Wigner phase-space representation of thermal excitations

The Wigner phase-space representation of thermal excitations used in thermo-field-dynamics (TFD) is investigated. The phase-space description is obtained by introducing a reduced density operator in which the fictitious degrees of freedom of TFD are traced over. A new closed-form expression for the Wigner function of the harmonic oscillator thermal excitation is derived, thereby showing that the thermal excitation may be described as a temperature-dependent radial spreading of the coordinates and momenta in phase space. The population distribution in these stationary excitations is determined and shown to reduce to a Boltzmann distribution for the thermal vacuum only.