Spacetime from Operator Algebras

Abstract: Under suitable assumptions, geometric objects such as the spacetime metric and curvature tensor can be reconstructed from the algebra of operators of quantized matter fields in the limit of vanishing Newton's constant. In this framework, the full non-linear Einstein equations can be expressed in the language of operator algebras, extending Jacobson's derivation without invoking the area law for Bekenstein-Hawking entropy. These assumptions can then be used as a criterion for determining whether the semiclassical limit of a given quantum theory admits an emergent gravitational description. Going in the other direction, the discrete spectrum of a holographic theory at finite N can be modelled by adding non-perturbative corrections to semiclassical operator algebras. The type III von Neumann algebra that arises in the vanishing Newton's constant limit can be enlarged by adjoining its modular Hamiltonian. A random matrix theory completion of this enlarged algebra, followed by ensemble averaging, results in a type I von Neumann algebra whose minimal projectors approximate those of the underlying microstates. In the case of an eternal black hole, the dimension of the type I algebra equals the Bekenstein-Hawking entropy with universal logarithmic corrections. The complexity of probe operators in the boundary theory provides a diagnostic of the validity of the corresponding bulk semiclassical effective field theory.

• The paper reconstructs the metric and Ricci curvature from Hadamard two-point functions, local Rindler algebras, and modular flows, without assuming a pre-existing geometry.
• The paper derives the nonlinear Einstein equations from locally stationary states and relative entropy in a crossed-product algebra, without assuming the Bekenstein–Hawking area law.
• The paper shows that random matrix completion can produce an ensemble-averaged type I factor with dimension approximately e^(S_BH+S_log), linking finite-N black hole microstates to spectral discreteness and complexity growth.

Overview

"Spacetime from Operator Algebras" by Mohan and Thorlacius develops a program in which spacetime geometry and gravitational dynamics are reconstructed from the von Neumann algebraic structure of quantized matter fields, without assuming a boundary dual or invoking Euclidean path integrals. The paper has two main results. First, under three purely algebraic assumptions, the metric and curvature tensors of a $D$-dimensional Lorentzian manifold can be extracted from the triple $(\mathcal{A}, \mathcal{H}, |\omega\rangle)$, where $\mathcal{A}$ is the operator algebra of a free massive scalar field, $\mathcal{H}$ its Fock space, and $|\omega\rangle$ a preferred state; with one further condition — the existence of locally stationary states — the full nonlinear Einstein equations follow, extending Jacobson's thermodynamic derivation while dispensing with the Bekenstein–Hawking area law as an input. Second, for an eternal AdS black hole dual to a thermofield double (TFD) state, a random matrix theory (RMT) completion of the large-$N$ crossed product algebra produces an ensemble-averaged type I$_d$ factor whose Hilbert space dimension is $e^{S_{\text{BH}} + S_{\text{log}}}$, i.e., the Bekenstein–Hawking entropy including universal logarithmic corrections.

The framework is motivated by Connes' reconstruction theorem, which recovers a compact Riemannian spin manifold from the commutative spectral triple $(C^\infty(M), L^2(M,S), \mathcal{D})$, and by recent work showing that Killing horizons emerge already at strict $G_N \to 0$ from the algebra of bulk fields in holography [Leutheusser:2021frk, Leutheusser:2021qhd]. The paper's contribution is to show that not merely horizons but all local geometric data — metric, Ricci tensor, and ultimately the field equations — are encoded algebraically.

Geometry from the length function

Working in the $(\mathcal{A}, \mathcal{H}, |\omega\rangle)$0 limit, where the bulk is free fields on a fixed background, the authors associate each spacetime point with an operator insertion $(\mathcal{A}, \mathcal{H}, |\omega\rangle)$1 and define a length function from the state's two-point functions. In the large-mass limit, which corresponds to the geometric optics limit of the wave equation, the Wightman function behaves as

$(\mathcal{A}, \mathcal{H}, |\omega\rangle)$2

and $(\mathcal{A}, \mathcal{H}, |\omega\rangle)$3 is extracted by differentiating with respect to $(\mathcal{A}, \mathcal{H}, |\omega\rangle)$4 after removing the divergent piece. The construction is verified explicitly for free scalars in Minkowski space (via the Bessel-function form of the Wightman function) and in AdS$(\mathcal{A}, \mathcal{H}, |\omega\rangle)$5 (via the hypergeometric expression), reproducing the correct geodesic distance in both cases. For states with geometric modular flow, the length function must satisfy the reverse triangle inequality on timelike-separated points, and saturating configurations identify operators inserted along timelike geodesics.

