Asymptotic completeness in interacting relativistic QFT
For a specified massive, local, interacting relativistic QFT in dimensions, prove that the vacuum and scattering states of stable particles span the chosen physical sector; formulate separate completeness criteria when massless particles or charges inval
Problem statement
For a specified massive, local, interacting relativistic QFT in dimensions, prove that the vacuum and scattering states of stable particles span the chosen physical sector; formulate separate completeness criteria when massless particles or charges invalidate this particle description.
Lean 4
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Background
In a massive particle-scattering setting, asymptotic completeness says that scattering states built from the specified stable particles exhaust the physical sector under consideration. It is stronger than existence of an S-matrix on a scattering subspace. Different superselection sectors or theories without a complete particle interpretation require an explicitly different target.
What is known: Haag–Ruelle theory constructs scattering states in massive theories under spectral assumptions; existence of these states does not prove completeness. Complete results exist for particular factorizing -dimensional models. Duell–Dybalski establish results for massive wedge-local theories in higher dimensions, which are not the same target as strictly local theories. Earlier constructive results in restricted energy sectors or space-cutoff models must not be read as a general completeness theorem for or interacting 2D/3D QFT. Many-body Schrödinger completeness theorems likewise concern a different setting.
The four-dimensional local target can be pursued by conditional axiomatic theorems as well as explicit constructions (QF1/QF3). With massless quanta, especially charged infraparticle sectors, ordinary finite-particle Fock completeness is not the right unrestricted statement; QF7 tracks alternative asymptotic observables and sectors.
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
- [1]Haag, 'Local Quantum Physics: Fields, Particles, Algebras' (Springer, 2nd ed., 1996) link
- [2]Buchholz, 'Collision theory for massless bosons', Comm. Math. Phys. 52 (1977) link
- [3]Duell & Dybalski, 'Asymptotic Completeness in a Class of Massive Wedge-Local Quantum Field Theories in Any Dimension', CMP (2023) link
Recorded progress
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