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Relative entropy between vacuum and a coherent state in interacting λφ^{4} theory equals the classical boost Noether charge to first order in λ.

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2026-07-10 17:36 UTC pith:ZRJBEMIQ

load-bearing objection Solid first-order extension of the free-field relative-entropy = boost-charge identity to λφ^{4}; the calculation is careful and the result is clean.

arxiv 2607.07810 v1 pith:ZRJBEMIQ submitted 2026-07-08 hep-th math-phmath.MP

Relative entropy for λ φ⁴ in the Rindler wedge

Markus B. Fr\"ob , Albert Much , Kyriakos Papadopoulos This is my paper
classification hep-th math-phmath.MP
keywords relative entropyRindler wedgeλφ^{4} theorymodular HamiltonianNoether chargeBekenstein boundcoherent statesperturbative QFT
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper asks whether the free-field fact that relative entropy between the Minkowski vacuum and a coherent state equals the classical boost energy still holds once a λφ^{4} interaction is turned on. Working inside the Rindler wedge, the authors construct the first-order interacting field, stress tensor and Weyl algebra, then use modular theory to evaluate the Araki–Uhlmann formula. They find that the relative entropy is exactly 2π times the classical interacting boost Noether charge evaluated on a causal classical solution of the nonlinear Klein–Gordon equation, plus higher-order terms. Because the classical energy density is positive and supported inside a strip of width R, the same expression automatically obeys the Bekenstein bound. The result shows that the free-theory identification survives the first interaction correction and supplies an explicit, positive expression that can be used for further information-theoretic studies of interacting quantum fields.

Core claim

To first order in the coupling λ the relative entropy between the Minkowski vacuum and an interacting coherent state W(f)Ω, with f supported in the Rindler wedge, equals the classical boost Noether charge of the interacting theory: S_rel(Ω∥W(f)Ω)=2π∫_{x^{0}=0} x^{1} T_{00}(φ_int_f) d^{3}x + O(λ^{2}), where φ_int_f is the unique causal classical solution whose support lies in the causal diamond of supp f.

What carries the argument

The Araki–Uhlmann formula for relative entropy together with the Bisognano–Wichmann theorem, which identify S_rel with the vacuum expectation value of the interacting boost generator; the latter is evaluated by expanding the interacting Weyl operators to first order and reducing the result to the classical Noether charge via an extended Green identity.

Load-bearing premise

The renormalization of products of Feynman propagators that defines the first-order stress tensor must leave no residual local counterterms that would spoil the exact match with the classical Noether charge.

What would settle it

An independent computation of the same relative entropy to order λ that produces a non-vanishing contact-term contribution proportional to δ^{4}(x) or that fails to recover the classical energy density would falsify the claimed identification.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper computes the relative entropy S_rel(Ω∥W(f)Ω) between the Minkowski vacuum and a coherent excitation generated by an interacting Weyl operator W(f)=e^{iΦ(f)} for λφ^{4} theory in the Rindler wedge, to first order in λ. After constructing the perturbatively interacting field (3.36), the renormalized stress tensor (3.16) (with conservation enforced by the Hadamard-type subtraction of products of Feynman propagators), the associated Noether charges (3.34), and the interacting Weyl algebra of the wedge (3.46) (shown to be local for spacelike-separated supports), the authors use the Araki–Uhlmann formula together with the Bisognano–Wichmann theorem to reduce the relative modular expectation value to a surface integral. The result is S_rel=2π∫_{x^{0}=0} x^{1} T_{00}(φ_int_f) d^{3}x + O(λ^{2}), where φ_int_f is the causal classical solution of the interacting Klein–Gordon equation (2.22)/(4.31). Positivity and a rigorous Bekenstein bound for excitations supported in a strip of width R follow immediately from the positivity of the classical energy density.

