Spacetime Emergence and Quantum Causal Structure

Claims that spacetime “emerges” are scientifically useful only after the emergent target is specified. Metric distance, topology, dimension, causal order, local quantum fields, Einstein dynamics, and a semiclassical state are distinct achievements. This chapter develops operational tests for each target and compares entanglement, tensor-network, causal, and thermodynamic mechanisms without treating a suggestive analogy as a derivation of our spacetime.

Helpful background. Flat-Spacetime Quantum Field Theory supplies locality and operator algebras; Einstein Equations supplies the long-distance dynamical target; Quantum Information supplies entanglement, channels, and error correction; and Quantum Gravity supplies the comparison criteria for candidate microscopic programmes.

Separate the emergence targets

A microscopic Hilbert space does not arrive with a preferred tensor factorization, distance function, or causal cone. A proposed construction should therefore identify its fundamental degrees of freedom and then define which observables diagnose adjacency, dimension, topology, and time. What Must Emerge? turns this demand into a checklist. The checklist also prevents circularity: if a background metric is used to define the microscopic Hamiltonian, recovering that same metric is not yet background-independent emergence.

Entanglement can encode geometry when the state, factorization, and entropy–area relation are controlled. The entanglement first law $\delta S=\delta\langle K\rangle$ is a kinematic identity for nearby states; obtaining a gravitational equation requires additional input relating modular energy to geometric area. Entanglement and Geometric Relations derives that restricted logic. Tensor networks make factorization and connectivity concrete, and quantum error correction clarifies reconstruction, but graph distance is not automatically proper distance. Tensor Networks and Discrete Geometry states what network features model area, radial scale, and reconstructible regions.

Causal and thermodynamic mechanisms

Ordinary quantum circuits assume a definite order of operations. Process-matrix frameworks instead characterize correlations without presupposing a single global order, while maintaining local quantum operations and probabilities. Quantum Causal Order asks which such correlations can arise from a superposition of gravitational geometries and which are merely laboratory control systems on a fixed background. Causal nonseparability, violation of a causal inequality, and indefinite metric order are not equivalent claims.

Thermodynamic approaches begin elsewhere. Local horizon entropy, energy flux, and equilibrium conditions can yield Einstein’s equation as an equation of state under strong locality and universality assumptions. Jacobson’s argument relates the Clausius relation on all local Rindler horizons to the Einstein equation Jacobson 1995, pp. 1260–1263. Hydrodynamic and Thermodynamic Gravity compares this logic with entanglement equilibrium and induced dynamics. It explains why an infrared constitutive equation can be universal even when the microphysics is unknown—and why that universality does not reconstruct the microstates.

Recovering the joint low-energy limit

A successful pregeometric model must recover both local QFT and classical gravity in one regime. Approximate commuting algebras at spacelike separation, Lorentz symmetry, a massless spin-2 excitation with universal coupling, and controlled corrections must coexist. Recovery of Local QFT and General Relativity develops those simultaneous tests.

The strongest consistency check links targets rather than verifying each separately. The causal cones governing commutators should agree with those of the emergent metric; the same metric should enter the spin-2 dynamics; and matter should couple universally enough to reproduce the equivalence principle. Failure of this “same geometry” test can leave several individually attractive but mutually incompatible emergent structures.

This chapter uses tensor networks as general quantum-information models. Explicit holographic dualities, string backgrounds, and bulk reconstruction in established gauge/gravity duality belong to the string-theory volume; the present discussion uses them only as motivation and does not claim that an arbitrary network constructs quantum gravity.

Results and open questions through 2026-08-20

This discussion reflects results available through 2026-08-20. Competing positions differ over how much the existing examples establish. A structural view treats entanglement identities, error-correcting reconstruction, process matrices, and horizon thermodynamics as converging clues. A constructive view requires a microscopic dynamics, continuum limit, and joint recovery of locality and Einstein gravity before calling spacetime emergent. The general claim remains unresolved and depends on the state, operator algebra, causal structure, and semiclassical regime.

Mechanism or result	Evidence at the cutoff	Scope and limitation
Entanglement first-law routes	Restricted variations reproduce linearized gravitational equations in controlled settings	State, region, modular Hamiltonian, and background assumptions are essential
Tensor-network geometry	Bond cuts and code subspaces model areas and reconstructible regions Swingle 2012, pp. 4–8	Usually kinematic and discretized; realistic dynamics and continuum Lorentz symmetry are incomplete
Indefinite causal processes	Process matrices allow causal nonseparability Oreshkov, Costa, and Brukner 2012, pp. 2–5	Operational indefiniteness need not be produced by dynamical quantum geometry
Thermodynamic gravity	Local-equilibrium arguments recover Einstein-like constitutive equations	Microscopic degrees of freedom and nonequilibrium corrections are not uniquely fixed

The conclusions could change if a pregeometric model derives a common continuum metric for commutators, universal spin-2 coupling, and relational observables, or if a no-go result excludes that joint limit; the interpretation could also change with an experiment that distinguishes indefinite gravitational causal order from ordinary coherent control, or by a counterexample to the assumed entropy–geometry relation.

Check your understanding

Diagnostic. A tensor network has a minimal cut whose bond count scales with the boundary of a region, and its code subspace corrects erasure of some boundary factors. Which spacetime-emergence targets have been modeled, and which central targets remain unestablished?

Solution

The minimal cut models an area-like quantity, while the code property models redundant reconstruction of bulk information. Connectivity may also provide a discrete notion of adjacency or radial organization. These facts do not by themselves establish a Lorentzian causal cone, continuum dimension and topology, a dynamical metric obeying Einstein’s equation, a local QFT operator algebra, or a semiclassical state with controlled fluctuations. One must derive those structures from a specified microscopic Hamiltonian and continuum limit, then check that they refer to the same emergent geometry.

References

• Jacobson, Ted (1995), “Thermodynamics of Spacetime: The Einstein Equation of State,” Physical Review Letters 75, 1260–1263, arXiv:gr-qc/9504004.
• Oreshkov, Ognyan, Fabio Costa, and Časlav Brukner (2012), “Quantum Correlations with No Causal Order,” Nature Communications 3, 1092, arXiv:1105.4464.
• Swingle, Brian (2012), “Entanglement Renormalization and Holography,” Physical Review D 86, 065007, arXiv:0905.1317.