Review and concrete description of the irreducible unitary representations of the universal cover of the complexified Poincaré group

Abstract

We give a pedagogical presentation of the irreducible unitary representations of ${\mathbb{C}}^4⋊\mathbf{Spin}(4,\mathbb{C})$, that is, of the universal cover of the complexified Poincaré group ${\mathbb{C}}^4⋊\mathbf{SO}(4,\mathbb{C})$. 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 ${\mathbb{R}}^4⋊{\mathbf{Spin}}^0(1,3)$. Our analysis is motivated by the interest in the Euclidean formulation of Fermionic theories.

26 pages