We associate a gravitational Hilbert space
Hσ to any closed compact
(d−1)-manifold
σ with real metric. A quantum state
J(σ) is a
d-manifold bounded by
σ and equipped with elliptic data. An inner product is defined by gluing states pairwise across
σ and evaluated by viewing the resulting closed
d-manifold as a boundary condition on the gravitational path integral (GPI) over
(d+1)-manifolds.
If
σ is nonempty and the GPI is dominated by a single
(d+1)-manifold
M in the
GN→0 limit, then
M contains a Lorentzian CRT fixed-point set, providing
J(σ) with a classical spacetime interpretation. Conversely, given a finite Lorentzian domain with edge
σ, a state
J(σ) may be associated to it by deforming its initial data off the real Lorentzian section and retaining only elliptic data. This establishes a broad correspondence between non-asymptotic spacetime regions and quantum states.
Assuming that
Hσ factorizes over connected components of
σ, our framework admits operators and partial traces. This allows us to explore the information-theoretic structure of the states we define. As an example, we construct a family of states by deforming partial Cauchy slices
Σ that straddle a two-sided black hole;
σ consists of two spheres. We construct the reduced state on one sphere and find that its Rényi entropies are positive, monotonic, and sensitive to all aspects of
Σ and its complex deformation. The von Neumann entropy, however, is controlled only by the maximin surface in the causal domain of
Σ, independently of other parameters, so long as the complex deformation does not vanish. Our proposal may thus explain the efficacy of tensor network toy models of holography while transcending their limitations.
We show that the quantum entanglement between the conformal field theories (CFTs) living on the future and past boundaries in the de Sitter/conformal field theory (dS/CFT) correspondence can be described by Wick rotating to a geometry with two timelike directions. We propose a new realization of the dS/CFT correspondence in which quantum gravity on this two-time geometry is holographically dual to a CFT defined on a Lorentzian torus. We show that this duality reproduces key features of dS holography, including the dS entropy, correlation functions, and pseudoentropy. Finally, by extending the framework of path-integral optimization, we explain how dS spacetime emerges from the CFT on the Lorentzian torus.
We propose that any compact
d-manifold with elliptic data,
J, prepares a quantum state
∣J⟩ on its
(d−1)-boundary
σ. Elliptic data consists of metric and field values, or their conjugates, but not both. No asymptotic structure is required. Inner products and traces are evaluated by the gravitational path integral with closed boundary conditions obtained by gluing elliptic data manifolds. In particular, we give a prescription for the Rényi entropies
Sn of a subregion of
σ. In a class of examples, we find that
Sn is nonnegative and nonincreasing with
n, as required for consistency. We obtain the von Neumann entropy by analytic continuation and find agreement with the minimal surface prescription of Bousso and Penington.
We introduce a new optimization procedure for Euclidean path integrals, which compute wave functionals in conformal field theories (CFTs). We optimize the background metric in the space on which the path integration is performed. Equivalently, this is interpreted as a position-dependent UV cutoff. For two-dimensional CFT vacua, we find the optimized metric is given by that of a hyperbolic space, and we interpret this as a continuous limit of the conjectured relation between tensor networks and Anti-de Sitter (AdS)/conformal field theory (CFT) correspondence. We confirm our procedure for excited states, the thermofield double state, the Sachdev-Ye-Kitaev model, and discuss its extension to higher-dimensional CFTs. We also show that when applied to reduced density matrices, it reproduces entanglement wedges and holographic entanglement entropy. We suggest that our optimization prescription is analogous to the estimation of computational complexity.
We investigate timelike entanglement in quantum field theory using spacetime density matrices and provide a microscopic lattice realization. For a two-dimensional free real scalar field, we extend Gaussian diagonalization methods to the generally non-Hermitian reduced spacetime density matrix and determine its complete nonzero spectrum in the generic regular case, together with all integer Rényi moments. The real-time replica construction identifies these moments with Lorentzian branch-point twist-operator correlation functions. We test this identification against the full four-point function on a circle, boundary two-point functions with Dirichlet and Neumann boundary conditions, and massive form-factor predictions, finding quantitative agreement in both magnitude and phase across distinct causal regimes. The boundary setup exhibits a finite causally connected window in which every integer Rényi entropy is real, showing that reality is not equivalent to causal disconnection. These results provide a microscopic lattice foundation for timelike entanglement and for Lorentzian twist-operator methods beyond equal-time regions.
A bstract In this paper, we investigate the general formalism and properties of the spacetime density matrix, which captures correlations among different Cauchy surfaces and can be regarded as a natural generalization of the standard density matrix defined on a single Cauchy surface. We present the construction of the spacetime density matrix in general quantum systems and its representation via the Schwinger-Keldysh path integral. We further introduce a super-operator framework, within which the spacetime density matrix appears as a special case, and discuss possible generalizations from this perspective. We also show that the spacetime density matrix satisfies a Liouville-von Neumann type equation of motion. When considering subsystems, a reduced spacetime density matrix can be defined by tracing over complementary degrees of freedom. We study the general properties of its moments and, in particular, derive universal short-time behavior of the second moment. We find that coupling between subsystems plays a crucial role in obtaining nontrivial results. Assuming weak coupling, we develop a perturbative method to compute the moments systematically.