This repository contains an exploratory theoretical framework at the intersection of Algebraic Quantum Field Theory (AQFT) and information geometry for faithful normal states on Type III₁ von Neumann algebras. The construction is based on second Gâteaux variations of Araki relative entropy and introduces a modular Fisher quadratic form on admissible state perturbations, formulated on the Tomita algebra of analytic elements. A strict logical Kernel (Protocol v3.1.2) is employed to enforce assumption monotonicity, operator-domain discipline, explicit statement typing, and explicit conditionality of all nontrivial claims. This work is an independent, exploratory research project developed with the assistance of AI-based formalization tools. All definitions, assumptions, and claims were manually validated under the Kernel protocol. The framework has not undergone formal academic peer review and is published to archive the methodology and invite rigorous mathematical validation by the AQFT and mathematical physics community.
This repository contains an exploratory theoretical framework at the intersection of Algebraic Quantum Field Theory (AQFT) and information geometry for faithful normal states on Type III₁ von Neumann algebras. The construction is based on second Gâteaux variations of Araki relative entropy and introduces a modular Fisher quadratic form on admissible state perturbations, formulated on the Tomita algebra of analytic elements. A strict logical Kernel (Protocol v3.1.2) is employed to enforce assumption monotonicity, operator-domain discipline, explicit statement typing, and explicit conditionality of all nontrivial claims. This work is an independent, exploratory research project developed with the assistance of AI-based formalization tools. All definitions, assumptions, and claims were manually validated under the Kernel protocol. The framework has not undergone formal academic peer review and is published to archive the methodology and invite rigorous mathematical validation by the AQFT and mathematical physics community.
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.
This work presents an exploratory theoretical framework in algebraic quantum field theory, developed by an independent researcher using AI-assisted formalization as a technical aid. The construction is explicitly conditional and assumption-driven. No claims are made beyond the stated hypotheses, and no global geometric or manifold structure is asserted without explicit regularity conditions. The purpose of this work is to provide a mathematically controlled structural framework and to identify clearly formulated directions for future validation, model testing, and comparison with the existing literature.
We develop Interval Algebraic Quantum Field Theory (IAQFT), a finite-precision operational formulation of algebraic quantum field theory (AQFT). Physical experiments provide only finitely many observations at finite resolution and therefore determine regions of compatible states rather than uniquely specified point states. IAQFT represents this information by quantum parcels---non-empty convex weak∗ open regions of state space---with observational parcels determined by finitely many finite-precision expectation constraints. Exact AQFT states remain indispensable mathematical objects and are recovered under appropriate parcel refinement. We establish reduction to exact states, measurement update and no-signalling, and formulate locality through compatible parcel nets. In finite dimensions an exact Jacobian formula for invertible Lüders updates gives a volume-contraction criterion. Spacelike vacuum correlations and strict CHSH violations persist throughout sufficiently small observational parcels. Reeh--Schlieder cyclicity becomes finite-precision local reachability, while parcel equivalence of vacuum refinements implies vacuum GNS equivalence and hence does not evade Haag-type obstructions. Tomita--Takesaki modular automorphisms act on parcels, measurement update is modularly covariant, and modular conjugation induces a duality between normal parcels of an algebra and its commutant. State-dependent modular correlations are continuous under predual-norm variation of faithful normal states; shrinking parcels with vanishing KMS defect recover a KMS state. Bisognano--Wichmann yields a parcel formulation of the Unruh effect, recovering the exact KMS relation and Unruh temperature under refinement. Finally, limiting projection faces and their conjugation actions recover the Murray--von Neumann factor classification.
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.
In this thesis, we study aspects of entanglement theory of quantum field theories from an algebraic point of view. The main motivation is to gain insights about the general structure of the entanglement in QFT, towards a definition of an entropic version of QFT. In the opposite direction, we are also interested in exploring any consequence of the entanglement in algebraic QFT. This may help us to reveal unknown features of QFT, with the final aim of finding a dynamical principle which allows us to construct non-trivial and rigorous models of QFT. The algebraic approach is the natural framework to define and study entanglement in QFT, and hence, to pose the above inquiries. After a self-contained review of algebraic QFT and quantum information theory in operator algebras, we focus on our results. We compute, in a mathematically rigorous way, exact solutions of entanglement measures and modular Hamiltonians for specific QFT models, using algebraic tools from modular theory of von Neumann algebras. These calculations show explicitly non-local features of modular Hamiltonians and help us to solve ambiguities that arise in other non-rigorous computations. We also study aspects of entanglement entropy in theories having superselection sectors coming from global symmetries. We follow the algebraic perspective of Doplicher, Haag, and Roberts. In this way, we find an entropic order parameter that "measures" the size of the symmetry group, which is made out of a difference of two mutual informations. Moreover, we identify the main operators that take account of such a difference, and we obtain a new quantum information quantity, the entropic certainty relation, involving algebras containing such operators. This certainty relation keeps an intrinsic connection with subfactor theory of von Neumann algebras.