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International Journal of Geometric Methods in Modern Physics

The geometrization of quantum mechanics, frozen stars, the Bohm–Poisson and nonlinear Klein–Gordon equations

Carlos Castro Perelman

Published May 21, 2024• 0 citations
Publisher

Abstract

We revisit the nonlinear Klein–Gordon-like equation that was proposed by us which captures how quantum mechanical probability densities curve spacetime, and find an exact solution that may appear to be “trivial” but with important physical implications related to the physics of frozen stars and with Mach’s principle. The nonlinear Klein–Gordon-like equation is essentially the static spherically symmetric relativistic analog of the Newton–Schrödinger equation. We finalize by studying the higher-dimensional generalizations of the nonlinear Klein–Gordon-like equation and examine the relativistic Bohm–Poisson equation as yet another equation capturing the interplay between quantum mechanical probability densities and gravity.

DOI
10.1142/s0219887824502335

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We revisit the nonlinear Klein–Gordon-like equation that was proposed by us which captures how quantum mechanical probability densities curve spacetime, and find an exact solution that may appear to be “trivial” but with important physical implications related to the physics of frozen stars and with Mach’s principle. The nonlinear Klein–Gordon-like equation is essentially the static spherically symmetric relativistic analog of the Newton–Schrödinger equation. We finalize by studying the higher-dimensional generalizations of the nonlinear Klein–Gordon-like equation and examine the relativistic Bohm–Poisson equation as yet another equation capturing the interplay between quantum mechanical probability densities and gravity.
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Abstract This work presents an alternative approach to obtain the quantum field equations in curved spacetime, considering that sufficiently small particles follow stochastic trajectories around geodesic. Our proposal is based on a stochastic differential equation in which the noise term experienced by the quantum particles is a consequence of the stochastic background in spacetime. This fact allows the particles to describe erratic movements and locally the universe exhibits characteristics akin to a lake with gentle ripples rather than a flat unyielding surface. Building upon this foundational understanding, we investigate the influence of this background on quantum-scale particles without considering the metric to be stochastic, rather we let test particles move randomly around the geodesic of macroscopic particles. Their behavior aligns with solutions to the Klein-Gordon (KG) equation specific to this curved spacetime. As the KG equation, in its non-relativistic limit within a flat spacetime, reduces to the Schrödinger equation, consequently, we propose a compelling connection: the Schrödinger equation may emerge directly from a spacetime lacking local smoothness.
Madelung's hydrodynamical forms of the Schrodinger equation and the Klein-Gordon equation are presented. The physical nature of the quantum potential is explored. It is demonstrated that the geometrical origin of the quantum potential is in the scalar curvature of the of the metric that defines the kinetic energy density for an extended particle and that the quantization of circulation (Bohr-Sommerfeld) is a consequence of associating an SO(2)-reduction of the Lorentz frame bundle with wave motion. The Madelung equations are then cast in a basis-free form in terms of exterior differential forms in such a way that they represent the equations for a timelike solution to the conventional wave equations whose rest mass density satisfies a differential equation of the "Klein-Gordon minus nonlinear term" type. The role of non-zero vorticity is briefly examined.
The Klein-Gordon equation is interpreted in the de Broglie-Bohm manner as a single-particle relativistic quantum mechanical equation that defines unique time-like particle trajectories. The particle trajectories are determined by the conserved flow of the intrinsic energy density which can be derived from the specification of the Klein-Gordon energy-momentum tensor in an Einstein-Riemann space. The approach is illustrated by application to the simple single-particle phenomena associated with square potentials.
The Klein-Gordon equation is interpreted in the de Broglie-Bohm manner as a single-particle relativistic quantum mechanical equation that defines unique time-like particle trajectories. The particle trajectories are determined by the conserved flow of the intrinsic energy density, which can be derived from the specification of the Klein-Gordon energy-momentum tensor in an Einstein-Riemann space. The approach is illustrated by application to the simple single-particle phenomena associated with square potentials.