A relativistic variant of the Wigner function

The Wigner function is a quasiprobability density over phase space (for reviews, see Lee[1], Cohen[2], or Hillery et al.[3]; a quasiprobability is generally not positive semi-definite), so in a quantum field theory setting it is not manifestly Lorentz covariant. Quasiprobabilities are conceptually interesting partly because Arthurs-Kelly type interpretations of positive definite Husimi functions derived by smoothing from Wigner functions are available[4–6], but the lack of manifest Lorentz covariance in the quantum field theory setting makes Wigner functions, in this respect, a poor relation to Feynman path integrals.