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REVIEW 3 major objections 4 minor 84 references

Relating the modular Hamiltonian to two-point functions

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read For a free scalar field in any Gaussian state, the restriction of the modular Hamiltonian to a region is given by an explicit arcoth formula in terms of the equal-time two-point functions, and the same operator follows from the KMS…

desk verdict The main formula is real and the derivation is solid; the soft spot is the unproved uniqueness claim in the KMS section, which makes the 'arbitrary Gaussian state' extension overreach as written but does not break the central result. read the letter →

arxiv 2501.09669 v2 pith:7D3GQCHD submitted 2025-01-16 math-ph math.MP

Markus B. Fröb This is my paper · ORCID
classification math-phmath.MP MSC 81T0581T2046L1046L60
keywords modularHamiltonianTomita-Takesakitheoryfreescalarfieldtwo-pointfunctionGaussianstatestandardsubspaceKMSconditionCCRalgebra
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes a direct formula for the modular Hamiltonian of a free scalar field in an arbitrary Gaussian state on a globally hyperbolic spacetime, working at the one-particle level. The formula expresses the restriction of the modular Hamiltonian to a standard subspace entirely in terms of the equal-time two-point functions of the field and its conjugate momentum inside that region. This matters because modular Hamiltonians encode entanglement structure and relative entropy, and they are usually hard to compute; here they become functions of correlation data. The paper also shows that the same result follows from the KMS condition and that it reproduces the known Peschel and Casini–Huerta expressions in the discretized case.

What carries the argument

The load-bearing object is the one-particle modular operator $\Delta$ and its logarithm, expressed first through the symplectic projection $P$ onto the standard subspace by $\ln\Delta = 2\operatorname{arcoth}(1-P+IPI)$. The key mechanism is rewriting the projected complex structure $PIP$ in terms of the two-point function $G$ via $2G = \epsilon PIP + i\epsilon$, so that all operators act on $L^2(R)\oplus L^2(R)$. The integral representation $\operatorname{arcoth} z = \int_1^\infty \frac{z}{t^2z^2-1}\,dt$ then turns the abstract logarithm into a concrete function of the operator $2(i\epsilon)^{-1}G - 1$, yielding the $\operatorname{arcoth}(B)$ form after block decomposition. The operator $B = \sqrt{\Pi^{1/2}X\Pi^{1/2}}$ appears as the square root of the symmetrized product of the two correlation functions; the bound $B \geq 1/2$, following from positivity of the state, guarantees that $\operatorname{arcoth}$ is well defined.

What would settle it

Take a free scalar field on a compact spatial manifold in a Gaussian state where the conjugate-momentum two-point function $\Pi$ has a zero mode (for example a constant mode on a torus); the right-hand side of the main formula is then undefined because $\Pi^{-1/2}$ and $B^{-1}$ do not exist, while the modular Hamiltonian itself remains well defined, so if the formula is claimed for all Gaussian states this case would disprove it. More quantitatively, computing both sides for the exactly solvable massless scalar on an interval in 1+1 dimensions and comparing with the known modular Hamiltonian would either confirm the identity or expose a discrepancy.

Watch

Extended reading notes

Core claim

The central result is that for a free scalar field, the restriction of the one-particle modular Hamiltonian to the standard subspace associated with a region $R$ takes the block-matrix form $I\ln\Delta|_R = \begin{pmatrix}0 & 2M\\ -2N & 0\end{pmatrix}$, with $M = \Pi^{1/2}B^{-1}\operatorname{arcoth}(2B)\Pi^{1/2}$ and $N = \Pi^{-1/2}B\operatorname{arcoth}(2B)\Pi^{-1/2}$, where $B = \sqrt{\Pi^{1/2}X\Pi^{1/2}}$ and $X$, $\Pi$ are the equal-time two-point functions of the field and its conjugate momentum restricted to $R$. This is derived by rewriting the abstract projector expression $\ln\Delta = 2\operatorname{arcoth}(1-P+IPI)$ in terms of the two-point function $G$ through $2G = \epsilon PIP + i\epsilon$, and then using an integral representation of $\operatorname{arcoth}$ for self-adjoint operators. The paper further shows that the KMS condition, applied to the modular flow of Weyl operators, yields the same modular Hamiltonian in the form $K = -\ln(G^{-1}G^T)$, and that in the discretized case the formulas reduce exactly to the Peschel and Casini–Huerta expressions.

Load-bearing premise

The load-bearing premise is that the equal-time two-point functions $X$ and $\Pi$, restricted to the region, are positive, invertible, self-adjoint operators on the complex $L^2$ space with domains large enough for $G^{-1}$, $\Pi^{-1/2}$, and $B^{-1}$ to exist; states with zero modes or insufficiently regular two-point functions are not covered by the formula.

