Modular automorphisms generated by Tomita–Takesaki theory are canonical butstate-dependent structures. In special QFT regimes (Bisognano–Wichmann wedgesectors, holographic entanglement wedges), modular flow coincides with geometriccausal evolution (Lorentz boosts, bulk modular flow). We define a state-independentsignaling defect ηt(A → B) that measures how much a candidate evolution αt (inparticular modular flow) fails to preserve the commutation structure between twocommuting algebras A and B. The defect is the operator-norm distance from theevolved algebra αt(B) to the commutant A′. We prove a sharp equivalence: ηt(A →B) = 0 if and only if no operational signaling channel exists from lab A to lab B underevolution αt, for all input states and all A-localized interventions. Quantitatively, ηt ≤ε implies the maximal signaling advantage is at most 2ε. In geometric regimes wheremodular flow is causal (Bisognano–Wichmann), the defect must vanish; nontrivialdefect is a hard falsification of the localization–evolution assignment. This yieldsa universal operational causality test applicable to any candidate dynamics, withimmediate consequences for emergent spacetime and holography.
Modular automorphisms generated by Tomita–Takesaki theory are canonical butstate-dependent structures. In special QFT regimes (Bisognano–Wichmann wedgesectors, holographic entanglement wedges), modular flow coincides with geometriccausal evolution (Lorentz boosts, bulk modular flow). We define a state-independentsignaling defect ηt(A → B) that measures how much a candidate evolution αt (inparticular modular flow) fails to preserve the commutation structure between twocommuting algebras A and B. The defect is the operator-norm distance from theevolved algebra αt(B) to the commutant A′. We prove a sharp equivalence: ηt(A →B) = 0 if and only if no operational signaling channel exists from lab A to lab B underevolution αt, for all input states and all A-localized interventions. Quantitatively, ηt ≤ε implies the maximal signaling advantage is at most 2ε. In geometric regimes wheremodular flow is causal (Bisognano–Wichmann), the defect must vanish; nontrivialdefect is a hard falsification of the localization–evolution assignment. This yieldsa universal operational causality test applicable to any candidate dynamics, withimmediate consequences for emergent spacetime and holography.
We address the problem of time in quantum mechanics: time is an external classical parameter, not a quantum observable, yet quantum gravity requires it to be emergent. We show that a canonical preferred temporal flow arises from within the algebraic structure of quantum mechanics, without presupposing a Hamiltonian, a Schrödinger equation, or a background time parameter. The starting point is a pair (M, ω) — a von Neumann algebra of observables and a faithful normal state — with no other input. From these, Tomita–Takesaki modular theory derives a canonical strongly-continuous one-parameter group of automorphisms σ_t : M → M (the modular flow). The group parameter t labels the spectral flow of the modular operator Δ = S^*S constructed from ω, and is not assumed as background structure. Three results are established. First (Structural Inevitability): any non-tracial quantum system unavoidably carries a non-trivial modular flow — the Tomita–Takesaki modular automorphism group, derived from algebraic data alone. Under the Connes–Rovelli thermal time hypothesis (adopted as an explicit physical postulate, supported by exact agreement with Heisenberg evolution for Gibbs states and with Rindler time via the Bisognano–Wichmann theorem), this flow constitutes physical time. Second (Cocycle Colimit): the directed colimit of all subsystem modular flows exists in the category W*-Dyn, interacting or not; for interacting systems the colimit is twisted by Araki's Connes cocycle, and the assembled flow equals the modular flow of the limiting algebra. Third (Classical Recovery): a structural outline of how, in the decoherence limit under density, separability, and the second law, the framework recovers a separable dense linear order from which n-dimensional spatial structure is derived; full topological reconstruction is left as an open program. This paper is self-contained and requires no companion papers. It extends the Connes–Rovelli thermal time hypothesis and the Page–Wootters relational framework to a universal directed cocycle colimit.
The modular Hamiltonian KA := − log ∆A is canonical in modular theory butgenerically nonlocal in quantum field theory, with exact locality restricted to specialcases such as the Bisognano–Wichmann theorem for Rindler wedges and the Casini–Huerta–Myers result for balls in CFT vacua. We introduce an operational algebraicdiagnostic for quasi-local modular behavior based on nested collar algebras A(0) ⊂ A(δ)and define the modular leakage functional Λ(δ; t) measuring how far the outer modularflow σA(δ)t pushes inner observables out of A(0). From this we define the modularleakage exponent αT , quantifying the best power-law suppression of leakage as thecollar thickness δ → 0+ on a fixed modular-time window |t| ≤ T . Exact localitycorresponds to αT = ∞. Our main structural result is an interior–exterior obstruction:leakage controls the strength of modularly-induced couplings between the deep interiorA(0) and the exterior algebra A(δ)′. In particular, if αT > 1 (superlinear leakagesuppression), then the interior–exterior commutator witness decays superlinearly in δ,ruling out any effective modular generator whose leading interior–exterior couplingwould produce an O(δ) (or stronger) commutator signal. This converts qualitative“nonlocality of entanglement Hamiltonians” into a falsifiable algebraic constraintwithout reference to stress tensors or background manifolds.
