[Paper Review] Classical Gravity as an Eikonal Approximation to a Manifestly Lorentz Covariant Quantum Theory with Brownian Interpretation

L. P. Horwitz, O. Oron|ArXiv.org|Jul 20, 2004

Quantum Mechanics and Applications8 references3 citations

TL;DR

This paper proposes that classical gravity emerges as an eikonal approximation to a manifestly Lorentz-covariant quantum theory based on relativistic Brownian motion. By modeling spacetime correlations in a stochastic process, the theory derives a Stueckelberg-Schrödinger equation with a spacetime-dependent tensor $ g_{ ueta} $, whose eikonal limit reproduces the geodesic equations of general relativity, thus identifying gravity as a geometric consequence of quantum stochastic dynamics.

ABSTRACT

We discuss in this Chapter a series of theoretical developments which motivate the introduction of a quantum evolution equation for which the eikonal approximation results in the geodesics of a four dimensional manifold. This geodesic motion can be put into correspondence with general relativity. One obtains in this way a quantum theory on a flat spacetime, obeying the rules of the standard quantum theory in Lorentz covariant form, with a spacetime dependent Lorentz tensor $g_{μν}$, somewhat analogous to a gauge field, coupling to the kinetic terms. Since the geodesics predicted by the eikonal approximation, with appropriate choice of $g_{μν}$, can be those of general relativity, this theory provides a quantum theory which could be underlying to classical gravitation, and coincides with it in this classical ray approximation. In order to understand the possible origin of the structure of this equation, we appeal to the approach of Nelson in constructing a Schroedinger equation from the properties of Brownian motion. Extending the notion of Browninan motion to spacetime in a covariant way, we show that such an equation follows from correlations between spacetime dimensions in the stochastic process.

Motivation & Objective

• To establish a manifestly Lorentz-covariant quantum theory where classical gravity arises as an eikonal approximation.
• To resolve the self-interaction problem of relativistic charged particles by treating mass as a dynamical variable and introducing five gauge fields, including a fifth field sourced by matter density.
• To show that the fifth gauge field can be absorbed into a conformal metric, yielding a geodesic equation consistent with general relativity.
• To unify the radiation reaction force in the Abraham-Lorentz-Dirac equation with spacetime curvature via geometric arguments.
• To derive a quantum evolution equation from relativistic Brownian motion that reproduces general relativistic geodesics in the eikonal limit.

Proposed method

• Formulate a relativistic generalization of Nelson’s stochastic mechanics, introducing spacetime correlations in a Markov process parameterized by an invariant evolution parameter $ \tau $.
• Construct a Stueckelberg-Schrödinger equation with a spacetime-dependent tensor $ g_{\mu\nu} $, analogous to a gauge field, coupling to kinetic terms.
• Demonstrate that the eikonal approximation of the wavefunction yields geodesic motion in a pseudo-Riemannian manifold, matching general relativity when $ g_{\mu\nu} $ is identified with the Einstein metric.
• Show that the fifth gauge field—sourced by matter density—can be reinterpreted as a conformal factor in the metric, leading to a modified Lorentz force law with curvature effects.
• Use the Brownian motion framework to derive a quantum theory in flat spacetime that reproduces classical gravity in the ray limit.
• Apply the Fokker-Planck equation to the stochastic process, showing that while the microscopic dynamics are non-local in time $ t $, they remain local and causal in $ \tau $.

Experimental results

Research questions

• RQ1 Can classical gravity be derived as an eikonal approximation of a manifestly Lorentz-covariant quantum theory with a dynamical mass?
• RQ2 How does the inclusion of a fifth gauge field, sourced by matter density, lead to a geometric description of gravity?
• RQ3 In what way do spacetime correlations in a relativistic Brownian process generate a metric-compatible geodesic structure?
• RQ4 How does the eikonal limit of the Stueckelberg-Schrödinger equation reproduce the geodesic equations of general relativity?
• RQ5 Can the radiation reaction force in the Abraham-Lorentz-Dirac equation be interpreted as a manifestation of spacetime curvature in a relativistic quantum framework?

Key findings

• The eikonal approximation of the Stueckelberg-Schrödinger equation with a spacetime-dependent tensor $ g_{\mu\nu} $ reproduces the geodesic equations of general relativity for an arbitrary pseudo-Riemannian metric.
• The fifth gauge field, sourced by matter density, can be absorbed into a conformal metric, yielding a modified Lorentz force law that includes curvature effects.
• The radiation reaction force $ \Gamma^\mu $ in the Abraham-Lorentz-Dirac equation arises naturally from the geometric structure of the accelerating frame, suggesting a deep link between radiation and spacetime curvature.
• Relativistic Brownian motion with spacetime correlations leads to a quantum evolution equation whose eikonal limit corresponds to classical geodesic motion.
• The underlying stochastic process is local and causal in the invariant parameter $ \tau $, even though the resulting dynamics in physical time $ t $ may appear non-Markovian or non-local.
• The spacetime metric in the eikonal limit originates from correlations in the stochastic process, providing a dynamical origin for gravity in terms of quantum fluctuations.