The Derivation of Phase-Space Metric in a Geometric Quantization Approach: General Relativity with Quantized Phase-Space Metric and Relative Spacetime

Abstract

Various extensions to Riemann geometry have been proposed since the inception of general relativity (GR). The aim has been and continues to be to construct a quantum and dynamic spacetime that incorporates the well-known classical (static) spacetime. Apparently, this seems to enable the principles of GR and quantum mechanics (QM) to be reconciled into a coherent relativity and quantum theory. A canonical geometric quantization approach that presents kinematics of free-falling quantum particles within a tangent bundle, expands QM to incorporate relativistic gravitational fields, and generalizes the four-dimensional Riemann manifold into an eight-dimensional one likely discretizes, if not fully quantizes, the Finsler and Hamilton structures. The Finsler and Hamilton metrics can be directly derived from the Hessian matrix. As introduced in [Physics, 7 (2025) 52], the quantized four-dimensional metric tensor can be deduced by means of approximations including proper parameterization of coordinates and the equating line elements on all these manifolds including Riemann manifold. This research, on the contrary, goes beyond all these approximations and proposes the incorporation of a phase-space metric tensor into GR. The derivation of a quantized eight-dimensional metric tensor is not only presented, but also the implications of it and the corresponding relative spacetime are examined.

44 pages, 0 table, 0 figure, accepted for publication in Nuclear Physics B

References in corpus (22)

• Kaluza-Klein Gravity
• 3d Quantum Gravity and Effective Non-Commutative Quantum Field Theory
• Making the Case for Conformal Gravity
• Sasaki-Einstein Manifolds and Volume Minimisation
• Planck-scale modified dispersion relations and Finsler geometry
• Generalized Uncertainty Principle: Approaches and Applications
• Short Distance vs. Long Distance Physics: The Classical Limit of the Minimal Length Uncertainty Relation
• The Generalized Uncertainty Principle in (A)dS Space and the Modification of Hawking Temperature from the Minimal Length
• Review on Generalized Uncertainty Principle
• Symmetry, Gravity and Noncommutativity
• Finsler geometric extension of Einstein gravity
• Minimum Length from Quantum Mechanics and Classical General Relativity
• Influence of Generalized and Extended Uncertainty Principle on Thermodynamics of FRW universe
• Finsler gravity action from variational completion
• Cosmological Finsler Spacetimes
• The Most General Form of Deformation of the Heisenberg Algebra from the Generalized Uncertainty Principle
• On gravitational waves in Born-Infeld inspired non-singular cosmologies
• A fundamental problem in quantizing general relativity
• A Possible Solution of the Cosmological Constant Problem based on Minimal Length Uncertainty and GW170817 and PLANCK Observations
• Covariant formulation of Generalised Uncertainty Principle
• Notes on the Sasaki metric
• Quantum-Conditioned Curvatures in Spacetime Surrounding Kerr-Newmann Black Hole