  There are Poincare group representations on complex Hilbert spaces, like the
Dirac spinor field, or real Hilbert spaces, like the electromagnetic field
tensor. The Majorana spinor is an element of a 4 dimensional real vector space.
The Majorana spinor field is a space-time dependent Majorana spinor, solution
of the free Dirac equation.
  The Majorana-Fourier and Majorana-Hankel transforms of Majorana spinor fields
are defined and related to the linear and angular momenta of a spin one-half
representation of the Poincare group. We show that the Majorana spinor field
with finite mass is an unitary irreducible projective representation of the
Poincare group on a real Hilbert space.
  Since the Bargmann-Wigner equations are valid for all spins and are based on
the free Dirac equation, these results open the possibility to study Poincare
group representations with arbitrary spins on real Hilbert spaces.
