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

Abstract

In this paper, we give a pedagogical presentation of the irreducible unitary representations of [Formula: see text], that is, of the universal cover of the complexified Poincaré group [Formula: see text]. 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 [Formula: see text]. Our analysis is motivated by the interest in the Euclidean formulation of Fermionic theories.