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We show that the dynamics of a closed quantum system obeys the Hamiltonvariation principle. Even though quantum particles lack well-definedtrajectories, their evolution in the Husimi representation can be treated as aflow of multidimensional probability fluid in the phase space. By introducingthe classical counterpart of the Husimi representation in a close analogy tothe Koopman-von Neumann theory, one can largely unify the formulations ofclassical and quantum dynamics. We prove that the motions of elementary parcelsof both classical and quantum Husimi fluid obey the Hamilton variationalprinciple, and the differences between associated action functionals stem fromthe differences between classical and quantum pure states. The Husimi actionfunctionals are not unique and defined up to the Skodje flux gauge fixing [R.T. Skodje et al. Phys. Rev. A 40, 2894 (1989)]. We demonstrate that the gaugechoice can dramatically alter flux trajectories. Applications of the presentedtheory for constructing semiclassical approximations and hybridclassical-quantum theories are discussed.

Joint quantum-classical Hamilton variation principle in the phase space