Quantum and semiclassical dynamics as fluid theories where gauge matters

TL;DR

This paper develops a unified gauge theory for trajectory-based quantum dynamics, revealing that various existing methods are specific cases within an infinite family of hydrodynamic analogies, and proposes a scheme to optimize gauge choices for better simulations.

Contribution

It introduces a comprehensive gauge framework unifying diverse quantum trajectory methods and demonstrates how gauge choices influence semiclassical dynamics and simulations.

Findings

Wigner, Husimi, and Bohmian mechanics are special cases of the hydrodynamic analogy.

Different gauge choices lead to distinct definitions of semiclassical trajectories.

The proposed gauge scheme improves semiclassical initial value representations in many-body quantum simulations.

Abstract

The family of trajectories-based approximations employed in computational quantum physics and chemistry is very diverse. For instance, Bohmian and Heller's frozen Gaussian semiclassical trajectories seem to have nothing in common. Based on a hydrodynamic analogy to quantum mechanics, we furnish the unified gauge theory of all such models. In the light of this theory, currently known methods are just a tip of the iceberg, and there exists an infinite family of yet unexplored trajectory-based approaches. Specifically, we show that each definition for a semiclassical trajectory corresponds to a specific hydrodynamic analogy, where a quantum system is mapped to an effective probability fluid in the phase space. We derive the continuity equation for the effective fluid representing dynamics of an arbitrary open bosonic many-body system. We show that unlike in conventional fluid, the flux of the effective fluid 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 of quantum mechanics are particular cases of our generic hydrodynamic analogy, and all the differences among them reduce to the gauge choice. Infinitely many gauges are possible, each leading to a distinct quantum hydrodynamic analogy and a definition for semiclassical trajectories. We propose a scheme for identifying practically useful gauges and apply it to improve a semiclassical initial value representation employed in quantum many-body simulations.