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

TL;DR

This paper proposes a quantum theory that reproduces classical gravity as an eikonal approximation, using a Lorentz covariant framework with Brownian motion concepts to connect quantum and gravitational phenomena.

Contribution

It introduces a Lorentz covariant quantum evolution equation whose eikonal approximation yields general relativity geodesics, linking quantum mechanics and gravity.

Findings

The theory reproduces classical geodesics in the eikonal limit.

It formulates gravity within a flat spacetime quantum framework.

Brownian motion concepts are extended to spacetime, underpinning the quantum equation.

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 $g_{\mu\nu}$, somewhat analogous to a gauge field, coupling to the kinetic terms. Since the geodesics predicted by the eikonal approximation, with appropriate choice of $g_{\mu\nu}$, 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.