Ryu-Takayanagi Formula: In the AdS/CFT correspondence, the Ryu-Takayanagi formula connects the entanglement entropy of a quantum system to the area of a minimal surface in the bulk spacetime. This indicates that spacetime geometry is encoded in the quantum entanglement of the boundary theory.
If ER = EPR is true, we might extend this idea to say that entanglement between quantum particles (EPR pairs) gives rise to wormhole-like connections in spacetime. By finding a formalism that connects the entropy of entanglement with the geometry of wormholes, we could develop a framework that unifies quantum information (QFT) with spacetime geometry (GR).
4. Non-Traversable Wormholes as a Model for Entanglement:
Non-traversable wormholes, which are solutions to the Einstein field equations, do not allow physical objects to travel through them, but they can still represent a connection between two regions of spacetime. These are analogous to quantum entanglement, where information about one particle’s state is instantaneously connected to the other, but no information can travel faster than light.
To formalize ER = EPR, we need a mathematical framework where non-traversable wormholes correspond to quantum correlations (entanglement) between distant particles. This would require a modification of Einstein's equations to account for quantum degrees of freedom (like entanglement) as a source of spacetime geometry.
Possible Formalism for ER = EPR
1. Einstein's Equations with Quantum Entanglement Terms: We could modify Einstein’s field equations to include terms that account for quantum entanglement as a source of spacetime curvature. This would involve coupling the classical stress-energy tensor  with a term that represents quantum information or entanglement.
The modified Einstein equations might take the form:
R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R + \Lambda g_{\mu\nu} = 8\pi G \left( T_{\mu\nu} + T_{\mu\nu}^{\text{quantum}} \right)
2. Quantum Circuit Representation of Spacetime: Another formalism might involve representing spacetime as a quantum circuit, where each EPR pair is represented by a quantum gate in a tensor network, and wormholes correspond to specific connections between these gates. This could be realized through a discrete model of spacetime, where each quantum bit (qubit) represents a piece of spacetime, and the entanglement structure dictates the geometry.
3. Entanglement as the Fundamental Building Block of Spacetime: The ER = EPR conjecture suggests that spacetime might emerge from the entanglement structure of quantum fields. A formalism based on entanglement entropy and its relationship to spacetime curvature could serve as a bridge between GR and QFT, where spacetime geometry is reconstructed from quantum entanglement
