Classical Gravity as an Eikonal Approximation to a Manifestily Lorentz Covariant Quantum Theory with
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 correpondence with general relativity. The well-known problem of the self-interaction of a relativistic charged particle is studied from the point of view of a manifestly Lorentz co-variant classical theory admitting the particle mass as a dynamical variable, i.e., the theory is intrinsically off-shell. The evolution parameter, τ , is an invariant parameter that can be identified with Newton's time (sometimes called, as in Schwinger's formulation, " proper time "). Gauge invariance requires the definition of five gauge fields; the fifth field has for its source the matter density. Together with the results of Gupta and Padmanabhan, showing a connection between the radiation reaction force and geometry, this structure motivates an investigation into the connection between these dynamical equations and gravity. We show that the fifth gauge field can indeed be absorbed into a conformal metric in the kine-matic terms, which then results in a geodesic equation generated by the conformal metric and the standard Lorentz force. We then go on to show that the generalized radiation field passing through an optical medium with non-trivial dielectric tensor results in an analog gravity for the eikonal approximation for an arbitrary metric. A mathematically simpler system for which the eikonal approximation provides the geodesic motion on a four dimensional pseudo-Riemannian manifold is that of the Stueckelberg-Schrödinger equation with a spacetime dependent tensor g µν (of the form of the Einstein metric tensor), somewhat analogously to a gauge field, coupling to the kinetic terms. This theory can be realized as a quantum theory in a flat spacetime, obeying the rules of the standard quantum theory in Lorentz covariant form. 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 underlies classical gravitation, and coincides with it in this classical ray approximation. This result is the principal content of this work. In order to understand the possible origin of the structure of this form of the Stueckelberg-Schrödinger equation, we appeal to the approach of Nelson in constructing a Schrödinger equation from the properties of Brownian motion. Extending the notion of 1 Brownian motion to spacetime in a …