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

摘要

We discuss in this Chapter a series of theoretical developments whichmotivate the introduction of a quantum evolution equation for which the eikonalapproximation results in the geodesics of a four dimensional manifold. Thisgeodesic motion can be put into correspondence with general relativity. Oneobtains in this way a quantum theory on a flat spacetime, obeying the rules ofthe standard quantum theory in Lorentz covariant form, with a spacetimedependent Lorentz tensor $g_{\mu\nu}$, somewhat analogous to a gauge field,coupling to the kinetic terms. Since the geodesics predicted by the eikonalapproximation, with appropriate choice of $g_{\mu\nu}$, can be those of generalrelativity, this theory provides a quantum theory which could be underlying toclassical gravitation, and coincides with it in this classical rayapproximation. In order to understand the possible origin of the structure ofthis equation, we appeal to the approach of Nelson in constructing aSchroedinger equation from the properties of Brownian motion. Extending thenotion of Browninan motion to spacetime in a covariant way, we show that suchan equation follows from correlations between spacetime dimensions in thestochastic process.