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The family of trajectories-based approximations employed in computationalquantum physics and chemistry is very diverse. For instance, Bohmian andHeller's frozen Gaussian semiclassical trajectories seem to have nothing incommon. Based on a hydrodynamic analogy to quantum mechanics, we furnish theunified gauge theory of all such models. In the light of this theory, currentlyknown methods are just a tip of the iceberg, and there exists an infinitefamily of yet unexplored trajectory-based approaches. Specifically, we showthat each definition for a semiclassical trajectory corresponds to a specifichydrodynamic analogy, where a quantum system is mapped to an effectiveprobability fluid in the phase space. We derive the continuity equation for theeffective fluid representing dynamics of an arbitrary open bosonic many-bodysystem. We show that unlike in conventional fluid, the flux of the effectivefluid is defined up to Skodje's gauge [R. T. Skodje et. al. Phys. Rev. A 40,2894 (1989)]. We prove that the Wigner, Husimi and Bohmian representations ofquantum mechanics are particular cases of our generic hydrodynamic analogy, andall the differences among them reduce to the gauge choice. Infinitely manygauges are possible, each leading to a distinct quantum hydrodynamic analogyand a definition for semiclassical trajectories. We propose a scheme foridentifying practically useful gauges and apply it to improve a semiclassicalinitial value representation employed in quantum many-body simulations.

Quantum and semiclassical dynamics as fluid theories where gauge matters