Abstract
We consider the problem of existence of asymptotic observables in local relativistic theories of massive particles. Let \({\tilde{p}_1}\) and \({\tilde{p}_2}\) be two energy-momentum vectors of a massive particle and let \({\Delta}\) be a small neighbourhood of \({\tilde{p}_1 + \tilde{p}_2}\) . We construct asymptotic observables (two-particle Araki–Haag detectors), sensitive to neutral particles of energy-momenta in small neighbourhoods of \({\tilde{p}_1}\) and \({\tilde{p}_2}\) . We show that these asymptotic observables exist, as strong limits of their approximating sequences, on all physical states from the spectral subspace of \({\Delta}\) . Moreover, the linear span of the ranges of all such asymptotic observables coincides with the subspace of two-particle Haag–Ruelle scattering states with total energy-momenta in \({\Delta}\) . The result holds under very general conditions which are satisfied, for example, in \({\lambda{\phi}_{2}^{4}}\) . The proof of convergence relies on a variant of the phase-space propagation estimate of Graf.
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Communicated by Y. Kawahigashi
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Dybalski, W., Gérard, C. Towards Asymptotic Completeness of Two-Particle Scattering in Local Relativistic QFT. Commun. Math. Phys. 326, 81–109 (2014). https://doi.org/10.1007/s00220-013-1831-x
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DOI: https://doi.org/10.1007/s00220-013-1831-x