On the Stueckelberg Like Generalization of General Relativity

TL;DR

This paper explores a higher-dimensional extension of general relativity and quantum gravity, linking Klein-Gordon and Stueckelberg equations through Clifford space, and proposes a generalized Wheeler-DeWitt equation that addresses the problem of time.

Contribution

It introduces a 6D space framework connecting Klein-Gordon and Stueckelberg equations, and develops a generalized quantum gravity formalism incorporating an extra parameter τ.

Findings

Reduction of Klein-Gordon to Stueckelberg equation in 4D

Formulation of Einstein equations in 6D Clifford space

Derivation of a generalized Wheeler-DeWitt equation that resolves the problem of time

Abstract

We first consider the Klein-Gordon equation in the 6-dimensional space $M_{2,4}$ with signature $+ - - - - +$ and show how it reduces to the Stueckelberg equation in the 4-dimensional spacetime $M_{1,3}$. A field that satisfies the Stueckelberg equation depends not only on the four spacetime coordinates $x^\mu$, but also on an extra parameter $\tau$, the so called evolution time. In our setup, $\tau$ comes from the extra two dimensions. We point out that the space $M_{2,4}$ can be identified with a subspace of the 16-dimensional Clifford space, a manifold whose tangent space at any point is the Clifford algebra Cl(1,3). Clifford space is the space of oriented $r$-volumes, $r=0,1,2,3$, associated with the extended objects living in $M_{1,3}$. We consider the Einstein equations that describe a generic curved space $M_{2,4}$. The metric tensor depends on six coordinates. In the presence of an isometry given by a suitable Killing vector field, the metric tensor depends on five coordinates only, which include $\tau$. Following the formalism of the canonical classical and quantum gravity, we perform the 4 + 1 decomposition of the 5-dimensional general relativity and arrive, after the quantization, at a generalized Wheeler-DeWitt equation for a wave functional that depends on the 4-metric of spacetime, the matter coordinates, and $\tau$. Such generalized theory resolves some well known problems of quantum gravity, including "the problem of time".