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Estimates for modular theory in algebraic QFT

Adriano Chialastri

Published Oct 16, 2026• 0 citations
OpenAlex

Abstract

The general scope of this PhD thesis is to establish relations between Tomita-Takesaki modular theory, in particular modular Hamiltonians and relative modular Hamiltonians, and physically better understood quantities, such as the stress-energy tensor, energy densities or other modular data in cases where it is known explicitly. In Chapter 1, we will present the basics of operators algebras, Tomita-Takesaki theory, symplectic spaces, and link these concepts to discuss quasi-equivalence of Gaussian states. In Chapter 2, we will discuss algebraic QFTs both in the framework of the Haag-Kastler axioms and of locally covariant QFT, to then specialize to scalar fields and present the stress-energy tensor and the Hadamard condition. In Chapter 3, focussing on Gaussian states of the Weyl-CCR algebra, we study relations between quasi-equivalence of their GNS-representations, differences of the total energy (on some Cauchy surface), and differences between functions of the modular Hamiltonians. Following qualitative relations by Araki and Yamagami, Verch and Longo, we investigate quantitative relations, including in particular estimates of differences between functions of modular Hamiltonians in terms of energy differences. These results will make use of inequalities between p-norms of operators, which we will discuss in Appendix C. In Chapter 4, generalizing the picture beyond Gaussian states, we exploit locality properties of general algebraic QFTs to construct a scheme that allows to estimate relative modular Hamiltonians for suitable pairs of states on some local algebra in terms of modular Hamiltonians of a reference state on bigger or smaller algebras, which might be better understood. For pairs of states susceptible to this scheme, there exist unitaries linking them to the reference state which do not enable superluminal signalling in the sense of Sorkin's paradox. Such unitaries may not however compose the whole class of non-signalling operators, raising the question whether the existence of a non-signalling unitary is a sufficient condition to show the applicability of our estimate. To investigate the strength of our estimates we then consider coherent states for CCR systems, focussing particularly on free scalar fields, where the difference between the upper and lower bounds can be made arbitrarily small, recovering an exact result by squeezing. This yields an independent proof for the relative entropy formula in cases where the relative modular Hamiltonian has not been computed exactly and corroborates the strength of our general estimates. For these results, we will make use of several commutation properties of operators on Fock spaces, which we will present in Appendix A. Results about operator monotone functions will be used throughout this thesis and are explained in Appendix B.

