The conformal metric associated with the U(1) gauge of the Stueckelberg- Schr\"odinger equation

TL;DR

This paper explores the gauge covariance of the Stueckelberg-Schroedinger equation, introduces a conformal metric to eliminate a scalar field, and connects the resulting geometry with cosmological event densities in Friedmann-Robertson-Walker models.

Contribution

It introduces a conformal metric to remove the scalar compensation field in the Stueckelberg-Schroedinger framework and links this geometry to cosmological event densities.

Findings

The scalar field can be eliminated via a conformal metric.

The geodesic equation matches the Lorentz force.

Event density in conformal space aligns with Friedmann-Robertson-Walker models.

Abstract

We review the classical and quantum mechanics of Stueckelberg, and introduce the compensation fields necessary for the gauge covariance of the Stueckelberg- Schr\"odinger equation. To achieve this, one must introduce a fifth, Lorentz scalar, compensation field, in addition to the four vector fields which compensate the action of the space-time derivatives. A generalized Lorentz force can be derived from the classical Hamilton equations associated with this evolution function. We show that the fifth (scalar) field can be eliminated through the introduction of a conformal metric on the spacetime manifold. The geodesic equation associated with this metric coincides with the Lorentz force, and is therefore dynamically equivalent. Since the generalized Maxwell equations for the five dimensional fields provide an equation relating the fifth field with the spacetime density of events, one can derive the spacetime event density associated with the Friedmann-Robertson-Walker solution of the Einstein equations. The resulting density, in the conformal coordinate space, is isotropic and homogeneous, decreasing as the square of the Robertson-Walker scale factor. Using the Einstein equations, one sees that both for the static and matter dominated models, the conformal time slice in which the events which generate the world lines are contained becomes progressively thinner as the inverse square of the scale factor, establishing a simple correspondence between the configurations predicted by the underlying Friedmann-Robertson-Walker dynamical model and the configurations in the conformal coordinates.