Energy-momentum and dark energy in $\boldsymbol{SU(\infty)}$-QGR quantum gravity

This paper demonstrates that in the $SU(\infty)$-QGR quantum gravity framework, the fragmentation of the universal Hilbert space generates local symmetries and spin-1 gravitons, leading to an Einstein-like constraint equation and identifying fields associated with inflation and late-time acceleration as order parameters reflecting the evolution of the Universe's quantum states.

The Gist

The Big Idea: The Universe as a Giant Quantum Orchestra

Imagine the Universe not as a stage with actors (particles) moving around on it, but as a giant, self-contained orchestra. In this model, called SU(∞)-QGR, there is no "stage" (spacetime) to begin with. There is only the music (the quantum states) and the musicians (the particles).

The paper argues that what we perceive as "space," "time," and "gravity" are just the average sound of this orchestra playing together. When the music gets complex and the musicians start playing in small, isolated groups, we suddenly feel like we are in a room with walls and a floor.

1. The "Stage" is an Illusion

In standard physics, we think of space as a fixed grid where things happen. In this paper, the author says: No, the grid is made up of the things happening on it.

• The Analogy: Imagine a crowd of people in a dark room. If everyone stands still, there is no "room" you can measure. But if people start moving, bumping into each other, and forming groups, you can suddenly say, "The group on the left is far from the group on the right."
• The Science: The "distance" between things isn't a pre-existing thing. It emerges from how the quantum states of particles change relative to each other. If the quantum states are very similar, they are "close." If they are very different, they are "far."

2. Gravity is Just a "Tension" in the Music

Usually, we think of gravity as a force pulling things together. In this model, gravity is a symmetry (a rule of the music) called SU(∞).

• The Analogy: Think of the Universe as a giant, invisible drum skin. The "gravity" is the tension in that skin.
• The Twist: In this model, the drum skin isn't a physical object; it's just the mathematical relationship between the notes being played.
• The Result: The paper derives a rule that looks exactly like Einstein's famous equation ($E=mc^2$'s big brother). But here, it's not a law of nature written in stone; it's a balance sheet. It says: The energy of the "music" (matter) plus the energy of the "tension" (gravity) must always add up to zero.
  • Why zero? Because the Universe is a closed system. There is no "outside" to push against, so the total energy must be balanced.

3. What is Dark Energy? (The Mystery of the Accelerating Expansion)

We know the Universe is expanding faster and faster. We call the invisible push causing this "Dark Energy." Usually, scientists think this is "vacuum energy" (energy from empty space).

The Problem: In this model, there is no empty space. The Universe is never empty; it's always full of quantum activity. So, where does Dark Energy come from?

The author suggests three creative possibilities:

1. The "Static Hum": Imagine the gravity field has a constant, low-level hum (a zero-mode) that never stops. This hum acts like a constant pressure pushing everything apart.
2. The "Topological Knot": The structure of the Universe might have a knot in it from the very beginning (at the Big Bang). Untying or maintaining this knot requires energy, which looks like Dark Energy.
3. The "Entanglement Tax": This is the most fascinating idea.
   • The Analogy: Imagine the Universe is one giant, tangled ball of yarn. To keep all the threads connected (entangled), you need energy. But as the yarn gets pulled apart into separate balls (subsystems), it releases energy.
   • The Science: As the Universe breaks apart into smaller, isolated particles (a process called Hilbert Space Fragmentation), the "global entanglement" changes. The paper suggests that the energy required to maintain this global connection, or the energy released when it changes, manifests as the "push" we call Dark Energy.

4. The "Fragmentation" of the Universe

The paper uses a concept from quantum computing and condensed matter physics called Hilbert Space Fragmentation.

• The Metaphor: Imagine a giant, perfectly clear lake (the early Universe). The water is one big, connected wave.
• The Event: Suddenly, ice starts forming. The lake breaks into thousands of separate ice floes.
• The Result: Even though the water is still there, it's now divided into isolated chunks.
• Why it matters: The paper argues that our Universe is like that lake. It started as one big quantum wave. As it cooled and evolved, it "froze" into isolated particles (electrons, quarks, etc.).
  • Inflation: The rapid expansion of the early Universe happened because the "ice" was breaking apart so fast.
  • Dark Energy: The current acceleration is the result of the "ice floes" drifting further apart, changing the way the "water" (the quantum state) behaves.

