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arXiv:2605.19106 · Physical review. D/Physical review. D.

Modular Self-Duality, Symmetrized Relative Entropy, and Bogoliubov--Kubo--Mori Susceptibility in Quantum Field Theory

Rupak Chatterjee

hep-thquant-phSubmitted May 18, 2026• Announced May 20, 2026• 0 citations• Revised Aug 27, 2026 (v2)
v1 · May 18, 2026v2 · Aug 27, 2026
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Abstract

We develop an operator-algebraic framework for modular self-duality, symmetrized relative entropy, and Bogoliubov--Kubo--Mori (BKM) susceptibility of local states in quantum field theory. In finite dimensions, modular self-duality singles out fixed points at which a state coincides with its modularly reflected partner. At such points, the natural comparison functional is the symmetrized Umegaki relative entropy. It vanishes at coincidence, and its Hessian is governed by the BKM quantum Fisher information along the reflected tangent direction. We then extend this fixed-point construction to the local type~III von Neumann algebras that arise in quantum field theory. Here, a local state is compared with the modular pullback of its commutant restriction, and the intrinsic comparison functional is the symmetrized Araki relative entropy. For sufficiently regular state deformations, the fixed-localization Hessian at the self-dual point defines a type~III BKM susceptibility. This coefficient is obtained by evaluating the BKM bilinear form on the tangent selected by the modular pairing. Exact coherent-state realizations are obtained for the free scalar field on wedge algebras and for the chiral $U(1)$ current on half-line algebras. In both examples, the comparison functional is exactly quadratic in the deformation parameter, and the susceptibility coefficients admit explicit boost-energy, stress-tensor, or half-line integral representations. We also apply the construction to a ball-shaped boundary region in the vacuum state of a conformal field theory with a semiclassical anti-de Sitter (AdS) gravitational dual. For linear bulk coherent excitations in an exact complementary-recovery model of this sector, bulk--boundary modular intertwining identifies the modular partner with the AdS--Rindler-reflected coherent state, and the symmetrized relative entropy is exactly quadratic.

DOI
10.1103/vxkq-xb7l
Comments
31 pages, 1 table. The revised version adds a holographic AdS/CFT application relating modular-reflection susceptibility to the AdS-Rindler canonical energy, together with clarifications and an expanded discussion. Accepted for publication in Physical Review D

