What Must Emerge?

“Spacetime emerges” is not one claim. A microscopic model may reproduce an adjacency graph yet lack causal order, or recover an Einstein-like equation while assuming the metric used to define its degrees of freedom. A precise proposal must name its fundamental objects, the state or phase considered, the map to geometric quantities, and the approximation in which each target is recovered.

Required background. Quantum Information supplies Hilbert-space factorization, operator algebras, correlations, channels, and state-dependent reconstruction used to formulate the microscopic side.

Helpful background. Flat-Spacetime Quantum Field Theory supplies local observables and microcausality; Einstein Equations supplies the dynamical target; Quantum Gravity supplies programme comparison; and Causality and Global Structure distinguishes local cones from global domains of dependence.

Eight distinct spacetime targets

The first target is relational localization: a way to say that two events or subsystems are near without importing a background coordinate distance. The second is a metric, including signature and scale, whose invariant intervals agree with clock, light-propagation, and free-fall measurements. An adjacency matrix or correlation strength may seed a distance, but triangle inequalities, continuum scaling, and Lorentzian signature require separate checks.

Third, a model must account for dimension and its scale dependence. Spectral, Hausdorff, and topological dimensions need not agree away from a smooth continuum. Fourth are topology and differentiable structure: local neighborhood data do not determine global connectedness, handles, or boundary conditions. Fifth is causal order. It must identify which interventions can influence which outcomes and recover stable light cones rather than merely an ordering parameter in an algorithm.

Sixth is a local quantum-field regime: approximately local operator algebras, cluster behavior, a vacuum or semiclassical state, and controlled violations at the microscopic scale. Seventh is gravitational dynamics. A metric-like variable must not merely exist; its long-wavelength response should reproduce the Einstein equation with the correct constraints, propagating polarizations, and universal coupling. Eighth is the semiclassical limit, in which geometric fluctuations are small enough for one effective spacetime to organize experiments while quantum corrections remain calculable.

These targets are linked but not identical. Entanglement can change without changing spatial topology. A causal set supplies order but needs dynamics and continuum reconstruction. A thermodynamic derivation may recover an equation of state while leaving microscopic localization unknown.

Fundamental data and reconstruction maps

A useful specification has four layers:

Layer	Required statement	Example of a discriminating check
Microscopic kinematics	Hilbert space or algebra, factorization, constraints, and observables	Remove gauge factors and count only physical degrees of freedom
Microscopic dynamics	Hamiltonian, path-integral weight, update rule, or process	Show stability under small allowed perturbations
Reconstruction	Map from states and operators to distances, areas, cones, and fields	Recover the same metric from two independent observables
Controlled regime	Large-size, continuum, code-subspace, equilibrium, or semiclassical limit	Bound corrections and identify when the map fails

The reconstruction map is often state dependent. For a network, graph cuts may represent area only inside a chosen code subspace. For correlations, defining $d(i,j)=-\ell\ln[I(i:j)/I_0]$ is a convention until one proves that the result approaches a geometric distance and remains stable under changes of local factorization. For causal processes, operational influence can be defined without a background metric, but a continuum cone requires additional structure.

The most powerful independent check is redundancy: reconstruct geometry from entanglement, propagation, and response, then compare. If correlation distance predicts one cone while commutator support predicts another, the proposal has not recovered one spacetime.

Dynamics, universality, and circularity

Jacobson’s local thermodynamic argument shows that an area–entropy relation, local boost energy, and the Clausius relation applied to every local Rindler horizon imply the Einstein equation with an undetermined cosmological constant Jacobson 1995, pp. 1260–1263. It is a powerful universality result: many microphysics could share the same infrared constitutive law. It does not identify those microphysics or derive the entropy postulate.

Circularity arises when the target already appears in the premises. A lattice placed in a fixed spatial metric can yield relativistic quasiparticles, but its background geometry has not emerged. A regulator defined by a background Laplacian may preserve covariance without removing background dependence. A network designed from a known geometry can demonstrate an encoding of that geometry, not its dynamical selection.

A claim is strongest when the microscopic definition avoids the target structure, a phase or state selects it dynamically, multiple relational probes agree, and deviations are bounded. Failure to recover exact locality at the microscopic level is not automatically fatal; what matters is whether violations decouple consistently in the continuum regime.

A compact evaluation protocol

Ask, in order: What is fundamental? What is the physical state space after constraints? Which quantity is claimed to emerge? Which reconstruction map defines it? In what limit is the map controlled? What independent observable checks it? What determines the dynamics? What correction would falsify the reconstruction?

This protocol also enforces scope. A model can legitimately claim “area-like connectivity in a code subspace” without claiming “four-dimensional Lorentzian spacetime.” Precision makes partial success more valuable because it shows exactly which problem has been solved.

Exercise

A quantum many-body ground state has mutual information that decays exponentially with graph separation, and the graph has a four-dimensional spectral dimension over one decade of scales. Classify what has and has not emerged.

Solution

The state supplies a correlation-based adjacency relation, and the chosen graph diffusion supplies evidence for an effective spectral dimension over the stated scale window. Neither fact establishes a Lorentzian metric: no clock or causal-cone observable has fixed signature or time. The result also does not establish topology, a local QFT algebra, universal spin-2 coupling, Einstein dynamics, or a semiclassical state. To strengthen it, one should derive the graph rather than assume it, reconstruct distance from an independent propagation observable, show a controlled continuum limit, and demonstrate that the same geometry governs causal influence and low-energy dynamics.

References

• Jacobson, Ted (1995), “Thermodynamics of Spacetime: The Einstein Equation of State,” Physical Review Letters 75, 1260–1263, arXiv:gr-qc/9504004.