[Paper Review] Fundamental Klein-Gordon equation from stochastic mechanics in curved spacetime

Eric S. Escobar-Aguilar, Tonatiuh Matos+1 more|arXiv (Cornell University)|Mar 9, 2023

Cosmology and Gravitation Theories4 citations

TL;DR

This paper derives the fundamental Klein-Gordon equation in curved spacetime from a stochastic mechanics framework, showing that quantum particles follow stochastic trajectories due to a universal Gravitational Wave Background (GWB). The stochastic formulation leads to the generalized Schrödinger equation in the non-relativistic limit, suggesting that quantum mechanics may emerge from spacetime fluctuations inherent in the GWB.

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.

Motivation & Objective

• To establish a theoretical link between quantum mechanics and spacetime fluctuations via a stochastic trajectory model.
• To investigate how the presence of a Gravitational Wave Background (GWB) disrupts geodesic motion of quantum particles.
• To derive the Klein-Gordon equation in curved spacetime from stochastic dynamics, using a Markovian Langevin-type approach.
• To demonstrate the emergence of the Schrödinger equation as a non-relativistic limit of the generalized stochastic formalism.
• To provide a foundational framework for relativistic stochastic quantum mechanics within general relativity.

Proposed method

• Introduces a stochastic term into Newton’s second law, modeling particle trajectories as Markovian diffusion processes with Gaussian white noise.
• Uses the Langevin equation with a Wiener process $ dW(t) $ to describe the stochastic force acting on quantum particles.
• Applies the Madelung transformation to convert the stochastic dynamics into hydrodynamic equations: continuity and Navier-Stokes-type equations.
• Derives the Klein-Gordon equation in curved spacetime by combining the continuity equation and the stochastic momentum equation.
• Applies the non-relativistic limit to the generalized Schrödinger equation, recovering the standard Schrödinger equation under specific conditions.
• Utilizes stochastic differential operators and Fokker-Planck equations to describe the evolution of particle density and velocities.

Experimental results

Research questions

• RQ1 Can quantum mechanics emerge from a stochastic description of particle trajectories in a spacetime permeated by a Gravitational Wave Background?
• RQ2 How does the presence of a GWB with Compton wavelength $ ho $ modify the standard equations of motion for quantum particles?
• RQ3 To what extent can the Klein-Gordon equation in curved spacetime be derived from fundamental stochastic dynamics rather than postulated?
• RQ4 Under what conditions does the generalized stochastic formalism reduce to the standard Schrödinger equation?
• RQ5 Can the stochastic mechanics framework be consistently extended to general relativity, particularly in the Markovian formulation?

Key findings

• The Klein-Gordon equation in curved spacetime is derived from a stochastic formulation of Newton’s second law with a Markovian fluctuating force.
• The generalized Schrödinger equation is obtained in the non-relativistic limit, confirming the emergence of standard quantum mechanics from stochastic dynamics.
• The stochastic trajectory model leads to a physically real, Markovian description of quantum particles, consistent with Nelson’s stochastic mechanics framework.
• The electromagnetic, gravitational, and quantum forces naturally emerge within the theoretical formulation as consequences of the stochastic dynamics.
• The method successfully solves the open problem of Markovian stochastic mechanics in the context of general relativity, as posed by Nelson.
• The derivation confirms that the Schrödinger equation is not fundamental but arises as a limiting case of a deeper stochastic process in a GWB-embedded spacetime.