  The density operator for a quantum system in thermal equilibrium with its
environment depends on Planck's constant, as well as the temperature. At high
temperatures, the Weyl representation, that is, the thermal Wigner function,
becomes indistinguishable from the corresponding classical distribution in
phase space, whereas the low temperature limit singles out the quantum ground
state of the system's Hamiltonian. In all regimes, thermal averages of
arbitrary observables are evaluated by integrals, as if the thermal Wigner
function were a classical distribution.
  The extension of the semiclassical approximation for quantum propagators to
an imaginary thermal time, bridges the complex intervening region between the
high and the low temperature limit. This leads to a simple quantum correction
to the classical high temperature regime, irrespective of whether the motion is
regular or chaotic. A variant of the full semiclassical approximation with a
real thermal time, though in a doubled phase space, avoids any search for
particular trajectories in the evaluation of thermal averages. The double
Hamiltonian substitutes the stable minimum of the original system's Hamiltonian
by a saddle, which eliminates local periodic orbits from the stationary phase
evaluation of the integrals for the partition function and thermal averages.
