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

TL;DR

This paper reviews and provides a modern, detailed description of the irreducible unitary representations of the universal cover of the complexified Poincaré group, extending classical results with explicit realizations relevant for Euclidean Fermionic theories.

Contribution

It offers a comprehensive, modern formulation and explicit constructions of these representations, building on Roffman's 1967 work and extending classical Wigner theory to complexified groups.

Findings

Provides a modern formulation of Roffman's results

Constructs explicit realizations for non-zero complex mass

Extends classical Wigner representation theory to complexified Poincaré group

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.