  In nonrelativistic quantum mechanics, Hudsons theorem states that a Gaussian
wavefunction is the only pure state corresponding to a positive Wigner function
(WF). We explicitly construct non Gaussian Dirac spinors with positive
relativistic WF. These pure states are coherent superpositions of particles and
antiparticles, while the existence of positive WF exclusive composed of
particles is conjectured. These observations challenge a common belief that
states with positive WF are classical or have classical analogues, and may open
new venues in relativistic quantum information.
