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

This paper proposes a geometric quantization approach that extends General Relativity from a four-dimensional Riemannian manifold to an eight-dimensional phase-space framework, deriving a quantized metric tensor without relying on previous approximations to reconcile quantum mechanics with relativistic gravitational fields.

Original authors: Kareem Mubaidin, Debarshi Mukherjee, Saleh O. Allehabi, Azzah A. Alshehri, Mahmoud Nasar, Ashraf F. El-Sherif, Abdel Nasser Tawfik

Published 2026-07-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kareem Mubaidin, Debarshi Mukherjee, Saleh O. Allehabi, Azzah A. Alshehri, Mahmoud Nasar, Ashraf F. El-Sherif, Abdel Nasser Tawfik

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Problem: Two Different Rulebooks

Imagine the universe is a giant game. For the last century, physicists have been trying to play this game using two different rulebooks that don't agree with each other.

The "Smooth Road" Rulebook (General Relativity): This describes gravity and huge things like stars and galaxies. It says space and time are like a smooth, continuous fabric (a trampoline) that bends when heavy objects sit on it. It assumes everything is predictable and certain.
The "Pixelated Fog" Rulebook (Quantum Mechanics): This describes tiny things like atoms and electrons. It says the universe is "rough," fuzzy, and full of uncertainty. You can't know exactly where a particle is and how fast it's moving at the same time.

The problem is that when you try to combine these two rulebooks (like trying to describe a black hole, which is both huge and tiny), the math breaks down. The smooth fabric of gravity clashes with the fuzzy fog of quantum mechanics.

The Authors' Solution: A New Map

The authors of this paper propose a new way to draw the map of the universe. Instead of trying to force the two rulebooks to fit, they suggest expanding the game board itself.

1. Adding a "Speedometer" to the Map

In our normal world, to describe where you are, you just need a location (Latitude and Longitude). In physics, this is called Position.

However, the authors argue that to truly understand the universe at a quantum level, you can't just know where something is; you also need to know how fast and in what direction it is moving. This is called Momentum.

Think of it like this:

Old Map (Riemann Geometry): A flat piece of paper showing only cities and roads (Position).
New Map (Phase Space Geometry): A 3D hologram where every city has a "speedometer" attached to it. You are now navigating a space that includes both Location and Momentum.

The paper argues that the universe isn't just a 4-dimensional stage (3 space + 1 time); it's actually an 8-dimensional stage (4 space + 4 momentum).

2. The "Rough" Fabric (Finsler Geometry)

The old rulebook (Einstein's) assumes the fabric of space is perfectly smooth and the same in every direction (like a perfect circle).

The authors introduce a concept called Finsler Geometry. Imagine a road that feels different depending on which way you drive.

If you drive North, the road is smooth.
If you drive East, the road is bumpy.
If you drive fast, the road feels different than if you drive slow.

This "direction-dependent" roughness is what the authors call anisotropy. They suggest that at the quantum level, space isn't a smooth circle; it's more like a complex, bumpy terrain that changes based on how you move through it. This allows them to mix the "fuzziness" of quantum mechanics with the "gravity" of Einstein.

3. The "Quantum Deformation"

The paper uses a mathematical tool called the Relativistic Generalized Uncertainty Principle (RGUP).

Think of this as a "shrink-ray" for the rules of the game.

In the old rules, you could measure a particle's position perfectly if you had a big enough microscope.
In the new rules, the universe has a "minimum size" (a pixel limit). You can't zoom in forever.
The authors show that because of this limit, the coordinates of space (where things are) get "deformed" or warped based on the particle's momentum (how fast it's going).

It's like trying to take a photo of a speeding car. In the old world, the photo is just a bit blurry. In this new world, the road itself bends and twists depending on how fast the car is going.

What Did They Actually Do?

The paper is a heavy mathematical derivation. Here is the step-by-step of their "recipe":

Start with the basics: They took the standard equations for gravity (General Relativity) and the standard equations for uncertainty (Quantum Mechanics).
Expand the dimensions: They moved from a 4D world to an 8D world (adding momentum as a coordinate).
Create a new "Metric": In physics, a "metric" is a ruler used to measure distance. They derived a new, complex ruler (the Quantized Phase-Space Metric) that measures distance in this 8D world.
- This ruler changes depending on where you are and how fast you are moving.
- It accounts for the "roughness" of the quantum world.
Check the math: They showed that if you turn off the quantum effects (make the universe "smooth" again), their new ruler turns back into Einstein's old ruler. This proves their new theory includes the old one as a special case.
Derive the consequences: They used this new ruler to write down new equations for:
- Geodesics: The path a particle takes (like a straight line, but curved by this new "rough" space).
- Curvature: How space bends.
- Einstein's Field Equations: The master equations of gravity, now updated for this 8D, momentum-dependent universe.

The Bottom Line

The authors haven't built a new engine or cured a disease. They have built a new theoretical framework.

They claim that by viewing the universe as an 8-dimensional space where position and momentum are equally important, and by accepting that space is "rough" and direction-dependent (Finsler geometry), we can finally write a single set of math rules that describes both the smooth gravity of stars and the fuzzy uncertainty of atoms.

They call this "Relative Spacetime," suggesting that where and when an event happens might depend on the observer's momentum, not just their location. This is a purely mathematical proposal to solve the century-old puzzle of unifying gravity and quantum mechanics.

Technical Summary: The Derivation of Phase-Space Metric in a Geometric Quantization Approach

Problem Statement
The fundamental incompatibility between General Relativity (GR) and Quantum Mechanics (QM) remains a primary challenge in theoretical physics. GR relies on a smooth, certain, and static spacetime manifold, whereas QM incorporates inherent roughness, uncertainty, and non-commutativity. This orthogonality is particularly problematic in regimes involving extreme gravitational fields, such as black holes and cosmological singularities. While previous attempts to unify these theories have often extended spacetime geometry into phase space to integrate momentum space, existing approaches frequently rely on approximations that equate line elements between higher-dimensional manifolds and four-dimensional Riemannian space, or they fail to fully quantize the spacetime coordinates themselves, often restricting quantization to momentum space only.