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We develop an operator-algebraic framework for modular self-duality, symmetrized relative entropy, and Bogoliubov--Kubo--Mori (BKM) susceptibility of local states in quantum field theory. In finite dimensions, modular self-duality singles out fixed points at which a state coincides with its modularly reflected partner. At such points, the natural comparison functional is the symmetrized Umegaki relative entropy. It vanishes at coincidence, and its Hessian is governed by the BKM quantum Fisher information along the reflected tangent direction. We then extend this fixed-point construction to the local typeIII von Neumann algebras that arise in quantum field theory. Here, a local state is compared with the modular pullback of its commutant restriction, and the intrinsic comparison functional is the symmetrized Araki relative entropy. For sufficiently regular state deformations, the fixed-localization Hessian at the self-dual point defines a typeIII BKM susceptibility. This coefficient is obtained by evaluating the BKM bilinear form on the tangent selected by the modular pairing. Exact coherent-state realizations are obtained for the free scalar field on wedge algebras and for the chiral U(1)U(1)U(1) current on half-line algebras. In both examples, the comparison functional is exactly quadratic in the deformation parameter, and the susceptibility coefficients admit explicit boost-energy, stress-tensor, or half-line integral representations. We also apply the construction to a ball-shaped boundary region in the vacuum state of a conformal field theory with a semiclassical anti-de Sitter (AdS) gravitational dual. For linear bulk coherent excitations in an exact complementary-recovery model of this sector, bulk--boundary modular intertwining identifies the modular partner with the AdS--Rindler-reflected coherent state, and the symmetrized relative entropy is exactly quadratic.
This preprint investigates structural relations between relative entropy, modular theory, and information geometry in algebraic quantum field theory (AQFT). In finite‑dimensional quantum systems, the Hessian of Umegaki relative entropy coincides with the Bogoliubov–Kubo–Mori (BKM) metric, and modular automorphisms act by isometries of this geometry; these standard results are reviewed as a mathematically controlled reference model. Motivated by this setting, the work formulates minimal structural conditions under which a perturbative modular information‑geometric programme for Type III1_11 von Neumann algebras could become mathematically meaningful. The analysis focuses on admissible perturbation sectors, Araki relative entropy, modular covariance, and the functional‑analytic obstructions caused by unbounded modular operators. A general consequence of positive semidefiniteness of the perturbative entropy Hessian is recorded: whenever the Hessian defines a well‑defined positive semidefinite bilinear form, its null directions annihilate the full form. In the Rindler wedge model, the corresponding regularity problem for candidate perturbations is formulated in terms of spectral properties and domain issues of the Lorentz boost generator. Throughout, explicit separation is maintained between rigorous results, conditional statements, heuristic motivations, and unresolved analytic problems. A conjecture concerning degenerate modular directions is stated together with a necessary falsifiable condition. This document is exploratory in nature and has not been peer reviewed.
This manuscript establishes a mathematically controlled one-parameter modular-time coordinate for the vacuum representation of a free neutral scalar field on the right Rindler wedge. By restricting the state space strictly to the finite-entropy Weyl-coherent sector, the vacuum relative entropy is shown to be exactly quadratic based on rigorous external results by Longo and Casini–Grillo–Pontello. Consequently, the entropy Hessian exists without remainder estimates (O(t^3)), is strictly positive, and remains invariant under the modular group. By applying the Bisognano–Wichmann theorem, the modular parameter directly identifies with the Rindler boost rapidity up to orientation and the standard KMS-Unruh normalization (s = -2\pi\tau). The manuscript explicitly addresses the separate status of unbounded Araki perturbations (\phi(f)) and defines the exact boundary between proven theorems (⁠[RIGOROUS]⁠) and open matching hypotheses (⁠[OPEN]⁠), avoiding any completing-the-square assumptions for the modular Hamiltonian.
An early result of algebraic quantum field theory is that the algebra of any subregion in a QFT is a von Neumann factor of type III _{1} 1 , in which entropy cannot be well-defined because such algebras do not admit a trace or density states. However, associated to the algebra is a modular group of automorphisms characterizing the local dynamics of degrees of freedom in the region, and the crossed product of the algebra with its modular group yields a type II _∞ ∞ factor, in which traces and hence von Neumann entropy can be well-defined. In this work, we generalize recent constructions of the crossed product algebra for the TFD to, in principle, arbitrary spacetime regions in arbitrary QFTs, formally paving the way to the study of entanglement entropy without UV divergences. In contrast to previous works, we emphasize that this construction is independent of gravity. In this sense, the crossed product construction represents a refinement of Haag’s assignment of nets of observable algebras to spacetime regions by providing a natural construction of a type II factor. We present several concrete examples: a QFT in Rindler space, a CFT in an open ball of Minkowski space, and arbitrary boundary subregions in AdS/CFT. In the holographic setting, we provide a novel argument for why the bulk dual must be the entanglement wedge, and discuss the distinction arising from boundary modular flow between causal and entanglement wedges for excited states and disjoint regions.
We compute the vacuum local modular Hamiltonian associated with a space ball region in the free scalar massive Quantum Field Theory. We give an explicit expression on the one particle Hilbert space in terms of the massive Legendre differential operator and the Green integral for the Helmholtz operator. The quadratic form of the massive modular Hamiltonian is expressed in terms of an integral of the energy density with parabolic distribution and of a Yukawa potential, that here appears intrinsically. We then get the formula for the local entropy of a Klein-Gordon wave packet. This gives the vacuum relative entropy of a coherent state on the double cone von Neumann algebras associated with the free scalar QFT. Among other points, we have a criterion for a Bogoliubov automorphism to be locally weakly inner and we provide the passivity characterisation of the modular Hamiltonian within the standard subspace set up.
This work presents an exploratory framework for modular information geometry associated with faithful normal states on Type III₁ von Neumann algebras. The construction is based on the second Gâteaux variation (Hessian) of Araki relative entropy for bounded analytic perturbations and leads to a modular Fisher form defined on an explicitly specified admissible domain. Under stated regularity and covariance assumptions, this form is symmetric and positive semidefinite and exhibits canonical null directions generated by the modular derivation. The physically relevant structure is obtained by quotienting the resulting degeneracy space. In the context of algebraic quantum field theory, and assuming the Bisognano–Wichmann property for wedge algebras, the modular derivation coincides with the Lorentz boost generator, so that modular flow directions correspond to gauge/null directions of the information-geometric structure. This note is exploratory and conjectural in nature. It was developed with the assistance of AI-based tools and has not undergone independent mathematical or peer validation. No claim of new theorems, analytical completeness, full differentiability theory, or spacetime reconstruction is made. All nontrivial results are conditional on explicitly stated assumptions and are intended as structural observations rather than definitive conclusions.
We prove a controlled completion of dual-affine entropy geometry connecting a supplied amplitude-carrier complex structure, constructed tangent-bundle Born complex structures, non-Abelian compact transport, and endpoint action through typed relative-entropy reconstruction and defects. On the supplied Hermitian carrier, Umegaki relative entropy reconstructs the entropy potential up to affine gauge and the relative logarithmic cotangent class; diagonal jets yield the Bogoliubov-Kubo-Mori (BKM) metric, Hessian cubic, and dual connections, which construct Born lifts on real doubled statistical carriers. Determinant normalization selects a principal-log traceless Gibbs representative. A quantum channel induces support-reduced output geometry and a Bregman loss whose Hessian and cubic are metric and response losses with sequential cocycles; one CPTP map recovering every state of an embedded faithful model annihilates its restricted loss and intertwines its intrinsic geometry and holonomy. Proximal families yield Rényi and Hopf-Lax identities; BKM relaxation needs mobility and scale. From a separately chosen affine-invariant metric, the compact branch constructs canonical principal SU(n) and associated tangent PSU(n) transport, compared with BKM transport by connection and curvature defects rather than conjugacy. The indexed architecture is dimension-stratified: its general branches are available for each fixed finite n≥2 under their stated inputs, full BKM tangent holonomy is proved only for n≥3, and the qutrit and independently graded AIII packages occur only at n=3 and n=5, respectively. For qutrits, the tracial BKM metric-cubic pair has oriented stabilizer PSU(3); alignment with normalized affine-invariant holonomy lets the central cubic seed a distinct, unique parallel comparison cubic defining a global tangent-frame reduction. A fixed action conversion gives an exact endpoint canonical relation; the parameterized Rényi-Moreau family gives a branch-local scale-Hamiltonian isotopy. Mechanical interpretation needs inertia and a clock. Relative entropy does not construct the amplitude-carrier complex structure or supply physical calibrations.