  The formulation of quantum mechanics with a complex Hilbert space is
equivalent to a formulation with a real Hilbert space and particular density
matrix and observables. We study the real representations of the Poincare
group, motivated by the fact that the localization of complex unitary
representations of the Poincare group is incompatible with causality, Poincare
covariance and energy positivity.
  We review 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 parity,
in contrast with many complex representations.
  We show that any localizable unitary representation of the Poincare group,
compatible with Poincare covariance, verifies: 1) it is a direct sum of
irreducible representations which are massive or massless with discrete
helicity. 2) it respects causality; 3) if it is complex it contains necessarily
both positive and negative energy subrepresentations 4) it is an irreducible
representation of the Poincare group (including parity) if and only if it is:
a)real and b)massive with spin 1/2 or massless with helicity 1/2. Finally, the
energy positivity problem is discussed in a many-particles context.
