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We give a pedagogical presentation of the irreducible unitary representationsof $\mathbb{C}^4\rtimes\mathbf{Spin}(4,\mathbb{C})$, that is, of the universalcover of the complexified Poincar\'e group$\mathbb{C}^4\rtimes\mathbf{SO}(4,\mathbb{C})$. These representations werefirst investigated by Roffman in 1967. We provide a modern formulation of hisresults together with some facts from the general Wigner-Mackey theory whichare relevant in this context. Moreover, we discuss different ways to realizethese representations and, in the case of a non-zero "complex mass", we give adetailed construction of a more explicit realization. This explicit realizationparallels 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 theinterest in the Euclidean formulation of Fermionic theories.

Review and concrete description of the irreducible unitary representations of the universal cover of the complexified Poincar\'e group