Methodology
The paper employs a geometric quantization approach that generalizes the four-dimensional pseudo-Riemannian geometry of GR into an eight-dimensional phase-space geometry (specifically, a Finsler or Hamilton geometry on the cotangent bundle $T^*M$ ). The methodology proceeds through the following steps:

Relativistic Generalized Uncertainty Principle (RGUP): The authors utilize the RGUP to generalize QM, incorporating relativistic gravitational fields. This introduces non-commutative relations between position and momentum operators that depend on the metric tensor $g_{\mu\nu}$ , ensuring applicability in curved spacetime without violating Lorentz covariance.
Geometric Extension: The framework extends Riemann geometry to Finsler geometry (where the metric depends on both position and direction/velocity) and subsequently to Hamilton geometry (on the cotangent bundle). This allows for the inclusion of quantum mechanical revisions within the geometric structure.
Scalar Potential and Deformation: A scalar potential $S(x, p)$ is introduced, defined as a quantum-deformed Finsler function $F^2(x, p)$ . To address the limitation of previous models where only momentum space was quantized, the authors introduce a quantum-deformed spacetime coordinate $x^\mu = x^\mu_0 + \psi^\mu(x_0, p_0)$ . This deformation $\psi^\mu$ is derived from RGUP corrections, introducing anisotropy (direction dependence) while maintaining Lorentz covariance.
Metric Derivation: The eight-dimensional quantized metric tensor $\tilde{g}_{AB}$ $\tilde{g}_{A B}$ (where $A, B \in \{0, \dots, 7\}$ $A, B \in {0, \dots, 7}$ ) is derived using the Hessian matrix of the scalar potential $S(x, p)$ $S (x, p)$ . The metric is decomposed into three blocks:
- Spacetime-spacetime ( $\tilde{g}^{(xx)}_{\mu\nu}$ ): Derived from second derivatives with respect to coordinates.
- Spacetime-momentum ( $\tilde{g}^{(xp)}_{\mu\nu}$ ): Derived from mixed derivatives.
- Momentum-momentum ( $\tilde{g}^{(pp)}_{\mu\nu}$ ): Derived from second derivatives with respect to momenta.
  The authors utilize a Sasaki-type lift to construct the full metric, incorporating a nonlinear connection $N^\mu_\nu$ to ensure covariance under bundle coordinate transformations.

Key Contributions and Results

Derivation of the Quantized Eight-Dimensional Metric: The paper presents a explicit derivation of the quantized metric tensor $\tilde{g}_{AB}$ in phase space. Unlike previous four-dimensional approximations, this metric is defined directly on the eight-dimensional cotangent bundle $T^*M$ , with coordinates $X = [x^\alpha, p^\beta]$ .
Anisotropic Spacetime Deformation: By retaining specific RGUP correction terms (specifically the $\alpha'$ term in the coordinate deformation), the authors demonstrate that the resulting spacetime is anisotropic (Finslerian) rather than isotropic. This allows for coordinate deformations that are explicitly dependent on momentum without violating Lorentz invariance.
Block Structure of the Metric: The derived metric is shown to have a block structure characteristic of a Sasaki metric:
$\tilde{g}_{AB} = \begin{pmatrix} \tilde{g}^{(xx)}_{\mu\nu} & \tilde{g}^{(xp)}_{\mu\nu} \\ \tilde{g}^{(px)}_{\mu\nu} & \tilde{g}^{(pp)}_{\mu\nu} \end{pmatrix}$
The components are expressed in terms of the deformation functions $\phi(p)$ and $\psi(x, p)$ , and the underlying Finsler function $F$ .
Formulation of Phase-Space Dynamics: The authors derive the line element, geodesic equations, and curvature tensors (Ricci tensor and scalar) for this eight-dimensional geometry.
Einstein Field Equations in Phase Space: By applying the variational principle to the eight-dimensional Einstein-Hilbert action, the paper derives the Einstein Field Equations (EFE) for phase space:
$\tilde{G}_{AB} - \Lambda \tilde{g}_{AB} = \frac{8\pi G}{c^4} \tilde{T}_{AB}$
where $\tilde{T}_{AB}$ is the phase-space energy-momentum tensor.

Significance and Claims
The paper claims that this approach offers a coherent framework for reconciling GR and QM by treating spacetime as "relative" and embedded within momentum space, rather than as an absolute background. Key claims regarding significance include:

Resolution of Approximation Limitations: The work moves beyond the approximations required to map eight-dimensional metrics back to four-dimensional Riemannian metrics (such as equating line elements or ignoring anisotropic correction terms). It proposes a direct formulation within the eight-dimensional phase space.
Relative Locality: The framework aligns with the concept of "relative locality," suggesting that the conventional four-dimensional spacetime emerges from a higher-dimensional phase space where events are observer-dependent.
Unification via Geometry: By generalizing the metric tensor to include both position and momentum dependencies, the approach attempts to "gravitize" QM and "quantize" GR simultaneously. The metric tensor itself becomes the vehicle for this unification, carrying quantum mechanical features (via RGUP parameters) while retaining the geometric structure necessary for gravity.

The authors conclude that while the derivation involves perturbative expansions (linear order in deformation parameters), it establishes a new theoretical framework for representing gravity in phase space geometry, potentially offering a more accurate description of spacetime at quantum scales without the need for the specific approximations used in previous four-dimensional quantization attempts.