We propose a mathematically rigorous construction of the scattering matrix and the interacting fields in models of relativistic perturbative quantum field theory with massless fields and long-range interactions. We consider quantum electrodynamics and a certain model of interacting scalar fields in which the standard definition of the scattering matrix is not applicable because of the infrared problem. We modify the Bogoliubov construction using the ideas of Dollard, Kulish and Faddeev. Our modified scattering matrix and modified interacting fields are constructed with the use of the adiabatic limit which is expected to exist in arbitrary order of perturbation theory. In the paper, we prove this assertion in the case of the first- and the second-order corrections to the modified scattering matrix and the first-order corrections to the modified interacting fields. We study the physical properties of our construction. We conclude that the electrons and positrons are always surrounded by irremovable clouds of photons. Moreover, the physical energy-momentum operators do not coincide with the standard ones and their joint spectrum does not contain the mass hyperboloid.

中文翻译：

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微扰量子场论中的红外问题

我们在具有无质量场和长程相互作用的相对论微扰量子场论模型中提出了散射矩阵和相互作用场的数学严格构造。我们考虑量子电动力学和相互作用标量场的某种模型，其中散射矩阵的标准定义由于红外问题而不适用。我们使用 Dollard、Kulish 和 Faddeev 的思想修改了 Bogoliubov 结构。我们修改的散射矩阵和修改的相互作用场是使用绝热极限构建的，绝热极限预计存在于任意阶摄动理论中。在论文中，我们在修改后的散射矩阵的一阶和二阶校正以及修改的相互作用场的一阶校正的情况下证明了这一断言。我们研究建筑的物理特性。我们得出结论，电子和正电子总是被不可移动的光子云包围。此外，物理能量-动量算子与标准算子不一致，其联合谱不包含质量双曲面。