Emergent Space-Time Geometry arising from Quantum Entanglement / Emergente Spaziotemporale Geometrie aus Quantenverschränkung (Study I)

Emergent Space-Time Geometry arising from Quantum Entanglement Mathematical derivation of the Ryu-Takayanagi formula, Einsteinian dynamics from entanglement thermodynamics, complex entanglement patterns of the four fundamental forces Dr. Theodor Heutschi Date: August 2026 FIELD: Quantum gravity, AdS/CFT correspondence, quantum information theory, theoretical particle physics KEY TERMS: Ryu–Takayanagi formula, entanglement entropy, modular Hamiltonian, emergent spacetime, replica trick, fundamental forces, tensor networks, quantum error correction, gauge theories, gauge entanglement, entanglement topology, Higgs phase transition Abstract The unification of quantum mechanics with the general theory of relativity is one of the central challenges facing modern theoretical physics. Whilst traditional approaches such as perturbative quantum gravity fail due to non-renormalisation, and string theory is criticised for its mathematical hyper-complexity, a paradigm shift has taken place: Spacetime and gravity are no longer understood as fundamental building blocks, but as macroscopic emergent phenomena resulting directly from microscopic quantum entanglement (“It from Qubit”). This scientific study presents a mathematically precise and physically profound elaboration of the Ryu-Takayanagi formula (RT formula) within the framework of AdS/CFT duality. We show in detail how the geometric minimal surface 𝛾𝐴 in a higher-dimensional bulk spacetime calculates the von Neumann entanglement entropy 𝑆_𝐴 of a boundary quantum field theory subsystem. Furthermore, using the First Law of Entanglement Thermodynamics (δ𝑆_𝐴 = δ⟨𝐻_𝐴⟩ ), we derive that the linearised Einstein field equations necessarily follow from the consistency conditions of quantum entanglement. A key innovation of this work lies in the systematic derivation of the four fundamental forces - gravity, electromagnetism, the strong and weak interactions - from different topologies and knot invariants of complex quantum entangled states. We show that the SU(3) x SU(2) x U(1) symmetry of the Standard Model can be reconstructed as an emergent property of specific entanglement patterns in the boundary Hilbert space, whereby the respective range and coupling strength correlate directly with the entanglement length and the topological entanglement entropy. Finally, we formalise quantum entanglement in the presence of local gauge symmetries (charged entanglement entropy), propose a unified MERA spin-network architecture, and model the electroweak Higgs mechanism as a topological entanglement phase transition. Solutions for de Sitter holography ( 𝑇𝑇̅-deformations) and bulk reconstruction via quantum error correction (QECC) provide a robust theoretical foundation for expanding emergent spacetime geometry to realistic cosmologies.