Loop Quantum Gravity in Ashtekar and Lagrange-Finsler Variables and Fedosov Quantization of General Relativity
arXiv:0801.4942
Abstract
We propose an unified approach to loop quantum gravity and Fedosov quantization of gravity following the geometry of double spacetime fibrations and their quantum deformations. There are considered pseudo-Riemannian manifolds enabled with 1) a nonholonomic 2+2 distribution defining a nonlinear connection (N-connection) structure and 2) an Arnowitt-Deser-Misner 3+1 decomposition. The Ashtekar-Barbero variables are generalized and adapted to the N-connection structure which allows us to write the general relativity theory equivalently in terms of Lagrange-Finsler variables and related canonical almost symplectic forms and connections. The Fedosov results are re-defined for gravitational gauge like connections and there are analyzed the conditions when the star product for deformation quantization is computed in terms of geometric objects in loop quantum gravity. We speculate on equivalence of quantum gravity theories with 3+1 and 2+2 splitting and quantum analogs of the Einstein equations.
latex 2e, 11pt, 48 pages, 1 latex figure, discussion added
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Cited by in corpus (6)
- Principles of Einstein-Finsler Gravity and Perspectives in Modern Cosmology
- Spectral Functionals, Nonholonomic Dirac Operators, and Noncommutative Ricci Flows
- Nonholonomic Ricci Flows: II. Evolution Equations and Dynamics
- Einstein Gravity in Almost Kahler Variables and Stability of Gravity with Nonholonomic Distributions and Nonsymmetric Metrics
- Nonholonomic Clifford and Finsler Structures, Non-Commutative Ricci Flows, and Mathematical Relativity
- Curve Flows and Solitonic Hierarchies Generated by Einstein Metrics