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The general scope of this PhD thesis is to establish relations between Tomita-Takesaki modular theory, in particular modular Hamiltonians and relative modular Hamiltonians, and physically better understood quantities, such as the stress-energy tensor, energy densities or other modular data in cases where it is known explicitly. In Chapter 1, we will present the basics of operators algebras, Tomita-Takesaki theory, symplectic spaces, and link these concepts to discuss quasi-equivalence of Gaussian states. In Chapter 2, we will discuss algebraic QFTs both in the framework of the Haag-Kastler axioms and of locally covariant QFT, to then specialize to scalar fields and present the stress-energy tensor and the Hadamard condition. In Chapter 3, focussing on Gaussian states of the Weyl-CCR algebra, we study relations between quasi-equivalence of their GNS-representations, differences of the total energy (on some Cauchy surface), and differences between functions of the modular Hamiltonians. Following qualitative relations by Araki and Yamagami, Verch and Longo, we investigate quantitative relations, including in particular estimates of differences between functions of modular Hamiltonians in terms of energy differences. These results will make use of inequalities between p-norms of operators, which we will discuss in Appendix C. In Chapter 4, generalizing the picture beyond Gaussian states, we exploit locality properties of general algebraic QFTs to construct a scheme that allows to estimate relative modular Hamiltonians for suitable pairs of states on some local algebra in terms of modular Hamiltonians of a reference state on bigger or smaller algebras, which might be better understood. For pairs of states susceptible to this scheme, there exist unitaries linking them to the reference state which do not enable superluminal signalling in the sense of Sorkin's paradox. Such unitaries may not however compose the whole class of non-signalling operators, raising the question whether the existence of a non-signalling unitary is a sufficient condition to show the applicability of our estimate. To investigate the strength of our estimates we then consider coherent states for CCR systems, focussing particularly on free scalar fields, where the difference between the upper and lower bounds can be made arbitrarily small, recovering an exact result by squeezing. This yields an independent proof for the relative entropy formula in cases where the relative modular Hamiltonian has not been computed exactly and corroborates the strength of our general estimates. For these results, we will make use of several commutation properties of operators on Fock spaces, which we will present in Appendix A. Results about operator monotone functions will be used throughout this thesis and are explained in Appendix B.
The relative entropy between two states is a key concept in quantum information theory and quantum field theory. In the setting of quantum field theory, its computation requires the handling of relative modular Hamiltonians, which are typically very difficult to compute explicitly. In this paper, we exploit locality properties of general algebraic QFTs to estimate relative modular Hamiltonians between two states, ωωω and ω~\tildeωω~, and hence also their relative entropy, in terms of the modular Hamiltonian of a reference state ω^\hatωω^, which might be better understood. For suitable pairs of states we can estimate the relative modular Hamiltonian for the algebra of a region V2V_2V2​ from above, resp. from below, in terms of the modular Hamiltonian of ω^\hatωω^ on a larger region V3V_3V3​, resp. a smaller region V1V_1V1​. Pairs of states and choices of regions which are susceptible to our scheme are related to the presence of superluminal signalling in the sense of Sorkin's paradox. If ω=ω^ω=\hatωω=ω^, then there exists a unitary that maps ωωω to ω~\tildeωω~ on V3V_3V3​ and that does not allow superluminal signalling from the spacelike complement V3′V_3'V3′​ to V2V_2V2​, resp. from V1V_1V1​ to V2′V_2'V2′​, if our upper, resp. lower, bound applies. To investigate the strength of our estimates we consider coherent states for CCR systems, focussing particularly on free scalar fields. Our estimates apply even if the relative modular Hamiltonian cannot be computed exactly. For sufficiently regular excitations we recover an exact result by squeezing. Our method thus yields an independent proof for the relative entropy formula in cases where the relative modular Hamiltonian cannot be computed exactly. For massless fields we establish the analogous result also for double cone regions. These results indicate that our estimates do not lose too much information.
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.
The Tomita-Takesaki modular operator for local algebras plays an important role in quantum field theory, and more recently in the study of relative entropy. However, the explicit expression of this operator, except for the case of wedges, is difficult to describe mathematically. We have obtained instead numerical results for the form of the modular Hamiltonian for a double cone in a massive scalar free field in (1+1)- and (3+1)-dimensional Minkowski space, which shows how it differs from the wedge case, in particular regarding the dependence of the modular Hamiltonian on the mass of the field.
This preprint develops a rigorous operator-algebraic extension of Tomita-Takesaki modular theory to pseudo-Hermitian quantum field systems. We construct a twisted Tomita operator S_η for PT-symmetric fields equipped with a positive-definite metric operator η, establishing its closability and polar decomposition S_η = J_η Δ_η^(1/2) within the η-inner product GNS representation. The central result demonstrates that the twisted modular conjugation J_η squares not to the identity (as in standard theory) but to a non-trivial central unitary W = J_η², where W ∈ Z(A). We prove W is unitary and central using purely algebraic methods, generalizing the standard relation J² = I. A finite-dimensional construction explicitly exhibits W = -I for specific pseudo-Hermitian operator algebras, providing concrete evidence of this stable algebraic phase. This central modular phase represents an intrinsic degree of freedom absent in Hermitian quantum field theory, persisting under pseudo-Hermitian similarity transformations. We discuss potential physical implications, including connections to CP violation and baryon asymmetry, where the phase W may serve as an algebraic source of matter-antimatter asymmetry emerging from field algebra structure rather than explicit Lagrangian symmetry breaking. The framework maintains mathematical consistency with Tomita-Takesaki axioms while extending modular theory beyond Hermitian constraints. This work bridges pseudo-Hermitian similarity transformations and modular conjugation into unified algebraic language, opening directions for PT-symmetric quantum field theory and non-Hermitian approaches to fundamental physics. STATUS: Verison 1.0 and 2.0 completed rest will be available soon .
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)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.
Appendix V‑A develops a modular-theoretic extension of R-layer Mode Theory (RLMT) by introducing relative modular Hamiltonians and relative layered entropy associated with the tension operator E and its eigenvalue hierarchy {λn}. Building on the spectral foundations of Appendix V, we formulate relative modular theory in a stratified Hilbert space where the spectral projections Π(n) define discrete layers. We construct the relative Tomita operator, the relative modular operator, and the relative modular Hamiltonian Kψ,ϕ, and define the relative layered entropy H(ψ,ϕ). For nested layer subsets L1⊂L2⊂L3, we establish monotonicity bounds HL1(ψ,ϕ)≤HL2(ψ,ϕ)≤HL3(ψ,ϕ), providing a layered analog of entropy inequalities in algebraic QFT. We further analyze coherent layered states in the CCR framework and derive explicit expressions for their relative modular Hamiltonians and entropies. Numerical models with Weyl-type eigenvalue growth illustrate the monotonicity and convergence of layered entropy bounds. This appendix provides the modular-theoretic backbone of RLMT, linking spectral hierarchy, modular flow, and information-theoretic structure across all RLMT volumes.