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.

与其环境处于热平衡状态的量子系统的密度算符取决于普朗克常数以及温度。在高温下，Weyl 表示，即热 Wigner 函数，变得与相空间中相应的经典分布无法区分，而低温极限则挑出系统哈密顿量的量子基态。在所有情况下，任意观测值的热平均值都是通过积分来评估的，就好像热 Wigner 函数是经典分布一样。

将量子传播器的半经典近似扩展到虚热时间，在高温和低温极限之间架起了复杂的中间区域。这导致对经典高温状态的简单量子校正，无论运动是规则的还是无序的。具有真实热时间的完全半经典近似的变体，虽然在双相空间中，但避免了在评估热平均值时对特定轨迹的任何搜索。双哈密顿量用鞍代替原始系统哈密顿量的稳定最小值，这从分配函数和热平均值的积分的固定相评估中消除了局部周期轨道。