Asymptotic completeness in quantum field theory

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The property of asymptotic completeness is a central concept of scattering theory. It requires that all states of a given physical system can be interpreted in terms of particles. After decades of research our understanding of this fundamental property is still rather limited: Only in quantum mechanics asymptotic completeness is under control, and even there only for systems of particles with quadratic dispersion relations. In non-relativistic quantum field theory we have a good understanding of collision processes involving one electron and photons, e.g. of Rayleigh and Compton scattering, but collisions of several electrons have received much less attention. Also, infrared problems and mass renormalisation often call for restrictive assumptions on the model parameters. In relativistic quantum field theory and for quantum spin systems asymptotic completeness is known only in several special cases. It is the goal of the present project to bridge the gap between relativistic and non-relativistic scattering theory. On the non-relativistic side, our aim is to better understand scattering processes involving one and several electrons. New functional analytic methods will be developed to deal with non-quadratic dispersion relations of the electrons without restrictive assumptions. The treatment of infrared problems requires major advances in spectral theory of self-adjoint operators, especially in the context of eigenvalues embedded in the continuous spectrum. The infravacuum picture of the electron, originating from relativistic quantum field theory, will be implemented in the non-relativistic setting. For relativistic quantum field theories and for quantum spin systems we will formulate general criteria for asymptotic completeness and test them in models.