Three assumptions underpin the reconstruction: (1) the state $(\mathcal{A}, \mathcal{H}, |\omega\rangle)$6 satisfies the Hadamard condition, interpreted as an operator-algebraic realization of the equivalence principle since it enforces the universal flat-space short-distance singularity structure; (2) the algebra contains local Rindler subalgebras $(\mathcal{A}, \mathcal{H}, |\omega\rangle)$7, each approximated by the algebra $(\mathcal{A}, \mathcal{H}, |\omega\rangle)$8 of a massive scalar on a full Rindler spacetime obtained by extending the locally flat patch; and (3) the two-point functions admit a well-defined large-mass limit. Crucially, these conditions are stated entirely in algebraic terms, so they double as criteria for whether an abstract triple admits an emergent geometric interpretation — no pre-existing geometry need be assumed.

Given the length function, Synge's world function $(\mathcal{A}, \mathcal{H}, |\omega\rangle)$9 yields the metric through the coincidence limit of its second derivatives. Since the "points" here are operator labels rather than manifold points, ordinary differentiation is unavailable; instead, the local Rindler assumption supplies approximate Poincaré generators. By the Bisognano–Wichmann theorem, modular operators of wedge algebras generate boosts, and half-sided modular inclusions generate null translations; combining wedges reconstructs the full Poincaré algebra (an explicit higher-dimensional construction is given in the appendices). Modular evolution then provides a continuous flow parameter $\mathcal{A}$0 along approximate Killing vectors $\mathcal{A}$1, so that

$\mathcal{A}$2

and varying over the available flows determines all metric components. Curvature enters through the subleading Hadamard-structure terms: expanding the coefficient $\mathcal{A}$3 in powers of $\mathcal{A}$4 brings in the van Vleck–Morette determinant, whose expansion contains $\mathcal{A}$5. Physically, this determinant encodes geodesic focusing via the Raychaudhuri equation, which explains why the Ricci tensor is accessible from two-point functions:

$\mathcal{A}$6

In the strict $\mathcal{A}$7 limit this also gives an algebraic diagnostic for the vacuum equations $\mathcal{A}$8, checked by contracting with independent approximate Killing vectors.

Einstein equations from locally stationary states

To derive rather than diagnose the field equations, the authors include perturbative $\mathcal{A}$9 corrections so that matter back-reacts. The key additional input is the existence of locally stationary states: quasifree Hadamard states stationary under affine dilations along the past branch $\mathcal{H}$0 of a local Rindler horizon. Restricted to the horizon algebra $\mathcal{H}$1, such a state satisfies the KMS condition at inverse temperature $\mathcal{H}$2, making the GNS vector separating and Tomita–Takesaki modular theory applicable; the modular operator acts as affine dilations by the extended Bisognano–Wichmann theorem.

Observables invariant under the modular flow are constructed by dressing with the associated charge, producing a type II crossed product algebra in which density matrices, traces, and entropies are well defined. For coherent excitations $\mathcal{H}$3 supported on $\mathcal{H}$4, the relative entropy obeys

$\mathcal{H}$5

Two technical caveats are noted: the individual right-hand-side terms are ill-defined at $\mathcal{H}$6, though their combination remains finite; and the area expectation value used here omits a perturbative charge correction that drops out of the relative entropy anyway. Evaluating both sides — the left side as horizon energy flux $\mathcal{H}$7, the right side as $\mathcal{H}$8 since coherent excitations leave entanglement entropy unchanged [Fiola:1994ir] — reproduces Jacobson's Clausius relation $\mathcal{H}$9. Imposing stress-tensor conservation and applying the Raychaudhuri equation then yields

$|\omega\rangle$0

The distinction from Jacobson's original argument is substantive: the area law for entropy is never assumed. It emerges instead from the algebraic structure of locally stationary states and their coherent excitations within the crossed product. The characterization is dimension-independent and imposes no conditions on asymptotics.

Type I algebras from RMT completion

The second half addresses finite-$|\omega\rangle$1 physics. In the $|\omega\rangle$2 limit of the boundary CFT (above the Hawking–Page temperature), the single-trace algebra is type III$|\omega\rangle$3 — admitting neither density matrices nor pure states — and adjoining the renormalized Hamiltonian $|\omega\rangle$4 via the crossed product $|\omega\rangle$5 yields a type II$|\omega\rangle$6 factor [Witten:2021unn, Chandrasekaran:2022eqq], which supports traces but still lacks minimal projectors and hence microstates. There is no canonical route from type III or II to type I; the paper constructs an approximate one.

The mechanism is an RMT completion: the smooth large-$|\omega\rangle$7 density of states $|\omega\rangle$8 (obtained by inverse Laplace transform of $|\omega\rangle$9) is replaced by the density of states of a random matrix ensemble with $N$0. Applied to time-shifted TFD states $N$1, the Gram matrix overlaps become the spectral form factor; the connected part scales as $N$2, and resolvent techniques give a completed Hilbert space of dimension $N$3. This restores the spectral discreteness lost at large $N$4 — level spacings there are suppressed as $N$5 — and is motivated independently by black hole fast scrambling and by the generic emergence of RMT behavior in chaotic systems, e.g., hard-sphere gases.

