Klein–Gordon equation in hydrodynamical form

Abstract

We follow and modify the Feshbach-Villars formalism by separating the Klein-Gordon equation into two coupled time-dependent Schrödinger equations for particle and antiparticle wave function components with positive probability densities. We find that the equation of motion for the probability densities is in the form of relativistic hydrodynamics where various forces have their classical counterparts, with the additional element of the quantum stress tensor that depends on the derivatives of the amplitude of the wave function. We derive the equation of motion for the Wigner function and we find that its approximate classical weak-field limit coincides with the equation of motion for the distribution function in the collisionless kinetic theory.

Quantum simulations of hydrodynamics via the Madelung transformation

“…The Dirac equation [20] plays a pivotal role in this approach. On one hand, the Dirac equation can be mapped into relativistic hydrodynamics by a generalization of the socalled Madelung transformation initially developed for the Schrödinger equation [21,22] and later extended to the Klein-Gordon (KG) equation [23][24][25] and quaternionic quantum mechanics [26]. On the other hand, quantum walks, which can be viewed as a quantum generalization of classical random walks [27][28][29][30], are a universal quantum primitive [31,32]; every quantum algorithm can be expressed as a quantum walk, and several quantum walks, usually called Dirac quantum walks, admit the Dirac equation as continuous limit [33,34].…”

Hybrid Quantum-Classical Algorithm for Hydrodynamics

“…The Dirac equation [11] plays a pivotal role in this new approach. On one hand, the Dirac equation can be mapped into relativistic hydrodynamics by a generalization of the so-called Madelung transformation initially developed for the Schrödinger equation [12,13] and later extended to the Klein-Gordon equation [14][15][16] and quaternionic quantum mechanics [17]. On the other hand, quantum walks, which can be viewed as a quantum generalization of classical random walks [18], are a universal quantum primitive [19]; every quantum algorithm can be expressed as a quantum walk, and several quantum walks, usually called Dirac quantum walks, admit the Dirac equation as continuous limit [20].…”

Thermohydrodynamics: Where Do We Stand?

“…Nevertheless, Feshbach & Villars (1958) were able to overcome this inconvenience by noting that the Klein-Gordon equation constrains both, particle and antiparticle degrees of freedom. That motivated Wong to write such equation in hydrodynamic form first for a single particle and then for N-body systems with strong interactions Wong (2010). Although one can indeed obtain rather complicated equations in which there are terms which may be identified, for instance, with a quantum stress tensor the whole scheme is still too formal to allow explicit calculations that may be related with transport properties, even less with transport coefficients.…”

“…The ensuing formalism may be loosely regarded as the relativistic generalization of Madelung's early attempt as described at the beginning of this section. The method itself, which has a long history Wong (2010), curiosly enough, started from the Klein-Gordon equation in spite of the fact that, as well known, has a probability density ρ = 2 Im Ψ ⋆ ∂Ψ ∂t which is not necessarily a positive quantity. Nevertheless, Feshbach & Villars (1958) were able to overcome this inconvenience by noting that the Klein-Gordon equation constrains both, particle and antiparticle degrees of freedom.…”