arXiv subjects

Markus B. Fröb

Publications and source records attributed to Markus B. Fröb.

At least 19 recordsLinked to original sources

arXiv2026

Finite relative entropy for locally squeezed states

Relative entropy serves as a fundamental measure of state distinguishability in both quantum information theory and relativistic quantum field theory. Despite its conceptual importance, however, explicit computations of relative entropy remain notoriously difficult. Thus far, results in closed form have only been obtained for ground states, coherent states, and, more recently, single-mode squeezed states. In this work, we extend the analysis to multi-mode squeezed states, imposing that the squeezing generators be local either in space or in spacetime, which results in a continuum of squeezed modes. We provide a detailed and self-contained analysis of such states for a free scalar quantum field on Minkowski spacetime, connecting also with older results on the essential self-adjointness of the Wick square, and showing that they lie in the folium of the Minkowski vacuum representation. We uncover that local-in-space squeezing leads to physically unreasonable states which have a divergent energy density and relative entropy, while local-in-spacetime squeezing provides a physically reasonable class of states with finite relative entropy.

quant-ph↗

arXiv2026

A generalization of the Powers-Størmer-Ogata inequality

We show that for any positive operator monotone function $f$ the inequality $\int_{[0,\infty)} f(t) \, \mathrm{d} \left\lVert E^{Δ_{φ,ψ}}(t) ξ_ψ\right\rVert^2 + f'_\infty φ(1-s(ψ)) \geq \frac{f(1)}{2} \Bigl( φ(1) + ψ(1) - \lVert φ- ψ\rVert \Bigr)$ holds, where $φ, ψ\in \mathcal{M}_{*,+}$ are two normal positive linear functionals on a von Neumann algebra $\mathcal{M}$, and $Δ_{φ,ψ}$ is the associated relative modular operator. Choosing $f(t) = t^s$ with $s \in [0,1]$, the Powers-Størmer-Ogata inequality is recovered.

math.OA↗

arXiv2026

Relative entropy for $λϕ^4$ in the Rindler wedge

We consider the relative entropy between the vacuum and a coherent state in the Rindler wedge for an interacting $λϕ^4$ theory to first order in $λ$. 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

Integral representations of $f$-divergences for general von Neumann algebras

We define and analyze hockeystick divergences and $f$-divergences for normal positive functionals on general von Neumann algebras, generalizing and unifying previous work in classical probability and finite-dimensional von Neumann algebras. All the main properties of these state distinguishability measures (including in particular monotonicity, convexity, semicontinuity, bounds, state discrimination, data processing inequality) are derived from properties of the Jordan decomposition of selfadjoint normal functionals. This is done by representing the $f$-divergences as integrals over hockeystick divergences, and their significance in quantum hypothesis testing is reviewed. The $f_0$-divergence given by the information function $f_0(t) = t \ln t$ is shown to coincide with Araki's relative entropy, extending results of Frenkel to general von Neumann algebras.

math.OA↗

arXiv2026

Bounding relative entropy for non-unitary excitations in quantum field theory

We show how one can use the convexity of non-commutative $L^p$ norms to bound the relative entropy between a faithful state on a von Neumann algebra and an arbitrary excitation thereof. Our results hold for general von Neumann algebras, including the local algebras of type III that are ubiquitous in quantum field theory, and do not require knowledge of the relative modular operator. As an application of our results, we prove that for the chiral current on a light ray, the relative entropy between the vacuum and a dense set of single-particle states is uniformly bounded.

math-ph↗

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

One-loop correction to primordial tensor modes during radiation era

The ability to infer properties of primordial inflation relies on the conservation of the superhorizon perturbations between their exit during inflation, and their re-entry during radiation era. Any considerable departure from this property would require reinterpreting the data. This is why it is important to understand how superhorizon perturbations interact with the thermal plasma driving the radiation dominated Universe. We model the plasma by free photons in a thermal state and compute the one-loop correction to the power spectrum of primordial tensor perturbations. This correction grows in time and is not suppressed by any small parameter. While one-loop result is not reliable because it invalidates perturbation theory, it signals potentially interesting effects that should be investigated further.

astro-ph.CO↗

arXiv2025

The Weyl anomaly in interacting quantum field theory on curved spacetimes

We define the notion of Weyl anomalies, measuring the violation of local scale invariance, in interacting quantum field theory on curved spacetimes in the framework of locally covariant field theory. We discuss some general properties of Weyl anomalies, such as their relation to the trace anomaly. We give a criterion for a theory to be conformal at the quantum level, and show that even for a conformal theory Weyl transformations in general obtain quantum corrections. We study the trace anomaly in detail for the $ϕ^4$ theory, in particular determining it up to second order in the interaction. We also show that at third order in the interaction a potential $\Box ϕ^2$ term can be removed by finite renormalization.

hep-th↗

arXiv2025

Measurements in stochastic gravity and thermal variance

We analyze the thermal fluctuations of a free, conformally invariant, Maxwell quantum field (photon) interacting with a cosmological background spacetime, in the framework of quantum field theory in curved spacetimes and semiclassical and stochastic gravity. The thermal fluctuations give rise to backreaction effects upon the spacetime geometry, which are incorporated in the semiclassical Einstein-Langevin equation, evaluated in the cosmological Friedmann-Lemaître-Robertson-Walker spacetime. We first evaluate the semiclassical Einstein equation for the background geometry sourced by the thermal quantum stress-energy tensor. For large enough temperature, the solution is approximated by a radiation-dominated expanding universe driven by the thermal bath of photons. We then evaluate the thermal noise kernel associated to the quantum fluctuations of the photon field using point-splitting regularization methods, and give its explicit analytic form in the limits of large and small temperature, as well as a local approximation. Finally, we prove that this thermal noise kernel corresponds exactly to the thermal variance of the induced fluctuations of the linearized metric perturbation in the local and covariant measurement scheme defined by Fewster and Verch. Our analysis allows to quantify the extent to which quantum fluctuations may give rise to non-classical effects, and thus become relevant in inflationary cosmology.