We establish a complete operator-theoretic and operational picture of Tomita-Takesaki modular flow on the qubit. For any pair of qubit states ρ_M (modular) and ρ_E (evaluation), with Bloch radii V_M, r, axes n̂, m̂, and rapidity η_M = arctanh(V_M), the variance of the modular Hamiltonian K_M = -log ρ_M admits the cross-state closed form Var_E(K_M) = η_M²(1 - r² cos²θ), yielding a Mandelstam-Tamm bound |d⟨O⟩/dt| ≤ 2η_M √(1 - r²cos²θ) · √Var_E(O) whose maximizing-observable saturation ratio is r sin θ / √(1 - r² cos²θ), equal to 1 exactly when r = 1. We prove a Heisenberg-picture data-processing inequality, an integrated Bures-distance bound, an operator-norm version ‖[K_M, O]‖op ≤ 2η_M ‖O‖op saturating for off-axis Pauli operators, and a minimum-time relation τ⊥ = π/(2η_M) for orthogonal evolution iff n̂·m̂ = 0. The Bloch vector of ρ_E precesses around n̂ at angular frequency 2η_M; the half-cycle work satisfies |W| ≤ 2r sin θ |h⊥| for any observable Hamiltonian H = h₀𝟙 + h·σ. The bound vanishes at V_M → 1 when ρ_E = ρ_M despite divergent η_M, a finite-dimensional third-law analog. The decomposition ρ_M^{it} = e^{iφ(t)} exp(it η_M n̂·σ) identifies ρ_M^{it} as an SL(2,ℂ) boost element with imaginary spinor rapidity ζ = 2itη_M, a finite-dimensional Bisognano-Wichmann analog in which the universal QFT factor 2π is replaced by the state-dependent 2η_M. All numerical coefficients verified below 4×10⁻¹⁵.
Flat space holography is an open and hard problem existing several different approaches, which may finally turn out to be consistent with each other, in the literature to tackle it. Focusing on how bulk emergent spacetime is encoded in quantum information of null boundaries, we choose a specific toy model called the flat3/BMSFT model, which conjectures the duality between boundary BMS3 invariant field theory and bulk quantum gravity in 3D asymptotic flat spacetimes (AFS), to explore. Aiming to find an entanglement wedge like quantity for single interval and a connected entanglement wedge for multi-intervals in flat3/BMSFT model, we explore the bulk causality structures related to the holographic swing surface proposal through both boundary and bulk local modular flow, make a corresponding decomposition of the global Minkowski spacetime and look at the entanglement phase transition. As a byproduct, we solve the problem about the existence of partial entanglement entropy (PEE) correspondence in this model which is a bit nontrivial due to the unusual behavior of boundary modular flow in BMS3 field theory. Among the literature considering quantum information aspects of flat3/BMSFT model, there are several substantial, unusual but overlooked phenomena which need to be emphasized and revisited to gain more deserved attention. Thus another motivation of this paper is to find where these unusual phenomena come from, and physically show in a manifest way what they may imply. After reading we hope readers can feel sincerely what we present about the above mentioned second aim is more valuable than the mathematical results in the present paper.
The general scope of this PhD thesis is to establish relations between Tomita-Takesaki modular theory, in particular modular Hamiltonians and relative modular Hamiltonians, and physically better understood quantities, such as the stress-energy tensor, energy densities or other modular data in cases where it is known explicitly. In Chapter 1, we will present the basics of operators algebras, Tomita-Takesaki theory, symplectic spaces, and link these concepts to discuss quasi-equivalence of Gaussian states. In Chapter 2, we will discuss algebraic QFTs both in the framework of the Haag-Kastler axioms and of locally covariant QFT, to then specialize to scalar fields and present the stress-energy tensor and the Hadamard condition. In Chapter 3, focussing on Gaussian states of the Weyl-CCR algebra, we study relations between quasi-equivalence of their GNS-representations, differences of the total energy (on some Cauchy surface), and differences between functions of the modular Hamiltonians. Following qualitative relations by Araki and Yamagami, Verch and Longo, we investigate quantitative relations, including in particular estimates of differences between functions of modular Hamiltonians in terms of energy differences. These results will make use of inequalities between p-norms of operators, which we will discuss in Appendix C. In Chapter 4, generalizing the picture beyond Gaussian states, we exploit locality properties of general algebraic QFTs to construct a scheme that allows to estimate relative modular Hamiltonians for suitable pairs of states on some local algebra in terms of modular Hamiltonians of a reference state on bigger or smaller algebras, which might be better understood. For pairs of states susceptible to this scheme, there exist unitaries linking them to the reference state which do not enable superluminal signalling in the sense of Sorkin's paradox. Such unitaries may not however compose the whole class of non-signalling operators, raising the question whether the existence of a non-signalling unitary is a sufficient condition to show the applicability of our estimate. To investigate the strength of our estimates we then consider coherent states for CCR systems, focussing particularly on free scalar fields, where the difference between the upper and lower bounds can be made arbitrarily small, recovering an exact result by squeezing. This yields an independent proof for the relative entropy formula in cases where the relative modular Hamiltonian has not been computed exactly and corroborates the strength of our general estimates. For these results, we will make use of several commutation properties of operators on Fock spaces, which we will present in Appendix A. Results about operator monotone functions will be used throughout this thesis and are explained in Appendix B.
We develop an operator-based framework in which quantum dynamics, interaction structure, and geometric features arise from the modular properties of a density operator constructed from a non-commuting pair of Hermitian operators. A positive state is defined via exponential coarse-graining, and time evolution is derived from the associated modular Hamiltonian rather than postulated. Using a systematic Baker–Campbell–Hausdorff expansion, interaction terms are shown to be organized by nested commutators, yielding a natural algebraic hierarchy analogous to effective field theory. Interpreting the coarse-graining parameter as a renormalization scale, we derive flow equations for operator-valued couplings, establishing a renormalization group structure in operator space. A continuum field-theoretic extension and a lattice regularization suitable for numerical investigation are constructed. Local automorphisms of the operator algebra give rise to gauge connections, while the quantum Fisher information metric defines an emergent notion of geometry. Under additional assumptions of locality and approximate equilibrium, the framework reproduces the structure of entropic approaches to gravitational dynamics. These results provide a unified algebraic perspective in which dynamics, interactions, and geometry are encoded in the modular structure of quantum states.