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

TL;DR

This paper derives a quantized phase-space metric tensor within a geometric quantization framework, extending general relativity to incorporate quantum effects and exploring the resulting implications for spacetime structure.

Contribution

It introduces a novel derivation of an eight-dimensional quantized phase-space metric tensor in GR, surpassing previous approximations and examining its implications for relative spacetime.

Findings

Derived a quantized eight-dimensional metric tensor from phase-space considerations.

Proposed a framework that extends GR to include quantum effects via phase-space metrics.

Examined the implications of the quantized metric on the structure of 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.