Semiclassical chaos, the uncertainty principle, and quantum dissipation

Using the Wigner method, it is shown that a classical-like equation of motion for a quasiprobability distribution ${\mathrm{\rho }}_{\mathit{W}}$ can be built up, ∂${\mathrm{\rho }}_{\mathit{W}}$/∂t=(L${\mathrm{^}}_{\mathrm{cl}}$+L${\mathrm{^}}_{\mathrm{QGD}}$)${\mathrm{\rho }}_{\mathit{W}}$, which is rigorously equivalent to the quantum von Neumann–Liouville equation. The operator L${\mathrm{^}}_{\mathrm{cl}}$ is equivalent to integrating classical trajectories, which are then averaged over an initial distribution, broadened so as to fulfill the requirements of the quantum uncertainty principle. It is shown that this operator produces semiclassical chaos and is responsible for quantum irreversibility and the fast growth of quantum uncertainty. Carrying out explicit calculations for a spin-boson Hamiltonian, the joint action of L${\mathrm{^}}_{\mathrm{cl}}$ and L${\mathrm{^}}_{\mathrm{QGD}}$ is illustrated. It is shown that the latter operator L${\mathrm{^}}_{\mathrm{QGD}}$ (where QGD stands for quantum generating diffusion), makes the 1/2-spin system ‘‘remember’’ its quantum nature, and competes with the irreversibility induced by the former operator. Some ambiguous aspects of ‘‘irreversibility’’ and ‘‘growth of quantum fluctuations’’ as indicators of semiclassical chaos are discussed.