Wigner distributions in Rindler spacetime and nonvacuum Minkowski states

In the 1970s, Fulling, Davies, and Unruh demonstrated that the vacuum state perceived by an inertial observer in Minkowski space appears as a thermal bath to a uniformly accelerated observer. We explore the transformation of the Wigner distribution of a real scalar field from an inertial to a Rindler frame, utilizing both Minkowski and Unruh modes. We present a general expression for the reduced Wigner distribution for a specific set of massless scalar field configurations, and validate it against known distributions within this set. This includes arbitrary Gaussian states of Unruh-Minkowski modes, the Minkowski vacuum state, the Rindler vacuum, and the thermal bath of Unruh particles. Additionally, we analyze several other distributions, such as a uniform momentum distribution, a slight deviation from the Minkowski vacuum, and a distribution with a Fermionic component in the Rindler frame. The conclusions are discussed.