👤 Janik Kruse

🏛️ Adam Mickiewicz University, Poland

📝 Asymptotic Completeness in Quantum Scattering Theory - Mourre Theory and Asymptotic Observables in Relativistic QFT

Abstract

The goal of scattering theory is to understand how a quantum system of interacting particles evolves asymptotically. A central concept in scattering theory is asymptotic completeness, which asserts that every state can be decomposed into a bound and a scattering state. While asymptotic completeness is well understood in non-relativistic quantum mechanics, it remains an open and challenging problem in local relativistic quantum field theory due to various conceptual and technical complications. In quantum mechanics, many proofs of asymptotic completeness depend on the convergence of asymptotic observables. In QFT, Huzihiro Araki and Rudolf Haag (1967, doi:10.1007/BF01645754) identified particle detectors as natural asymptotic observables. They demonstrated the convergence of these detectors on scattering states, but the convergence on arbitrary states, which is relevant for establishing asymptotic completeness, remains unproven. Relatively recently, Wojciech Dybalski and Christian Gérard (2014, doi:10.1007/s00220-014-2069-y) made progress in this area by adapting quantum mechanical propagation estimates to QFT. They covered coincident arrangements of multiple detectors sensitive to particles with distinct velocities, but they did not manage to establish the convergence of a single detector due to a missing low-velocity estimate. Typically, such an estimate is proved through Mourre's conjugate operator method, a powerful tool in quantum mechanics that has so far resisted adaptation to QFT. In a recent paper (2024, doi:10.1007/s00220-024-05091-7), we succeeded in applying Mourre's method to QFT through Haag-Ruelle scattering theory. This allowed us to prove the convergence of a single Araki-Haag detector on states of bounded energy below the three-particle threshold.