Annales Henri Poincaré最新文献

Relative entropy is a powerful measure of the dissimilarity between two statistical field theories in the continuum. In this work, we study the relative entropy between Gaussian scalar field theories in a finite volume with different masses and boundary conditions. We show that the relative entropy depends crucially on d, the dimension of Euclidean space. Furthermore, we demonstrate that the mutual information between two disjoint regions in (mathbb {R}^d) is finite if the two regions are separated by a finite distance and satisfies an area law. We then construct an example of “touching” regions between which the mutual information is infinite. We argue that the properties of mutual information in scalar field theories can be explained by the Markov property of these theories.

Drinfeld defined the Knizhnik–Zamolodchikov (KZ) associator (Phi _{textrm{KZ}}) by considering the regularized holonomy of the KZ connection along the droit chemin [0, 1]. The KZ associator is a group-like element of the free associative algebra with two generators, and it satisfies the pentagon equation. In this paper, we consider paths on ({mathbb {C}}backslash { z_1, dots , z_n}) which start and end at tangential base points. These paths are not necessarily straight, and they may have a finite number of transversal self-intersections. We show that the regularized holonomy H of the KZ connection associated with such a path satisfies a generalization of Drinfeld’s pentagon equation. In this equation, we encounter H, (Phi _{textrm{KZ}}), and new factors associated with self-intersections, tangential base points, and the rotation number of the path.

KMS states on ({mathbb {Z}}_2)-crossed products of unital (C^*)-algebras ({mathcal {A}}) are characterized in terms of KMS states and twisted KMS functionals of ({mathcal {A}}). These functionals are shown to describe the extensions of KMS states (omega ) on ({mathcal {A}}) to the crossed product ({mathcal {A}} rtimes {mathbb {Z}}_2) and can also be characterized by the twisted center of the von Neumann algebra generated by the GNS representation corresponding to (omega ). As a particular class of examples, KMS states on ({mathbb {Z}}_2)-crossed products of CAR algebras with dynamics and grading given by Bogoliubov automorphisms are analyzed in detail. In this case, one or two extremal KMS states are found depending on a Gibbs-type condition involving the odd part of the absolute value of the Hamiltonian. As an application in mathematical physics, the extended field algebra of the Ising QFT is shown to be a ({mathbb {Z}}_2)-crossed product of a CAR algebra which has a unique KMS state.

We consider a closed macroscopic quantum system in a pure state (psi _t) evolving unitarily and take for granted that different macro states correspond to mutually orthogonal subspaces ({mathcal {H}}_nu ) (macro spaces) of Hilbert space, each of which has large dimension. We extend previous work on the question what the evolution of (psi _t) looks like macroscopically, specifically on how much of (psi _t) lies in each ({mathcal {H}}_nu ). Previous bounds concerned the absolute error for typical (psi _0) and/or t and are valid for arbitrary Hamiltonians H; now, we provide bounds on the relative error, which means much tighter bounds, with probability close to 1 by modeling H as a random matrix, more precisely as a random band matrix (i.e., where only entries near the main diagonal are significantly nonzero) in a basis aligned with the macro spaces. We exploit particularly that the eigenvectors of H are delocalized in this basis. Our main mathematical results confirm the two phenomena of generalized normal typicality (a type of long-time behavior) and dynamical typicality (a type of similarity within the ensemble of (psi _0) from an initial macro space). They are based on an extension we prove of a no-gaps delocalization result for random matrices by Rudelson and Vershynin (Geom Funct Anal 26:1716–1776, 2016).

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.

Schrödinger operators with periodic potential have generally been shown to exhibit ballistic transport. In this work, we investigate whether the propagation velocity, while positive, can be made arbitrarily small by a suitable choice of the periodic potential. We consider the discrete one-dimensional Schrödinger operator (Delta +mu V), where (Delta ) is the discrete Laplacian, V is a p-periodic non-degenerate potential and (mu >0). We establish a Lieb–Robinson-type bound with a group velocity that scales like (mathcal {O}(1/mu )) as (mu rightarrow infty ). This shows the existence of a linear light cone with a maximum velocity of quantum propagation that is decaying at a rate proportional to (1/mu ). Furthermore, we prove that the asymptotic velocity, or the average velocity of the time-evolved state, exhibits a decay proportional to (mathcal {O}(1/mu ^{p-1})) as (mu rightarrow infty ).

We study four-dimensional asymptotically flat electrostatic electro-vacuum spacetimes with a connected black hole, photon sphere, or equipotential photon surface inner boundary. Our analysis, inspired by the potential theory approach by Agostiniani–Mazzieri, allows to give self-contained proofs of known uniqueness theorems of the sub-extremal, extremal, and super-extremal Reissner–Nordström spacetimes. We also obtain new results for connected photon spheres and equipotential photon surfaces in the extremal case. Finally, we provide, up to a restriction on the range of their radii, the uniqueness result for connected (both non-degenerate and degenerate) equipotential photon surfaces in the super-extremal case, not yet treated in the literature.

The classical singularity theorems of R. Penrose and S. Hawking from the 1960s show that, given a pointwise energy condition (and some causality as well as initial assumptions), spacetimes cannot be geodesically complete. Despite their great success, the theorems leave room for physically relevant improvements, especially regarding the classical energy conditions as essentially any quantum field theory necessarily violates them. While singularity theorems with weakened energy conditions exist for worldline integral bounds, so-called worldvolume bounds are in some cases more applicable than the worldline ones, such as the case of some massive free fields. In this paper, we study integral Ricci curvature bounds based on worldvolume quantum strong energy inequalities. Under the additional assumption of a—potentially very negative—global timelike Ricci curvature bound, a Hawking-type singularity theorem is proved. Finally, we apply the theorem to a cosmological scenario proving past geodesic incompleteness in cases where the worldline theorem was inconclusive.

We consider a homogeneous Bose gas in the Gross–Pitaevskii limit at temperatures that are comparable to the critical temperature for Bose–Einstein condensation. Recently, an upper bound for the grand canonical free energy was proved in Boccato et al. (SIAM J Math Anal 56(2):2611–2660, 2024) capturing two novel contributions. First, the free energy of the interacting condensate is given in terms of an effective theory describing the probability distribution of the number of condensed particles. Second, the free energy of the thermally excited particles equals that of a temperature-dependent Bogoliubov Hamiltonian. We extend this result to a more general class of interaction potentials, including interactions with a hard core. Our proof follows a different approach than the one in Boccato et al. (SIAM J Math Anal 56(2):2611–2660, 2024): We model microscopic correlations between the particles by a Jastrow factor and exploit a cancellation in the computation of the energy that emerges due to the different length scales in the system.