Mourre Theory and Asymptotic Observables in Local Relativistic Quantum Field Theory

We prove the convergence of Araki–Haag detectors in any Haag–Kastler quantum field theory with an upper and lower mass gap. We cover the case of a single Araki–Haag detector on states of bounded energy, which are selected from the absolutely continuous part of the energy-momentum spectrum sufficiently close to the lower boundary of the multi-particle spectrum. These states essentially encompass those states in the multi-particle spectrum lying below the three-particle threshold. In our proof, we draw on insights from proofs of asymptotic completeness in quantum mechanics. Notably, we apply Mourre’s conjugate operator method for the first time within the framework of Haag–Kastler quantum field theory. Furthermore, we discuss applications of our findings for the problem of asymptotic completeness in local relativistic quantum field theory.