The orthogonal representation of the Poincare group on the Majorana spinor field

The irreducibility of a representation of a real Lie algebra may depend on whether the representation space is a real or complex Hilbert space. The unitary projective representations of the Poincare group on complex Hilbert spaces were studied by Wigner and many others. Although the Poincare group has a real Lie algebra, we do not know of any study of the orthogonal projective representations of the Poincare group on real Hilbert spaces. The Majorana spinor field, a space-time dependent element of a 4 dimensional real vector space, is a solution of the free Dirac equation. Our goal is to study the projective representation of the Poincare group on the real Hilbert space of Majorana spinor fields. The Majorana-Fourier and Majorana-Hankel orthogonal transforms of Majorana spinor fields are defined and related to the linear and angular momentums of a spin one-half projective representation of the Poincare group. Then we show that the projective representation of the Poincare group on the Majorana spinor field, whether we include the parity and time reversal or not, is orthogonal and irreducible. This contrasts with the unitary projective representations of the Poincare group on the Dirac and Weyl spinor fields, whose properties change when including or excluding the parity and time reversal transformations.