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Albert Much

Publications and source records attributed to Albert Much.

At least 19 recordsLinked to original sources
arXiv2026

The two-sided Bogoliubov inequality in von Neumann algebras conceptualizes the free energy--quantum correlations link

The quantum-mechanical two-sided Bogoliubov inequality provides upper and lower bounds for the free energy required to separate a system of interacting particles into independent subsystems. The bounds can be calculated straightforwardly from the ensemble average of the interface energy, bypassing the direct evaluation of the free energy. In this work, we generalize the two-sided Bogoliubov inequality to arbitrary von Neumann algebras by employing the Araki-Uhlmann relative entropy and the framework of unbounded perturbation theory of KMS states. Furthermore, we obtain variational expressions for the relative free energy that extend existing bounded-perturbation principles to the unbounded setting. Crucially, these mathematical developments yield a physically well-founded thermodynamic criterion for the quantification of entanglement in infinite-dimensional systems.

math-ph↗
arXiv2026

Relative entropy for $\lambda \phi^4$ in the Rindler wedge

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.

hep-th↗
arXiv2026

A UV-Finite Ryu-Takayanagi Relation from Relative Entropy in AdS$_3$/CFT$_2$

We establish a Ryu-Takayanagi (RT) relation in AdS$_3$/CFT$_2$ using \emph{relative entropy} as the central object, in place of the ultraviolet-divergent von Neumann entanglement entropy. Adapting Hollands' exact result for the chiral relative entropy to a diamond region, we express the boundary relative entropy between the vacuum and a coherent state as a Schwarzian functional, which the Fefferman-Graham dictionary identifies with the asymptotic data of a Ba\~nados geometry; the rigidity of three-dimensional gravity promotes this boundary identification to the bulk. To linear order in the metric perturbation, the relative entropy then equals the variation of the RT geodesic length divided by $4G_N$. The construction rests only on the Bisognano-Wichmann/Borchers theorem and the holographic dictionary, giving a UV-finite, operator-algebraic counterpart to the RT relation.

hep-th↗
arXiv2026

Noncommutative QFT and Relative Entropy on Axisymmetric Bifurcate Killing Horizons

We construct a deformed algebraic quantum field theory on bifurcate Killing horizons in stationary axisymmetric spacetimes. The deformation is generated by the commuting actions of affine dilations along the null generators of the horizon and rotations about the axis of symmetry, analogously to the Moyal-Rieffel deformation. Physically, this effectively implements a noncommutative geometric structure of the horizon. Moreover, we compute the relative entropy between coherent states in the deformed horizon theory, which remains strictly positive and exhibits a novel second-order correction in the deformation parameter, which becomes particularly significant for black holes whose horizon area is sufficiently small for Planck-scale effects to become non-negligible.

hep-th↗
arXiv2026

A proposal for the algebra of a novel noncommutative spacetime

We investigate the quantum structure of spacetime at fundamental scales via a novel, Lorentz-invariant noncommutative coordinate framework. Building on insights from noncommutative geometry, spectral theory, and algebraic quantum field theory, we systematically construct a quantum spacetime algebra whose geometric and causal properties are derived from first principles. Using the Weyl algebra formalism and the Gelfand--Naimark--Segal (GNS) construction, we rigorously define operator-valued coordinates that respect Lorentz symmetry and encode quantum gravitational effects through nontrivial commutation relations. We show how the emergent quantum spacetime exhibits minimal length effects, which deliver both classical Minkowski distances and quantum corrections proportional to the Planck length squared. Furthermore, we establish that noncommutativity respects a fuzzy form of causality, where the quantum causal structure gives back the light cone in the classical limit, vanishing for spacelike separations and encoding a time orientation for timelike intervals.