Editorial extensions

If this is right

  • Modular Hamiltonians for free fields can be computed from equal-time correlation functions in the region, without constructing the full modular group explicitly.
  • The formula reproduces the known Peschel and Casini–Huerta expressions when $X$ and $\Pi$ are bounded, providing a continuum derivation of those lattice results.
  • The KMS condition gives an independent route: $K = -\ln(G^{-1}G^T)$, so the modular Hamiltonian is read off directly from the two-point function.
  • The derivation extends to general CCR algebras with a non-standard symplectic form, yielding $K = \ln\left(\frac{2V-1}{2V+1}\right)$ for a weakly positive operator $V$.
  • These results make quantities such as relative entropy and modular flow accessible from two-point functions in curved spacetimes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The formula suggests a practical numerical scheme: approximate $X$ and $\Pi$ on a lattice, compute $B$ and $\operatorname{arcoth}(2B)$ by spectral calculus, and take the continuum limit; the paper's equivalence with Casini–Huerta supports the validity of this limit for free fields.
  • One could test the formula against known exact modular Hamiltonians, such as the wedge vacuum or the de Sitter static patch, by computing $X$ and $\Pi$ in those states and comparing both sides of the identity; agreement would further confirm the derivation, and any discrepancy would pinpoint where the operator-domain assumptions fail.
  • The KMS derivation may extend to interacting theories or to states where the two-point function is not the full data, though the free-field Gaussian structure is essential to the present argument.
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Formalized claims in Lean

  1. Claim #1: The central result is that for a free scalar field, the restriction of the one-particle modular Hamiltonian to the standard subspace associated with a region $R$ takes the block-matrix form $I\ln\Delta|_R = \begin{pmatrix}0 & 2M\\ -2N & 0\end{pmatrix}$, with $M = \Pi^{1/2}B^{-1}\operatorname{arcoth}(2B)\Pi^{1/2}$ and $N = \Pi^{-1/2}B\operatorname{arcoth}(2B)\Pi^{-1/2}$, where $B = \sqrt{\Pi^{1/2}X

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper considers a free scalar field on a globally hyperbolic spacetime in a Gaussian (quasifree) state and derives a formula for the restriction of the one-particle modular Hamiltonian to a standard subspace in terms of the equal-time two-point function. Starting from the standard-subspace identity ln Δ = 2 arcoth(1−P+IPI), the author uses the operator integral representation of arcoth to obtain Eq. (2.27), then via an isometry U and the relation 2G = ϵPIP + iϵ arrives at Eq. (2.40) with M = Π^{1/2}B^{−1}arcoth(2B)Π^{1/2}, N = Π^{−1/2}B arcoth(2B)Π^{−1/2}, B = (Π^{1/2}XΠ^{1/2})^{1/2}. This is shown to reproduce the formulas of Peschel and Casini–Huerta in the bounded case. A generalization to general CCR algebras is given in §2.4, and §2.5 presents an alternative derivation from the KMS condition, claiming K = −ln(G^{−1}G^T).

Significance. The potential significance is high if the result is fully established: it gives a practical, correlation-function-only route to modular Hamiltonians for free bosonic fields, complements recent fermionic results, and supplies a derivation that does not rely on the explicit form of the modular flow. The §2.3 derivation is largely self-contained, starts from the standard Tomita–Takesaki data, and explicitly recovers known formulas; the positivity bound B ≥ 1/2 is derived rather than assumed. The KMS route in §2.5 is a useful cross-check, but its advertised scope depends on a uniqueness assertion that is not proved. The abstract's claim of 'arbitrary Gaussian state' is not matched by the pure-state restriction in the body.