5. The "Spin-1" Graviton

In standard physics, gravity is carried by a "spin-2" particle (the graviton). This model suggests gravity is actually a spin-1 field (like light or magnetism), but because it's part of the infinite SU(∞) symmetry, it behaves differently.

• The Analogy: Think of light (spin-1) as a radio wave. In this model, gravity is also a radio wave, but it's broadcasting on a frequency so low and vast that it connects everything in the Universe simultaneously.

Summary: What does this mean for us?

This paper proposes a radical shift in how we see reality:

1. Space and Time are not fundamental. They are "emergent" properties, like temperature. You can't have a single molecule be "hot," but a crowd of them can be. Similarly, you can't have a single particle have "distance," but a crowd of them creates the illusion of space.
2. Dark Energy is a side effect. It's not a mysterious new substance; it's the result of the Universe breaking into smaller pieces and the quantum rules changing as it does.
3. The Universe is a closed loop. Everything balances out to zero. The "push" of Dark Energy is just the Universe adjusting its internal balance sheet as it evolves.

In a nutshell: The Universe isn't a box with things inside it. It's a giant, evolving quantum dance. As the dancers (particles) separate and form their own groups, the "floor" (space) they stand on expands, and the "music" (gravity) changes, creating the effects we call Dark Energy.

Technical Summary

1. Problem Statement

The paper addresses the fundamental inconsistency between General Relativity (GR) and Quantum Mechanics (QM), specifically regarding the nature of spacetime and gravity.

• The Core Issue: In standard GR, spacetime is a dynamic background, while in QM, entities exist within a fixed background. Attempts to quantize gravity (e.g., string theory, loop quantum gravity) often struggle with renormalization, the definition of gravitational energy-momentum, and the origin of spacetime dimensionality.
• Specific Challenges:
  • The lack of a consistent definition for the energy-momentum tensor of gravity in classical GR (it is usually defined only perturbatively for gravitational waves).
  • The nature of Dark Energy: Whether it is a cosmological constant (vacuum energy) or a dynamical field, and how it arises in a quantum framework where "vacuum" might not exist.
  • The emergence of classical spacetime from a fundamentally quantum system without a pre-existing background.

2. Methodology

The author utilizes SU(∞)-QGR (SU($\infty$) Quantum Gravity), a fundamentally quantum approach where the Universe is treated as a single isolated quantum system with $SU(\infty)$ symmetry.

• Foundational Axioms:
  • The Hilbert space of the Universe ($H_U$) represents the $SU(\infty)$ symmetry group.
  • There is no background spacetime; spacetime emerges from the effective parameters characterizing the quantum states of subsystems.
  • The Universe fragments into approximately isolated subsystems (particles) via Hilbert space fragmentation, a concept borrowed from condensed matter physics (Many-Body Localization and Hilbert Space Fragmentation).
• Mathematical Framework:
  • Lagrangian Superoperators: The author defines Lagrangian functionals as $C$-numbers derived from traces of $SU(\infty)$ generators, then promotes them to Lagrangian superoperators acting on the space of density matrices ($B[H_U]$).
  • Metric Independence: The effective action for subsystems is defined on a 4D parameter space $\Xi$. The paper imposes the condition that the action must be independent of the metric of $\Xi$ (diffeomorphism invariance of the parameter space).
  • Quantum Information Measures: The study employs concepts from Quantum Information Theory (QIT), specifically coherence ($\ell_1$-norm) and state overlap (fidelity), to quantify the fragmentation of the Hilbert space and the emergence of classical geometry.

3. Key Contributions

A. Definition of Gravitational Energy-Momentum Tensor

The paper derives a constraint equation resembling the Einstein field equations but interpreted differently:

• Constraint vs. Field Equation: In SU(∞)-QGR, the equation derived from the metric independence of the Lagrangian is a consistency constraint, not a dynamical field equation for gravity (since gravity is an $SU(\infty)$ gauge field).
• Gravitational Energy-Momentum: By applying the standard definition of the energy-momentum tensor ($T_{\mu\nu} \propto \delta S / \delta g^{\mu\nu}$) to the $SU(\infty)$ Yang-Mills term, the author defines a gravitational energy-momentum tensor ($T_{\mu\nu}^{(SU(\infty))}$).
• The Constraint Equation: The result is a zero-total-energy constraint:
  $$\kappa T_{\mu\nu}^{(SU(\infty))} + T_{\mu\nu}^{(G)} + T_{\mu\nu}^{(m)} = 0$$
  Where $T_{\mu\nu}^{(SU(\infty))}$ is the energy-momentum of the spin-1 graviton field, $T_{\mu\nu}^{(G)}$ is for internal symmetries, and $T_{\mu\nu}^{(m)}$ is for matter. This implies the total energy-momentum of the closed Universe is zero.

