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.
Categories: math-ph math.MP quant-ph