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.
Building on the established one-particle formula for the Araki relative entropy of coherent states, we study how its value acquires a nontrivial dependence on the mass of the scalar field. For a localized vector
h belonging to the standard subspace
Hm of the one-particle Hilbert space, the known quadratic-form expression is:
SHm(h)=−⟨h,logδHmh⟩. Our contribution is to construct explicitly a mass-indexed family of vectors of the wedge standard subspace on which this expression is evaluated. The mass-shell map
hm=Emf organizes four structural conditions on rapidity representatives---on-shell dependence, a controlled massless boundary value, decay for large real rapidity, and Bisognano--Wichmann strip analyticity---and we exhibit an entire rapidity wave function, built from a doubled light-cone phase, two sinc factors, and a Gaussian pair, that satisfies them together with the sharp localization criterion: Hardy-type
L2 control throughout the Bisognano--Wichmann strip and the exact Tomita boundary relation. The family therefore belongs to $H_m(\W_R)$ for every
m>0, and its Araki relative entropy is finite and strictly positive, with an exact spectral representation that makes positivity manifest. The entropy is strongly suppressed at large mass, attains a maximum at intermediate mass in
1+1 dimension, and converges to a finite value along the modular flow as
m→0+. The construction extends fiberwise to
1+d dimensions through the transverse mass.
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.
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.
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 numerically investigate the Araki-Uhlmann relative entropy in Quantum Field Theory, focusing on a free massive scalar field in 1 + 1 -dimensional Minkowski spacetime. Using Tomita-Takesaki modular theory, we analyze the relative entropy between a coherent state and the vacuum state, with several types of test functions localized in the right Rindler wedge. Our results confirm that relative entropy decreases with increasing mass and grows with the size of the spacetime region, aligning with theoretical expectations.