math-ph↗
arXiv2025

From Quantum Relative Entropy to the Semiclassical Einstein Equations

We provide arguments indicating that the semiclassical Einstein equations follow from quantum relative entropy and its proportionality to an area variation. Using modular theory, we establish that the relative entropy between the vacuum state and coherent excitations of a scalar quantum field on a bifurcate Killing horizon is given by the energy flux across the horizon. Under the assumption of the Bekenstein-Hawking entropy-area formula, this energy flux is proportional to a variation in the surface area of the horizon cross section. The semiclassical Einstein equations follow automatically from this identification. Our approach provides a quantum field theoretic generalization of Jacobson's thermodynamic derivation of the Einstein equations, replacing classical thermodynamic entropy with the well-defined quantum relative (Araki-Uhlmann) entropy. This suggests that quantum information plays a central role in what is often seen as a zeroth order approximation of a theory of quantum gravity, namely quantum field theory in curved spacetimes.

hep-th↗
arXiv2025

Local Expansion Mechanisms for Quantum-Scale Wormholes

Quantum models of spacetime have been conjectured to hypothetically allow for the formation of Planck scale wormholes. Building on the proposal of Morris, Thorne, and Yurtsever that such microscopic spacetime structures might be enlarged to macroscopic size, we revisit Roman's analysis of a wormhole in an inflationary de Sitter background. In this context, we introduce a refined quasi-local toy mechanism, which we call the local inflation bubble. This construction inflates a compact region of spacetime and thereby magnifies the underlying microstructure. Using the Einstein equations we determine the required stress-energy to sustain the bubble and obtain intrinsic lower bounds for the corresponding energy density, while acknowledging the continued reliance on exotic matter.

gr-qc↗
arXiv2025

The Relative Fermionic Entropy in Two-Dimensional Rindler Spacetime

The fermionic relative entropy in two-dimensional Rindler spacetime is studied using both modular theory and the reduced one-particle density operators. The methods and results are compared. A formula for the relative entropy for general Gaussian states is derived. As an application, the relative entropy is computed for a class of non-unitary excitations.

math-ph↗
arXiv2025

A Quantum Energy Inequality for a Non-commutative QFT

We present a quantum energy inequality (QEI) for quantum field theories formulated in non-commutative spacetimes, extending fundamental energy constraints to this generalized geometric framework. By leveraging operator-theoretic methods inspired by the positivity map of Waldmann et al. \cite{waldmannpos}, we construct linear combinations of deformed operators that generalize the commutative spacetime techniques of Fewster et al., \cite{Few98}. These non-commutative analogs enable us the derivation of a lower bound on the deformed averaged energy density, ensuring the stability of the underlying quantum field theory. Our result establishes rigorous constraints on the expectation values of the deformed (non-commutative) energy density, reinforcing the physical consistency of non-commutative models while preserving core principles of quantum field theory.

hep-th↗
arXiv2024

Microlocal Analysis of a Deformed Quantum Field Theory

A deformation technique, known as the warped convolution, takes quantum fields in Minkowski spacetime to quantum fields in noncommutative Minkowski space-time. Since a quantum field is an operator valued regular distribution and the warped convolution is (weakly) an oscillatory integral of Rieffel type, we prove that the symbol classes introduced by Hormander admit extensions which are suited to the warped convolutions of scalar quantum field operators. We further show that, if a particular vector state on the undeformed algebra of field operators fulfills the microlocal spectrum condition, then every vector state on the deformed algebra generated by these warped convolutions fulfills the microlocal spectrum condition.

math-ph↗
arXiv2024

Global Hyperbolicity and Self-adjointness

We show that the spatial part of the Klein-Gordon operator is an essentially self-adjoint operator on the Cauchy surfaces of various classes of spacetimes. Our proof employs the intricate connection between global hyperbolicity and geodesically complete Riemannian surfaces, and concludes by proving global hyperbolicity of the spacetimes under study.

gr-qc↗
arXiv2024

Notions of Fermionic Entropies of a Causal Fermion System

The fermionic von Neumann entropy, the fermionic entanglement entropy and the fermionic relative entropy are defined for causal fermion systems. Our definition makes use of entropy formulas for quasi-free fermionic states in terms of the reduced one-particle density operator. Our definitions are illustrated in various examples for Dirac spinors in two- and four-dimensional Minkowski space, in the Schwarzschild black hole geometry and for fermionic lattices. We review area laws for the two-dimensional diamond and a three-dimensional spatial region in Minkowski space. The connection is made to the computation of the relative entropy using modular theory.

math-ph↗
arXiv2024

Quantum Spacetimes from General Relativity?