gr-qc↗

arXiv2025

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↗

arXiv2025

Local operators in the Sine-Gordon model: $\partial_μϕ\, \partial_νϕ$ and the stress tensor

We consider the simplest non-trivial local composite operators in the massless Sine-Gordon model, which are $\partial_μϕ\, \partial_νϕ$ and the stress tensor $T_{μν}$. We show that even in the finite regime $β^2 < 4 π$ of the theory, these operators need additional renormalisation (beyond the free-field normal-ordering) at each order in perturbation theory. We further prove convergence of the renormalised perturbative series for their expectation values, both in the Euclidean signature and in Minkowski space-time, and for the latter in an arbitrary Hadamard state. Lastly, we show that one must add a quantum correction (proportional to $\hbar$) to the renormalised stress tensor to obtain a conserved quantity.

math-ph↗

arXiv2025

Petz-Rényi relative entropy in QFT from modular theory

We consider the generalization of the Araki-Uhlmann formula for relative entropy to Petz-Rényi relative entropy. We compute this entropy for a free scalar field in the Minkowski wedge between the vacuum and a coherent state, as well as for the free chiral current in a thermal state. In contrast to the relative entropy which in these cases only depends on the sympletic form and thus reduces to the classical entropy of a wave packet, the Petz-Rényi relative entropy also depends on the symmetric part of the two-point function and is thus genuinely quantum. We also consider the relation with standard subspaces, where we define the Rényi entropy of a vector and show that it admits an upper bound given by the entropy of the vector.

math-ph↗

arXiv2025

Relating the modular Hamiltonian to two-point functions

We consider the modular Hamiltonian associated to standard subspaces for a free scalar field in a globally hyperbolic spacetime in an arbitrary Gaussian state. We show how the modular Hamiltonian is related to the two-point function of the theory. For the restriction of the modular Hamiltonian to the subspace, we recover formulas that were obtained previously by Peschel, Casini and Huerta. We also show how the same results can be obtained more directly from the KMS condition, and generalize our results to general CCR algebras.

math-ph↗

arXiv2025

De Sitter quantum gravity within the covariant Lorentzian approach to asymptotic safety

Recent technical and conceptual advancements in the asymptotic safety approach to quantum gravity have enabled studies of the UV completion of Lorentzian Einstein gravity, emphasizing the role of the state dependence. We present here the first complete investigation of the flow equations of the Einstein-Hilbert action within a cosmological spacetime, namely de Sitter spacetime. Using the newly derived graviton propagator for general gauges and masses in de Sitter spacetime, we analyze the dependence on the gauge and on finite renormalization parameters. Our results provide evidence of a UV fixed point for the most commonly used gauges.

hep-th↗

arXiv2024

Modular Hamiltonian for fermions of small mass

We consider the algebra of massive fermions restricted to a diamond in two-dimensional Minkowski spacetime, and in the Minkowski vacuum state. While the massless modular Hamiltonian is known for this setting, the derivation of the massive one is an open problem. We compute the small-mass corrections to the modular Hamiltonian in a perturbative approach, finding some terms which were previously overlooked. Our approach can in principle be extended to all orders in the mass, even though it becomes computationally challenging.

hep-th↗

arXiv2024

The Sine-Gordon QFT in de Sitter spacetime

We consider the massless Sine-Gordon model in de Sitter spacetime, in the regime $β^2 < 4 π$ and using the framework of perturbative algebraic quantum field theory. We show that a Fock space representation exists for the free massless field, but that the natural one-parameter family of vacuum-like states breaks the de Sitter boost symmetries. We prove convergence of the perturbative series for the S matrix in this representation, and construct the interacting Haag-Kastler net of local algebras from the relative S matrices. We show that the net fulfills isotony, locality and de Sitter covariance (in the algebraic adiabatic limit), even though the states that we consider are not invariant. We furthermore prove convergence of the perturbative series for the interacting field and the vertex operators, and verify that the interacting equation of motion holds.

math-ph↗

arXiv2024

Heat kernel coefficients for massive gravity

We compute the heat kernel coefficients that are needed for the regularization and renormalization of massive gravity. Starting from the Stueckelberg action for massive gravity, we determine the propagators of the different fields (massive tensor, vector and scalar) in a general linear covariant gauge depending on four free gauge parameters. We then compute the non-minimal heat kernel coefficients for all the components of the scalar, vector and tensor sector, and employ these coefficients to regularize the propagators of all the different fields of massive gravity. We also study the massless limit and discuss the appearance of the van Dam-Veltman-Zakharov discontinuity. In the course of the computation, we derive new identities relating the heat kernel coefficients of different field sectors, both massive and massless.

hep-th↗

arXiv2024

Modular Hamiltonian and modular flow of massless fermions on a cylinder

We determine explicitly the modular flow and the modular Hamiltonian for massless free fermions in diamonds on a cylinder in 1+1 dimensions. We consider both periodic and antiperiodic boundary conditions, the ground state in the antiperiodic case and the most general family of quasi-free zero-energy ground states in the periodic case, which depend on four parameters and are generally mixed. While for the antiperiodic ground state and one periodic ground state (the maximally mixed zero-temperature state) the modular data is known, our results for the generic ground state in the periodic case are completely new. We find that generically both the modular flow and the modular Hamiltonian are non-local, and we show that in the parametric limit where the state becomes pure the modular data becomes local. Moreover, even in the local case the modular flow generically mixes the two chiralities. This kind of behavior has not been observed previously.

math-ph↗