On the real representations of the Poincare group

We do a study of the real representations of the Poincare group, motivated by the following: i) the classical electromagnetic field —from which the Poincare group was originally defined— transforms as a real representation of the Poincare group; ii) the localization of complex unitary representations of the Poincare group is incompatible with causality, Poincare covariance and energy positivity, while the complex representation corresponding to the photon is not localizable. We start by reviewing the map from the complex to the real irreducible representations— finite-dimensional or unitary—of a Lie group on a Hilbert space. Then we show that all the finite-dimensional real representations of the identity component of the Lorentz group are also representations of the full Lorentz group, in contrast with many complex representations. We finally study the unitary irreducible representations of the Poincare group with discrete spin or helicity and show that: for each pair of complex representations with positive/negative energy, there is one real representation; the localization, compatible with causality and Poincare covariance, exists for representations with discrete spin or helicity.