Quantum Gravity Framework Based on Noncommutative Geometry and Quantum Entanglement Dynamics

This paper proposes a quantum gravity framework integrating noncommutative geometry (NCG) and dynamic quantum entanglement. By rigorously mapping noncommutative spacetime algebra to quantum bit networks, we demonstrate that spacetime geometry emerges spontaneously from the covariant gradient of entanglement entropy, deriving a natural unification mechanism for general relativity and quantum field theory. Key innovations include: 1. Geometrization of Noncommutative Spacetime and Entanglement: Spacetime coordinates satisfy [xµ,xν] = iθµν, where the noncommutative parameter θµν is dynamically generated by entanglement network topology, obeying θµν ∝ ∇µSent∇νSent. 2. Covariant Quantum Einstein Equations: Derived via noncommutative variational principles, the modified Einstein equations read Gµν +Λeffgµν = 8πG⟨Tµν⟩+κ∇µSent∇νSent, with Λeff driven by entanglement entropy fluctuations. 3. Experimental Predictions: • Noncommutative Corrections to Gravitational Wave Polarization: High-frequency gravitational waves (f ∼ 103 Hz) exhibit anisotropic polarization tensors, with ∆h+/h× ∝ (θ·f)2, testable by next-generation detectors (e.g., Einstein Telescope). • Entanglement-Induced Dark Matter Candidate: Ascalarfieldparticle (mϕ ∼ Λeff) emerges from entangled vacuum excitations, with coupling strength correlated to CMB anisotropies, fitting Planck data with errors below 1σ