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arXiv:2602.20347

A note on the conceptual problems on the Unruh effect

Hideyasu Yamashita

gr-qcmath-phquant-phSubmitted Feb 23, 2026• Announced Feb 25, 2026
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Abstract

This brief note is written with a somewhat similar purpose to Earman's 2011 paper on the conceptual problems on the Unruh effect. However, we confine ourselves to Sewell's modular approach to the Unruh effect, which is based on the theorems of Tomita-Takesaki and Bisognano-Wichmann. This approach is rigorous, and has an advantage of being model-independent. However, we will see that a number of conceptual problems remain unsolved on this approach.

Comments
10 pages

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This brief note is written with a somewhat similar purpose to Earman's 2011 paper on the conceptual problems on the Unruh effect. However, we confine ourselves to Sewell's modular approach to the Unruh effect, which is based on the theorems of Tomita-Takesaki and Bisognano-Wichmann. This approach is rigorous, and has an advantage of being model-independent. However, we will see that a number of conceptual problems remain unsolved on this approach.
Abstract Results about entanglement (or modular) Hamiltonians of quantum many‐body systems in field theory and statistical mechanics models, and recent applications in the context of quantum information and quantum simulation, are reviewed. In the first part of the review, what is known about entanglement Hamiltonians of ground states (vacua) in quantum field theory is summarized, based on the Bisognano–Wichmann theorem and its extension to conformal field theory. This is complemented with a more rigorous mathematical discussion of the Bisognano–Wichmann theorem, within the framework of Tomita–Takesaki theorem of modular groups. The second part of the review is devoted to lattice models. There, exactly soluble cases are first considered and then the discussion is extended to non‐integrable models, whose entanglement Hamiltonian is often well captured by the lattice version of the Bisognano–Wichmann theorem. In the last part of the review, recently developed applications in quantum information processing that rely upon the specific properties of entanglement Hamiltonians in many‐body systems are summarized. These include protocols to measure entanglement spectra, and schemes to perform state tomography.
The Reeh-Schlieder theorem says that every target vector can be approximated from the vacuum by an operator localized in an arbitrarily small spacetime region, but it gives no quantitative cost for doing so. This note isolates a standard Tomita-Takesaki estimate as a model-independent preparation bound. Targets with deeply negative modular energy require large local operators. After rescaling such an operator to a physical contraction, the same estimate becomes a lower bound on postselection overhead. In geometries where the modular Hamiltonian is known, the bound becomes explicit. Bisognano-Wichmann turns it into a boost energy statement for wedges, and the Casini-Huerta-Myers formula gives a stress-tensor version for bounded regions of conformal field theories. Local unitaries can only reach states of nonnegative modular energy. Negative modular sectors require nonunitary or postselected outcomes, giving a preparation cost bound that complements vacuum embezzlement in type III local algebras.
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.