Non-classicality of positive relativistic Wigner function

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.