On the Stueckelberg Like Generalization of General Relativity

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".

18 pages. Presented at "The 7th Biennial Conference on Classical and Quantum Relativistic Dynamics of Particles and Fields", 30 May - 1 June 2010, National Dong HWa University - Hualien, Taiwan; published version of the paper

References in corpus (7)

• The Standard Model of Particles and Forces in the Framework of 2T-physics
• Spin Gauge Theory of Gravity in Clifford Space: A Realization of Kaluza-Klein Theory in 4-Dimensional Spacetime
• Kaluza-Klein Theory without Extra Dimensions: Curved Clifford Space
• Clifford Space as a Generalization of Spacetime: Prospects for QFT of Point Particles and Strings
• A Novel View on the Physical Origin of E8
• On a Unified Theory of Generalized Branes Coupled to Gauge Fields, Including the Gravitational and Kalb-Ramond Fields
• The Landscape of Theoretical Physics: A Global View; From Point Particles to the Brane World and Beyond, in Search of a Unifying Principle

Cited by in corpus (2)

• Branes and Quantized Fields
• Extended Relativity: Beyond