Significance. The work supplies the first fully explicit first-order computation of relative entropy for a non-conformal interacting scalar theory between the vacuum and a coherent state. It confirms that the free-field identification of relative entropy with the classical boost Noether charge survives the inclusion of a λφ^{4} interaction (at least to O(λ)), and that the relevant classical solution is the causal one with support in J^{+}(supp f)∪J^{-}(supp f). The construction of the interacting Weyl algebra of the wedge, the verification of locality despite the non-local appearance of the retarded integrals, and the control of renormalization ambiguities by the conservation condition are technically non-trivial and useful for further studies of modular theory, Petz–Rényi divergences, capacity of entanglement, and entropy-area relations in interacting QFT. The derivation is parameter-free within the class of renormalizations that preserve ∂^µT_µν=0 and the free-field limit.

minor comments (4)
  1. In the classical section the non-uniqueness of the causal solution is correctly noted (after (2.22)), yet the quantum computation selects a unique combination (4.31). A short remark in the conclusion that this selection is forced by the modular expectation value (rather than by support properties alone) would make the logic clearer for readers who stop at the classical analysis.
  2. Appendix A introduces several finite renormalization constants that are later absorbed into mass and field-strength renormalization. Explicitly stating that none of these constants survive in the final surface integral (4.29) would remove any residual doubt about scheme dependence.
  3. The extended Green identity (C.4) is central to the reduction; a one-sentence physical motivation (why the surface terms at ±∞ must be retained when the test function is only space-compact) would help non-specialists.
  4. A few typographical inconsistencies appear (e.g., “λφ^{4}” vs. “λφ^{4}” in the title/abstract, occasional missing spaces around operators). They do not affect readability but should be cleaned in the final version.

Circularity Check

0 steps flagged

No circularity: explicit first-order computation from Araki–Uhlmann + Bisognano–Wichmann reduces to classical charge by direct expansion and Green identities, not by construction or self-fit.

full rationale

The central claim (eqs. 4.29/5.1) is obtained by expanding the interacting Weyl operators W(f) to O(λ) (3.38–3.39), inserting into the Araki–Uhlmann formula with the standard Bisognano–Wichmann modular Hamiltonian ln Δ_Ω = 2π M_01, evaluating the resulting free-field correlators (3.53), and reducing the surface integrals via the extended Green identity (C.4) and support properties of the causal solution φ_int_f (2.22). The renormalization of T^(1) is fixed solely by the conservation condition (3.15) so that the Noether charges satisfy the correct commutators (3.33); residual constants are absorbed into mass/field-strength renormalization (A.20–A.21) and do not reappear in the final surface integral. No parameter is fitted to data, no uniqueness theorem is imported from the authors’ prior work to force the result, and the free-theory precursors [10,27] are used only for analogy—the interacting calculation is self-contained and independent. The derivation therefore does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The central claim rests on standard modular theory, the Wightman axioms (which guarantee Bisognano–Wichmann), and a first-order perturbative expansion of λφ⁴ together with a renormalization condition that enforces stress-tensor conservation. No free parameters are fitted to data; renormalization constants are fixed by the usual mass and wave-function renormalization and drop out of the final charge. No new physical entities are postulated.

axioms (4)
  • domain assumption Bisognano–Wichmann theorem: modular Hamiltonian of the vacuum on the Rindler wedge is 2π M_{01} for any Wightman QFT
    Invoked at the start of §4 to replace ln Δ_Ω by the boost generator; holds for interacting theories under the Wightman axioms.
  • standard math Araki–Uhlmann formula S_rel(ω|ϕ)=−ω(ln Δ_{ϕ,ω}) and the unitary-excitation relation Δ_{UΩ,Ω}=U Δ_Ω U†
    Used in (1.1)–(1.4) and (4.1); standard modular theory.
  • domain assumption Existence of a conserved renormalized stress tensor to first order after Hadamard subtraction of products of Feynman propagators
    Imposed by condition (3.15) and realized in appendix A; required for the Noether charges to generate the correct infinitesimal transformations.
  • domain assumption Perturbative solution of the interacting Klein–Gordon equation admits a causal representative with support in J^{+}(supp f)∪J^{-}(supp f)
    Constructed classically in §2 and used quantum-mechanically in (4.31); non-uniqueness is acknowledged but the modular calculation selects a definite representative.

pith-pipeline@v1.1.0-grok45 · 36813 in / 2605 out tokens · 29433 ms · 2026-07-10T17:36:59.033330+00:00 · methodology

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read the original abstract

We consider the relative entropy between the vacuum and a coherent state in the Rindler wedge for an interacting $\lambda \phi^4$ theory to first order in $\lambda$. We construct the perturbatively interacting Weyl algebra of the wedge, and employ Tomita--Takesaki modular theory and the Araki--Uhlmann formula to compute the relative entropy. We verify that the relative entropy reduces to the classical (interacting) boost Noether charge, analogously to the free theory, and that the Bekenstein bound holds.

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