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AIP conference proceedings

A novel approach to the quantum hydrodinamic equations of motion by means of Fisher information

Juan Villacres

Published Jan 1, 2024• 1 citation
Publisher

Abstract

The quantum hydrodynamic equations are traditionally associated with the de Broglie-Bohm formulation of quantum mechanics in terms of hidden variables.Here, it is proposed a way to arrive to such equations by means of Fisher Information, departing from a conceptual framework that differs from the one posed in Bohmian mechanics.Based on this, a method to obtain quantum equations starting from classical ones is proposed.This, is used to obtain the Klein-Gordon equation departing from the relativistic Hamilton-Jacobi equation.

DOI
10.1063/5.0190214

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The quantum hydrodynamic equations are traditionally associated with the de Broglie-Bohm formulation of quantum mechanics in terms of hidden variables.Here, it is proposed a way to arrive to such equations by means of Fisher Information, departing from a conceptual framework that differs from the one posed in Bohmian mechanics.Based on this, a method to obtain quantum equations starting from classical ones is proposed.This, is used to obtain the Klein-Gordon equation departing from the relativistic Hamilton-Jacobi equation.
This work presents a reconstruction of quantum mechanics in which Schrödinger-type dynamics emerges from reversible probabilistic evolution rather than being postulated. Starting from a normalized probability density, we impose reversible probability dynamics, Fisher information geometry, and classical correspondence. These compatibility conditions yield coupled real equations equivalent to the continuity equation and the Hamilton–Jacobi equation with an additional information curvature (quantum potential) term. By combining density and phase into a complex amplitude, the Schrödinger equation appears as a compact encoding of reversible statistical flow. The result is mathematically equivalent to the Madelung hydrodynamic formulation and structurally connected to Bohmian mechanics, but derived without assuming Hilbert space, operator axioms, or measurement postulates. The work can therefore be viewed as an information-geometric derivation of quantum dynamics: quantum theory arises as the unique consistent representation of reversible probability flow. Keywords: quantum foundations, quantum reconstruction, emergent quantum mechanics, derivation of the Schrödinger equation, information geometry, Fisher information, Hamilton–Jacobi theory, Madelung formulation, Bohmian mechanics, reversible probability dynamics
The hydrodynamical form of Wigner–Dunkl quantum mechanics has been formulated. The main step to get such formulation was to derive a pair of Madelung-like equations (called Madelung–Wigner–Dunkl equations) from the Schrödinger equation appearing in Wigner–Dunkl quantum mechanics. Some aspects of both equations contained in the pair, i.e. the inhomogeneous continuity equation and the quantum Hamilton–Jacobi equation, have been investigated and discussed. With certain assumption, the relation between the expectation value of the quantum potential appearing in the quantum Hamilton–Jacobi equation and the so-called Fisher information has been derived. Then an uncertainty relation between momentum and position which may be tighter than the standard uncertainty relation has been derived by using Cramér–Rao inequality and Fisher information.
The Hamilton-Jacobi method is generalized, both, in classical and relativistic mechanics. The implications in quantum mechanics are considered in the case of Klein-Gordon equation. We find that the wave functions of Klein-Gordon theory can be considered as describing the motion of an ensemble of particles that move under the action of the electromagnetic field alone, without quantum potentials, hidden uninterpreted variables, or zero point fields. The number of particles is not locally conserved.
In this paper, we show that the Klein-Gordon quantum-mechanical operator, operating on the function ${e}^{\frac{\mathrm{iS}}{\ensuremath{\hbar}}}$, where SSS is the relativistic action of a free particle or of a particle in a field defined by a four vector, is completely equivalent to the relativistic Hamilton-Jacobi equation, provided one takes into account the vanishing of the divergence of the energy-momentum four vector. From this we see that classical relativistic mechanics can be formulated in terms of operators which are identical with those used in quantum mechanics.
The quantum hydrodynamic-like equations for two real variables (i.e., the phase and the amplitude of the wave function) of the relativistic Klein-Gordon equation are derived in the present paper. The paper also shows that in classical limit the hydrodynamic Klein-Gordon equations lead to the Madelung pseudo-potential [1] as well as to the quantum pseudo potential for a charged particle given by Janossy [2] and by Bialynicki et al [3]. The origin of the non-local interactions of quantum mechanics in the hydrodynamic model are discussed both for free and charged particles.