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On the Stueckelberg Like Generalization of General Relativity

Apr, 2011
16 pages
Published in:
  • J.Phys.Conf.Ser. 330 (2011) 012011
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Abstract: (arXiv)
We first consider the Klein-Gordon equation in the 6-dimensional space M2,4M_{2,4} with signature +−−−−++ - - - - + and show how it reduces to the Stueckelberg equation in the 4-dimensional spacetime M1,3M_{1,3}. A field that satisfies the Stueckelberg equation depends not only on the four spacetime coordinates xμx^\mu, but also on an extra parameter τ\tau, the so called evolution time. In our setup, τ\tau comes from the extra two dimensions. We point out that the space M2,4M_{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 rr-volumes, r=0,1,2,3r=0,1,2,3, associated with the extended objects living in M1,3M_{1,3}. We consider the Einstein equations that describe a generic curved space M2,4M_{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 τ\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 τ\tau. Such generalized theory resolves some well known problems of quantum gravity, including "the problem of time".
Note:
  • 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
  • space: Clifford
  • vector: Killing
  • field theory: vector
  • space-time
  • general relativity
  • quantum gravity
  • Wheeler-DeWitt equation
  • Klein-Gordon equation
  • Einstein equation
  • talk