The many-body problem is usually approached from one of two perspectives: the first originates from an action and is based on Feynman diagrams, the second is centered around a Hamiltonian and deals with quantum states and operators. The connection between results obtained in either way is made through spectral (or Lehmann) representations, well known for two-point correlation functions. Here, we complete this picture by deriving generalized spectral representations for multipoint correlation functions that apply in all of the commonly used many-body frameworks: the imaginary-frequency Matsubara and the real-frequency zero-temperature and Keldysh formalisms. Our approach separates spectral from time-ordering properties and thereby elucidates the relation between the three formalisms. The spectral representations of multipoint correlation functions consist of partial spectral functions and convolution kernels. The former are formalism independent but system specific; the latter are system independent but formalism specific. Using a numerical renormalization group method described in the accompanying paper, we present numerical results for selected quantum impurity models. We focus on the four-point vertex (effective interaction) obtained for the single-impurity Anderson model and for the dynamical mean-field theory solution of the one-band Hubbard model. In the Matsubara formalism, we analyze the evolution of the vertex down to very low temperatures and describe the crossover from strongly interacting particles to weakly interacting quasiparticles. In the Keldysh formalism, we first benchmark our results at weak and infinitely strong interaction and then reveal the rich real-frequency structure of the dynamical mean-field theory vertex in the coexistence regime of a metallic and insulating solution.

中文翻译：

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多点相关函数：光谱表示和数值评估

多体问题通常从两个角度之一来解决：第一个源自动作并基于费曼图，第二个以哈密顿量为中心并处理量子态和算子。以任何一种方式获得的结果之间的联系是通过光谱（或莱曼）表示建立的，众所周知，两点相关函数。在这里，我们通过推导适用于所有常用多体框架的多点相关函数的广义谱表示来完成这幅图：虚频 Matsubara 和实频零温度和 Keldysh 形式。我们的方法将频谱与时间排序属性分开，从而阐明了三种形式之间的关系。多点相关函数的谱表示由部分谱函数和卷积核组成。前者独立于形式主义，但特定于系统；后者是独立于系统但特定于形式主义的。使用随附论文中描述的数值重整化群方法，我们展示了选定量子杂质模型的数值结果。我们专注于为单杂质 Anderson 模型和单波段哈伯德模型的动态平均场理论解获得的四点顶点（有效相互作用）。在 Matsubara 形式主义中，我们分析了顶点到极低温度的演化，并描述了从强相互作用粒子到弱相互作用准粒子的交叉。在 Keldysh 形式主义中，