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arXiv:1902.08400 · International Journal of Quantum Information

Effects of entanglement on vortex dynamics in the hydrodynamic representation of quantum mechanics

Satoya Imai

Published Sep 1, 2020• 2 citations
arXivHTMLPDF

Abstract

The hydrodynamic representation of quantum mechanics describes virtual flow as if a quantum system were fluid in motion. This formulation illustrates pointlike vortices when the phase of a wavefunction becomes nonintegrable at nodal points. We study the dynamics of such pointlike vortices in the hydrodynamic representation for a two-particle wavefunction. In particular, we discuss how quantum entanglement influences vortex–vortex dynamics. For this purpose, we employ the time-dependent quantum variational principle combined with the Rayleigh–Ritz method. We analyze the vortex dynamics and establish connections with Dirac’s generalized Hamiltonian formalism.

DOI
10.1142/s0219749920500306

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The hydrodynamic representation of quantum mechanics describes virtual flow as if a quantum system were fluid in motion. This formulation illustrates pointlike vortices when the phase of a wavefunction becomes nonintegrable at nodal points. We study the dynamics of such pointlike vortices in the hydrodynamic representation for a two-particle wavefunction. In particular, we discuss how quantum entanglement influences vortex–vortex dynamics. For this purpose, we employ the time-dependent quantum variational principle combined with the Rayleigh–Ritz method. We analyze the vortex dynamics and establish connections with Dirac’s generalized Hamiltonian formalism.
We study the vortex dynamics in the hydrodynamic representation of quantum mechanics for two particles, in which a pointlike vortex comes from a nonintegrable phase of a wavefunction. In particular, we introduce an ansatz for an entangled state of the two particles and discuss how entanglement affects the dynamics of vortex-vortex interaction. For this purpose, we employ the time-dependent quantum variational principle combined with the Rayleigh-Ritz method and elucidate the effects of entanglement on the approximate vortex dynamics.
In quantum theory, vortex lines arise in the hydrodynamic interpretation of the wave equation. In this interpretation, which is originally due to Madelung, the flow of the probability density for a single particle is described in terms of the hydrodynamic variables. For the sake of simplicity, the standard time-dependent Schrödinger equation, and the related vortex lines embedded in the probability fluid of the quantum particle, are considered here. A vortex line in this case is simply the curve defined by equating the wave function to zero. The linearity of the Schrödinger equation enables us to obtain a large family of exact time-dependent analytic solutions for the wave functions with vortex lines. Moreover, the method is general enough to allow for various initial configurations of the vortex lines. Although the equation of motion of the quantum mechanical probability fluid is different in its literal form from the equations describing the real physical fluid, we believe that the evolution of the vorticity in the quantum and in the real fluid share the same qualitative features that can be described in terms of the topology of the vortex lines configurations. The general phenomena such as the switch-over, creation and annihilation of vortices can be observed in the quantum mechanical fluid. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Quantum entanglement represents one of the most profound and counterintuitive phenomena in modern physics, challenging our classical understanding of locality and causality. While the mathematical formalism of quantum mechanics accurately predicts entanglement correlations, the lack of a physically intuitive mechanism has led to numerous interpretational difficulties and philosophical debates. This article presents a comprehensive analysis of existing interpretations of quantum entanglement, from the Copenhagen interpretation to many-worlds and hidden variable theories, highlighting their limitations and motivating a novel vortex-based interpretation grounded in the hydrodynamic behavior of a superfluid vacuum, drawing upon the author’s published work on electron, quark, and gravitational force dynamics. This vortex model provides a physically intuitive and causally consistent explanation for nonlocal correlations without invoking superluminal signaling, offering a bridge between classical physics and quantum mechanics. The proposed interpretation suggests that entangled particles are surface manifestations of deeper, coherent vortex structures within a continuous vacuum medium, analogous to connected vortices in a fluid system. This framework not only resolves the conceptual difficulties associated with “spooky action at a distance” but also provides a foundation for understanding quantum phenomena through classical hydrodynamic principles.
It is shown that a natural extension of the hydrodynamical formalism of quantum mechanics for a Schrödinger particle to include vortical flows leads to the hydrodynamical formalism of quantum mechanics for a spinning particle. This latter formalism is then analysed in regard to its characteristic features, especially the subsidiary condition connecting the vorticity of flow with the inhomogeneity of spin field and the existence of spin stress. Also the formalism is brought to completion by establishing the global condition that quantizes circulation around a singular vortex line. The geometro-hydrodynamical formalism which is equivalent to the above hydrodynamical formalism but introduces a triad structure underlying the classical spin is reconstructed on its own footing. The geometrical property of the triad implies the invariance of theory with respect to the rotation of each triad around its symmetry axis by an arbitrary angle, and this necessitates the introduction of the electromagnetic potential, providing the geometrical interpretation of local gauge invariance. Applications of theory to various special cases and typical examples are deferred to Part II.
The Madelung equations offer a hydrodynamic description of quantum systems, from single particles to quantum fluids. In this formulation, the probability density is mapped onto the fluid density and the phase is treated as a scalar potential generating the velocity field. As examples of potential flows, quantum fluids described in this way are inherently irrotational, but quantum vortices may arise at discrete points where the phase is undefined. In this paper, starting from this irrotational description of a quantum fluid, a coarse-graining procedure is applied to arrive at a macroscopic description of the quantum fluid in terms of a hierarchy of moments in which the role of velocity is played by a Favre average of the microscopic velocity field. This hierarchy is truncated using an explicit closure derived from an expansion in a finite length scale. The resulting coarse-grained fields are shown to allow for finite vorticity at any point in the fluid. Furthermore, it is shown that this vorticity obeys a similar equation to the vorticity equation in classical hydrodynamics and includes a vortex-stretching term. The particular closure employed here also gives rise to a novel stress term in the fluid equations, which in the appropriate limit appears analogous to an artificial viscous stress from computational fluid dynamics.
Vortices are known to play a key role in many important processes in physics and chemistry. Here, we study vortices in connection with the quantum trajectories that can be defined in the framework provided by the de Broglie–Bohm formalism of quantum mechanics. In a previous work, it was shown that the presence of a single moving vortex is enough to induce chaos in these trajectories. Here, this situation is explored in more detail by discussing the relationship between Lyapunov exponents and the parameters characterizing the vortex dynamics. We also consider the issue when more than one vortex exists. In this case, the interaction among them can annihilate or create pairs of vortices with opposite vorticity. This phenomenon is analyzed from a dynamical point of view, showing how the size of the regular regions in phase space grows, as vortices disappear.