On the Stueckelberg-like generalization of general relativity

Abstract

We first consider the Klein-Gordon equation in the 6-dimensional space M2,4 with signature +−−−−+ and show how it reduces to the Stueckelberg equation in the 4-dimensional spacetime M1,3. A field that satisfies the Stueckelberg equation depends not only on the four spacetime coordinates xμ, but also on an extra parameter τ, the so called evolution time. In our setup, τ comes from the extra two dimensions. We point out that the space M2,4 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 M1,3. We consider the Einstein equations that describe a generic curved space M2,4. 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 τ. 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 τ. Such generalized theory resolves some well known problems of quantum gravity, including "the problem of time".