We introduce a non-commutative product for curved spacetimes, that can be regarded as a generalization of the Rieffel (or Moyal-Weyl) product. This product employs the exponential map and a Poisson tensor, and the deformed product maintains associativity under the condition that the Poisson tensor $\Theta$ satisfies $\Theta^{\mu\nu}\nabla_{\nu}\Theta^{\rho\sigma}=0$, in relation to a Levi-Cevita connection. We proceed to solve the associativity condition for various physical spacetimes, uncovering non-commutative structures with compelling properties.

gr-qc↗
arXiv2023

Entropy-area law and temperature of de Sitter horizons from modular theory

We derive an entropy-area law for the future horizon of an observer in diamonds inside the static patch of de Sitter spacetime, taking into account the backreaction of quantum matter fields. We prove positivity and convexity of the relative entropy for coherent states using Tomita--Takesaki modular theory, from which the QNEC for diamonds follows. Furthermore, we show that the generalized entropy conjecture holds. Finally, we reveal that the local temperature which is measured by an observer at rest exhibits subleading quantum corrections with respect to the well-known cosmological horizon temperature $H/(2\pi)$.

hep-th↗
arXiv2023

Relative Entropy in de Sitter is a Noether Charge

We compute the relative entropy between the vacuum and a coherent state for a massive scalar field in de Sitter spacetime, using Tomita-Takesaki modular theory and the Araki-Uhlmann formula for the relative entropy. Embedding de Sitter spacetime as a hyperboloid in the ambient Minkowski space, we can restrict the Minkowski wedge and the corresponding modular operator to de Sitter, and we verify that this construction gives the correct modular flow. We check that the relative entropy is positive and jointly convex, relate it to the Noether charge of translations along the trajectories of the modular flow, and determine the local temperature as seen by an observer that moves along these trajectories.

gr-qc↗
arXiv2023

Superluminal local operations in quantum field theory: A ping-pong ball test

It is known that in quantum field theory, localized operations, e.g.\ given by unitary operators in local observable algebras, may lead to non-causal, or superluminal, state changes within their localization region. In this article, it is shown that both in quantum field theory as well as in classical relativistic field theory, there are localized operations which correspond to ``instantaneous'' spatial rotations (leaving the localization region invariant) leading to superluminal effects within the localization region. This shows that ``impossible measurement scenarios'' which have been investigated in the literature, and which rely on the presence of localized operations that feature superluminal effects within their localization region, do not only occur in quantum field theory, but also in classical field theory. This article is part of a Special Issue on the 'Physics of Time Travel' in the journal Universe, edited by A. Alonso-Serrano, S. Schuster, J. Santiago and M. Visser.

quant-ph↗
arXiv2023

Relative Entropy of Fermion Excitation States on the CAR Algebra

The relative entropy of certain states on the algebra of canonical anticommutation relations (CAR) is studied in the present work. The CAR algebra is used to describe fermionic degrees of freedom in quantum mechanics and quantum field theory. The states for which the relative entropy is investigated are multi-excitation states (similar to multi-particle states) with respect to KMS states defined with respect to a time-evolution induced by a unitary dynamical group on the one-particle Hilbert space of the CAR algebra. If the KMS state is quasifree, the relative entropy of multi-excitation states can be explicitly calculated in terms of 2-point functions, which are defined entirely by the one-particle Hilbert space defining the CAR algebra and the Hamilton operator of the dynamical group on the one-particle Hilbert space. This applies also in the case that the one-particle Hilbert space Hamilton operator has a continuous spectrum so that the relative entropy of multi-excitation states cannot be defined in terms of von Neumann entropies. The results obtained here for the relative entropy of multi-excitation states on the CAR algebra can be viewed as counterparts of results for the relative entropy of coherent states on the algebra of canonical commutation relations (CCR) which have appeared recently. It turns out to be useful to employ the setting of a self-dual CAR algebra introduced by Araki.

math-ph↗