The algebraic version proceeds by decomposing the type II$N$6 factor into standard form $N$7 using bounded spectral projections $N$8 onto energy windows of width $N$9. After RMT completion and ensemble averaging — now requiring the full GUE sine-kernel two-point function rather than the Poisson approximation, since only energies near $_d$0 contribute — the authors show

$_d$1

up to corrections suppressed by $_d$2. The second tensor factor collapses because level repulsion makes the averaged projector behave minimally: any proper subprojector $_d$3 either satisfies $_d$4 or $_d$5, a dichotomy following from cancellation between the connected and sine-kernel contributions when the window narrows below the level-repulsion scale. The first tensor factors match because Fourier-projectors onto time-shifted states have energy resolution $_d$6 compatible with $_d$7 when $_d$8.

Fixing the trace ambiguity of the type II algebra canonically by requiring $_d$9 fixes the constant $e^{S_{\text{BH}} + S_{\text{log}}}$0, and the connected trace of products of time-evolved density matrices gives

$e^{S_{\text{BH}} + S_{\text{log}}}$1

where $e^{S_{\text{BH}} + S_{\text{log}}}$2 is the fluctuation entropy carrying the universal logarithmic correction to black hole entropy [Das:2001ic]. The same construction transposes to the bulk, with the Hartle–Hawking state replacing the TFD and boosts generated by the modular Hamiltonian inside the wedge.

Complexity as a diagnostic

Because the overlap of time-shifted TFD states is the spectral form factor, the validity of semiclassical bulk effective field theory can be read off from operator complexity. Krylov complexity grows linearly before saturating at times $e^{S_{\text{BH}} + S_{\text{log}}}$3, driven precisely by the ramp and plateau sourced by the sine-kernel and delta-function terms in the averaged density-of-states correlator. Under the Complexity=Volume prescription, wormhole corrections modify extremal-volume computations at exactly these exponential timescales [Iliesiu:2021ari], and the region where the semiclassical volume fails coincides with the appearance of trapped extremal surfaces implying null geodesic incompleteness [Mohan:2025acj] — suggesting that the non-perturbative corrections responsible for complexity saturation may also resolve the interior singularity. The observable-dependence of the probed geometry is framed as an instance of black hole complementarity: simple probes see smooth classical geometry, complex probes encounter wormholes or baby universes, and the descriptions are operationally complementary.

Limitations and open questions

Several restrictions are acknowledged. The geometric reconstruction is carried out for a single free massive scalar field; extension to realistic interacting matter content is left open, and the scalar mass must either be extractable from symmetry generators or assumed known. The derivation of Einstein's equations requires perturbative control around $e^{S_{\text{BH}} + S_{\text{log}}}$4 and observables dressed against a fixed-background symmetry, appropriate to perturbative quantum gravity but not to a fully diffeomorphism-invariant formulation. The type I result holds only up to corrections suppressed by $e^{S_{\text{BH}} + S_{\text{log}}}$5, relies on a diagonal approximation in the RMT integrals and a Gaussian approximation to the sine kernel, and the identification of the emergent type I algebra with genuine microstate projectors is established at the level of ensemble averages rather than for individual states. Whether the trapped-surface/singularity resolution hinted at by complexity saturation can be made precise within the algebraic framework remains an open question, as does a non-perturbative definition of the locally stationary state condition.

Conclusion

The paper consolidates an operator-algebraic account of emergent gravity in two directions. Kinematically, the Hadamard condition, local Rindler subalgebras, and a well-defined large-mass limit suffice to extract metric and curvature data from $e^{S_{\text{BH}} + S_{\text{log}}}$6 alone, providing a concrete criterion for when a semiclassical quantum theory admits a geometric description. Dynamically, locally stationary states replace the Bekenstein–Hawking area law as the input to Jacobson's argument, yielding the nonlinear Einstein equations from relative entropy in the crossed product algebra. On the finite-$e^{S_{\text{BH}} + S_{\text{log}}}$7 side, RMT completion of the type II$e^{S_{\text{BH}} + S_{\text{log}}}$8 crossed product furnishes an ensemble-averaged type I$e^{S_{\text{BH}} + S_{\text{log}}}$9 factor of dimension $(C^\infty(M), L^2(M,S), \mathcal{D})$0, connecting algebraic classification to black hole microstates and to complexity-based diagnostics of semiclassical breakdown, all without modifying the standard AdS/CFT dictionary or resorting to Euclidean path integrals.