Asymptotic completeness in quantum field theory

渐近完备性是散射理论的一个核心概念。它要求给定物理系统的所有状态都能用粒子来解释。经过几十年的研究，我们对这一基本性质的理解仍然相当有限：只有在量子力学中才可以控制渐近完备性，甚至只有在具有二次色散关系的粒子系统中才可以。在非相对论量子场论中，我们已经很好地理解了涉及一个电子和光子的碰撞过程，例如瑞利散射和康普顿散射，但几个电子的碰撞却很少受到关注。此外，红外问题和质量重整化通常需要对模型参数进行限制性假设。在相对论量子场论和量子自旋系统中，渐近完备性只在几种特殊情况下才已知。本项目的目标是弥合相对论和非相对论散射理论之间的差距。在非相对论方面，我们的目标是更好地理解涉及一个和几个电子的散射过程。将发展新的泛函分析方法来处理电子的非二次色散关系，而不需要限制性的假设。红外问题的处理需要自伴随算子的光谱理论取得重大进展，特别是在连续光谱中嵌入特征值的情况下。起源于相对论量子场论的电子的次真空图像将在非相对论设置中实现。对于相对论量子场论和量子自旋系统，我们将制定渐近完备性的一般准则，并在模型中进行检验。

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.