B. Resolution of the Vacuum Energy Problem

• No Global Vacuum: The model posits that a "global vacuum" (an empty Universe) does not exist because the $SU(\infty)$ symmetry requires infinite observables. An empty state would imply a trivial representation, meaning no Universe.
• Dark Energy Origin: Dark energy is not vacuum energy but arises from the dynamics of the quantum state of the Universe. The paper proposes three mechanisms:
  1. Static $SU(\infty)$ Field: A zero-mode condensate of spin-1 gravitons acting as a cosmological constant.
  2. $\Theta$-Vacuum: A topological term (Chern-Simons) in the initial state.
  3. Global Entanglement vs. Decoherence: An equilibrium between the energy required for global entanglement and the energy released by local decoherence, manifesting as negative pressure (dark energy).

C. Emergence of Spacetime and Cosmology

The paper links the evolution of the quantum state to the expansion of the Universe:

• Affine Separation: The "distance" between events in the emergent spacetime is defined by the variation of the quantum state (density matrix) in Hilbert space, related via the Mandelstam-Tamm inequality (Quantum Speed Limit).
• Hilbert Space Fragmentation: The expansion of the Universe is interpreted as the growth in the number of approximately isolated subsystems (fragmentation).
• Inflation and Acceleration:
  • Inflation: Corresponds to the rapid fragmentation of the Hilbert space into many subsystems. The "inflaton" is not a fundamental field but an order parameter describing this fragmentation.
  • Late-time Acceleration: Driven by the slow evolution of a spin-1 graviton condensate (a generalized Glauber coherent state) which remains entangled with the rest of the system, providing a dynamical dark energy component.

4. Key Results

1. Einstein Equation as a Constraint: The Einstein equation emerges not as a law of motion for geometry, but as a constraint ensuring the total energy-momentum (including gravity) vanishes. This resolves the ambiguity of defining gravitational energy in GR.
2. Spin-1 Gravitons: In this framework, the mediator of gravity is a spin-1 field (part of the $SU(\infty)$ Yang-Mills theory), contrasting with the spin-2 graviton of perturbative GR. The spin-2 behavior emerges only in the classical limit of fluctuations.
3. Dark Energy Dynamics: The model supports a dynamical dark energy ($w(z)$ varying with redshift) arising from the interplay between the graviton condensate and the fragmentation of the quantum state, rather than a static cosmological constant.
4. Black Hole Definition: A black hole is defined as a fully entangled, inseparable quantum system where the affine separation $ds \to 0$ (orthogonality of states), and Hawking radiation is the process of breaking this entanglement.
5. Weak Gravity Conjecture: The analysis of state overlap under $SU(\infty)$ generators shows that small-scale (UV) variations are amplified, implying local interactions are stronger than global gravity, consistent with the Weak Gravity Conjecture.

5. Significance

• Conceptual Unification: The paper provides a unified framework where spacetime, gravity, and matter emerge from a single quantum mechanical principle ($SU(\infty)$ symmetry) without a background.
• Renormalization: By treating gravity as a gauge field within a constrained system, the model potentially bypasses traditional renormalization issues associated with the quantization of the metric.
• Cosmological Insights: It offers a novel explanation for inflation and dark energy rooted in quantum information theory (entanglement, decoherence, and state overlap) rather than ad-hoc scalar fields.
• Testability: The model predicts that dark energy is dynamical and linked to the "order parameter" of Hilbert space fragmentation, offering potential observational signatures distinct from standard $\Lambda$CDM models.

In summary, Ziaeepour's work reinterprets gravity and cosmology through the lens of quantum information and $SU(\infty)$ symmetry, suggesting that the geometry of the Universe is a manifestation of the statistical properties and fragmentation of its underlying quantum state.