles reached only by continuation. Elasticity and factorization hypotheses provides the special two-dimensional setting in which strong completeness results are known.

Topics

1. Particles, Mass-Shell Spectrum, and One-Particle Subspaces — Identifies stable particles through isolated mass-shell spectral subspaces.
2. Haag–Ruelle Scattering-State Construction — Constructs scattering states using separated velocity supports.
3. Wave Operators and Asymptotic Fields — Defines asymptotic fields and the ranges of wave operators.
4. LSZ Reduction and Amputated Distributions — Relates suitable asymptotic particles to amputated correlation functions.
5. Araki–Haag Detectors and Particle Weights — Extracts asymptotic particle information from local observables.
6. Asymptotic Completeness: Definitions and Known Models — Distinguishes existence of scattering states from exhaustion of the physical space.
7. Massless Scattering and Radiation Fields — Treats radiation fields when massive-particle estimates do not apply.
8. Infraparticles and Velocity Superselection — Explains continuous charged spectra and velocity-dependent sectors.
9. Gauss-Law Charges and Infrared Sectors — Connects long-range electric flux to charge localization and representation.
10. Dressed, Inclusive, and Algebraic Infrared Observables — Compares three different ways of posing infrared-finite questions.
11. Scattering Analyticity, Crossing, and Rigorous Bounds — States the analytic domains and hypotheses behind rigorous amplitude constraints.

How the ideas fit together

The massive route runs from a stable one-particle subspace through scattering limits to isometric wave operators; surjectivity is an additional completeness claim. These steps are developed in Buchholz and Dybalski 2024, § 2, Theorem 1 (PDF). Local detectors provide a different asymptotic description Araki and Haag 1967, pp. 77–91. Gauss-law sectors can obstruct a sharp charged mass shell Buchholz 1986, pp. 331–334; analyticity also has a proved domain, rather than an unrestricted crossing rule Bros, Epstein, and Glaser 1965, pp. 240–264.

What the hypotheses actually imply

Hypotheses, licensed conclusions, and decisive failure tests in relativistic scattering theory

Object and domain	Spectral or scattering hypotheses	Licensed conclusion	Excluded converse or extension	Adversarial check
Joint spectral projection on a positive-energy mass hyperboloid	The shell is nonzero and isolated in the real translation spectrum; the Poincaré representation is strongly continuous.	A stable one-particle subspace carrying Wigner representations.	A resonance pole or threshold enhancement does not imply such a projection.	Replace the shell by a second-sheet pole: the spectral projection is zero, so the construction stops.
Finite Haag–Ruelle products applied to the vacuum	A unique gapped Reeh–Schlieder vacuum, isolated massive shell, spectrally filtered smooth almost-local operators with nonzero one-particle projection, smooth compact momentum packets, and pairwise disjoint velocity supports.	Norm limits of vacuum product vectors; asymptotic factorization gives Fock inner products for the constructed species.	Existence of these states does not imply that they span the physical Hilbert space.	Overlap two velocity supports, or remove shell isolation: the commutator estimate no longer proves convergence.
Møller maps from asymptotic Fock space	Haag–Ruelle limits exist and their inner products factorize on the finite-particle core.	Isometric wave operators and a scattering operator on the appropriate asymptotic ranges.	An isometry is not automatically unitary; range density is an additional completeness theorem.	Add a nonzero orthogonal bound-state or topological sector: the constructed map remains isometric but is not onto.
LSZ-smeared time-ordered distributions	Isolated stable-particle poles with nonzero residue, asymptotic states, tempered boundary values, and controlled amputation.	On-shell scattering matrix elements as limits of amputated distributions.	The formula does not extend unchanged to unstable particles or charged infraparticles without a sharp pole.	Apply electron LSZ in charged QED after the pole has become a continuous threshold: the external-state limit is absent.
Almost-local energy-decreasing detectors and particle weights	Bounded-energy states, energy–momentum transfer away from the forward cone, propagation estimates, and an appropriate large-time topology.	Asymptotic velocity-sensitive functionals or weights, sometimes without normalizable particle vectors.	A detector limit need not produce a sharp-mass state or a complete particle basis.	Use a strictly local positive operator with nonzero vacuum response: persistent vacuum contamination defeats particle selectivity.
Null radiation fields and charged Gauss-law sectors	For neutral radiation, suitable light-cone propagation and decay; for charge, the Gauss-law flux relation and noncompact localization are retained.	Neutral asymptotic radiation fields, or charged infrared sectors distinguished by long-range data.	Neither conclusion supplies an ordinary compactly localized charged Fock particle.	Insert a long-range charged field into a neutral radiation theorem: its localization and decay estimates fail.
Massive scattering amplitudes as analytic boundary values	Locality, positive energy, stable LSZ particles, a mass gap, unitarity, a specified complex domain, and polynomial growth where a bound requires it.	Qualified crossing continuations, dispersion relations, and rigorous bounds inside the proved domain.	There is no automatic global crossing identity or massless forward Froissart bound.	Add an unsubtracted massless exchange pole at zero momentum transfer: the gap and bounded-domain argument fail.

Structured table data (JSON).

Review the chapter

If a wave operator is isometric, must every physical state be a scattering state?

Solution

No. Isometry preserves inner products on the asymptotic input space. Its range may be a proper closed subspace of the physical Hilbert space. Asymptotic completeness requires the appropriate range to exhaust the sector under consideration.

Where to go next

Continue to local observable algebras for a formulation that remains meaningful when ordinary asymptotic particles are unavailable.

References

• Araki, H., and Haag, R. (1967). “Collision Cross Sections in Terms of Local Observables.” Communications in Mathematical Physics 4, 77–91. DOI.
• Bros, J., Epstein, H., and Glaser, V. (1965). “A Proof of the Crossing Property for Two-Particle Amplitudes in General Quantum Field Theory.” Communications in Mathematical Physics 1, 240–264. DOI.
• Buchholz, D. (1986). “Gauss’ Law and the Infraparticle Problem.” Physics Letters B 174, 331–334. DOI.
• Buchholz, D., and Dybalski, W. (2024). “Scattering in Relativistic Quantum Field Theory: Basic Concepts, Tools, and Results.” arXiv:math-ph/0509047, version 3, 3 November 2024. Versioned preprint. Open PDF.