Title: Review and concrete description of the irreducible unitary representations of the universal cover of the complexified Poincaré group Show Chinese title

Title: 对复化庞加莱群的万有覆盖的不可约酉表示的综述和具体描述

Abstract: We give a pedagogical presentation of the irreducible unitary representations of $\mathbb{C}^4\rtimes\mathbf{Spin}(4,\mathbb{C})$, that is, of the universal cover of the complexified Poincar\'e group $\mathbb{C}^4\rtimes\mathbf{SO}(4,\mathbb{C})$. These representations were first investigated by Roffman in 1967. We provide a modern formulation of his results together with some facts from the general Wigner-Mackey theory which are relevant in this context. Moreover, we discuss different ways to realize these representations and, in the case of a non-zero "complex mass", we give a detailed construction of a more explicit realization. This explicit realization parallels and extends the one used in the classical Wigner case of $\mathbb{R}^4\rtimes\mathbf{Spin}^0(1,3)$. Our analysis is motivated by the interest in the Euclidean formulation of Fermionic theories. Show Chinese abstract

Abstract: 我们对$\mathbb{C}^4\rtimes\mathbf{Spin}(4,\mathbb{C})$的不可约酉表示进行教学性介绍，即复化洛伦兹群的万有覆盖群$\mathbb{C}^4\rtimes\mathbf{SO}(4,\mathbb{C})$的不可约酉表示。 这些表示最早由 Roffman 在 1967 年进行研究。 我们提供他对这些结果的现代表述，并结合一些与本上下文相关的广义 Wigner-Mackey 理论中的事实。 此外，我们讨论了实现这些表示的不同方法，并在“复质量”非零的情况下，给出了一个更显式的实现的详细构造。 这种显式实现平行并扩展了经典 Wigner 情况下使用的实现，即$\mathbb{R}^4\rtimes\mathbf{Spin}^0(1,3)$的情况。 我们的分析是出于对费米理论的欧几里得表述的兴趣。