major comments (3)
  1. [§2.5, Eq. (2.60)] The statement that the system G^T L(t−i) = G L(t), L(s+t)=L(t)L(s), L(0)=1 has the unique solution L(t)=e^{itK} with K=−ln(G^{−1}G^T) is asserted but not proved, and without additional hypotheses it is false. If P is a spectral projection of K, then L'(t)=e^{it(K+2π i P)} satisfies the same algebraic equations, because e^{K+2π i P}=e^K and the factors commute in the required order; however L'(t) is not unitary for real t and is not the modular flow. The missing hypotheses are exactly the ones that the KMS condition should provide (strong continuity, unitarity, and the strip analyticity of the correlation functions), and they are not used in the text. Since Eq. (2.60) is the basis of the KMS derivation and of the claimed extension to arbitrary Gaussian states, this is a load-bearing gap. Please prove uniqueness under stated assumptions, or restrict the claim.
  2. [§2, after Eq. (2.2); Abstract] The abstract and introduction promise results for an arbitrary Gaussian state, but the body explicitly restricts to pure states, with the remark that mixed states can be purified. The purification argument is not spelled out, and it is not automatic that the modular Hamiltonian of a mixed state on the original Weyl algebra is obtained by restricting the modular Hamiltonian of the purified pure state on the doubled algebra. The only candidate for covering mixed states is the KMS derivation in §2.5, which is affected by the uniqueness gap in the previous comment. Please either state the pure-state restriction in the abstract, or provide the missing argument that the results extend to mixed Gaussian states.
  3. [§2.3, after Eq. (2.38)] The text asserts that X and Π are positive and 'in particular invertible on a suitable domain dense in \hat H', and the later formulas use Π^{−1/2}, B^{−1}, and G^{−1}. For an arbitrary Gaussian state this invertibility is not automatic; zero modes or insufficiently regular two-point functions would make Eqs. (2.40)–(2.41) and (2.60) ill-defined. The paper should either prove this invertibility from the standard subspace and quasifree assumptions, or state it explicitly as a standing hypothesis. This is not a purely cosmetic issue, because the advertised scope is 'arbitrary Gaussian state'.
minor comments (4)
  1. [§2.3, Eqs. (2.45)] The equivalence with the Casini–Huerta formula is shown only for bounded X and Π, using the Wigner weak-positivity decomposition; please state explicitly that the unbounded case is not covered by this comparison, or discuss the limiting argument.
  2. [§2.5, Eq. (2.54)] The notation iK for the one-particle modular Hamiltonian is potentially confusing because K should be self-adjoint while iK is anti-Hermitian; a one-line clarification of the convention would help.
  3. [Eq. (2.24)] The interchange of the t-integral and the spectral integral is justified on the vectors y = P_ϵ x; the claim that the conclusion extends by the core property would benefit from a reference to the relevant statement for unbounded self-adjoint operators.
  4. [References] The reference list is extensive, but a footnote or sentence connecting Refs. [65] and [67] to the bosonic result would make the relation to the fermionic literature easier to follow.

Circularity Check

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No circularity: the modular-Hamiltonian formula is derived from standard-subspace modular data and the defining relation of G, with external Casini-Huerta/Peschel results recovered only as checks.

full rationale

The paper's central derivation is self-contained. Section 2.3 obtains Eq. (2.40) from Eq. (2.19), which is derived in Section 2.1 directly from the Tomita operator S and the symplectic projection P, together with Eq. (2.36), which merely defines the two-point operator G in terms of the Hilbert-space inner product and symplectic form. No occurrence of the target modular Hamiltonian is inserted into these inputs. The Casini-Huerta and Peschel formulas are external results that the paper recovers at the end as consistency checks, not assumptions used to derive (2.40). The KMS section independently derives K = -ln(G^{-1}G^T) from the KMS condition and the group property of modular flow, and then shows algebraically that this agrees with the earlier expression; this is an independent route, not a self-referential loop. There are no fitted parameters or data subsets, so no prediction is forced by construction. The only caveat is technical rather than circular: the uniqueness of the solution L(t) = e^{itK} in Section 2.5 is asserted rather than proved, and invertibility of G is assumed on a suitable domain, but these are rigor/completeness gaps that do not make the argument equivalent to its inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The derivation relies on standard quasifree-state and standard-subspace assumptions, plus technical invertibility and domain conditions on the restricted two-point function. No free parameters are fitted and no new entities are postulated.

assumptions (4)
  • domain assumption The subspace L of initial data supported in R is standard and factorial, so H=L⊕IL and L∩(IL)^⊥={0}.
    Assumed at the start of Sec. 2 after Eq. (2.2); required for the Tomita operator S and for the identity lnΔ=2arcoth(1-P+IPI).
  • domain assumption The physical state is quasifree (Gaussian) and pure; mixed states are purified to a pure state on a larger system.
    Sec. 2, around Eqs. (2.2)-(2.3); the whole construction uses the one-particle Hilbert space of a pure quasifree state.
  • domain assumption The two-point function G and its diagonal blocks X and Π, restricted to R, extend to positive, invertible, self-adjoint operators on the L^2 space via Friedrichs extensions.
    Stated after Eq. (2.38) and used in Eqs. (2.36), (2.41), and (2.60); without invertibility K=-ln(G^{-1}G^T) is undefined.
  • standard math Wigner's spectral decomposition for weakly positive operators applies to XΠ and V.
    Used in Eqs. (2.44) and (2.51) to rewrite arcoth(2B) into the Casini-Huerta form with C=sqrt(XΠ).

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Pith. "Pith review of Relating the modular Hamiltonian to two-point functions." pith.science (2026). https://pith.science/paper/7D3GQCHD

@misc{pith2026250109669,
  author       = {Pith},
  title        = {Pith review of: Relating the modular Hamiltonian to two-point functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7D3GQCHD}},
  note         = {Machine review of arXiv:2501.09669}
}
read the original abstract

We consider the modular Hamiltonian associated to standard subspaces for a free scalar field in a globally hyperbolic spacetime in an arbitrary Gaussian state. We show how the modular Hamiltonian is related to the two-point function of the theory. For the restriction of the modular Hamiltonian to the subspace, we recover formulas that were obtained previously by Peschel, Casini and Huerta. We also show how the same results can be obtained more directly from the KMS condition, and generalize our results to general CCR algebras.

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