Authors:Michel Bauer; Denis Bernard Pages: 653 - 693 Abstract: Motivated by studies of indirect measurements in quantum mechanics, we investigate the effect, in the strong noise limit, of adding an infinitesimal repulsive perturbation to a stochastic differential equation with an attractive fixed point. We conjecture, and prove for an important subclass, that the solutions exhibit a universal behavior when time is rescaled appropriately: by fine-tuning of the time scale with the infinitesimal repulsive perturbation, the trajectories converge in a precise sense to spiky trajectories that can be reconstructed from an auxiliary time-homogeneous Poisson process. Our results are based on two main tools. The first is a time change followed by an application of Skorokhod’s lemma. We prove an effective approximate version of this lemma of independent interest. The second is an analysis of first passage times, which shows a deep interplay between scale functions and invariant measures. We conclude with some speculations of possible applications of the same techniques in other areas. PubDate: 2018-03-01 DOI: 10.1007/s00023-018-0645-y Issue No:Vol. 19, No. 3 (2018)

Authors:Sven Bachmann; Alex Bols; Wojciech De Roeck; Martin Fraas Pages: 695 - 708 Abstract: We provide a short proof of the quantization of the Hall conductance for gapped interacting quantum lattice systems on the two-dimensional torus. This is not new and should be seen as an adaptation of the proof of Hastings and Michalakis (Commun Math Phys 334:433–471, 2015), simplified by making the stronger assumption that the Hamiltonian remains gapped when threading the torus with fluxes. We argue why this assumption is very plausible. The conductance is given by Berry’s curvature and our key auxiliary result is that the curvature is asymptotically constant across the torus of fluxes. PubDate: 2018-03-01 DOI: 10.1007/s00023-018-0651-0 Issue No:Vol. 19, No. 3 (2018)

Authors:Gian Michele Graf; Clément Tauber Pages: 709 - 741 Abstract: Floquet topological insulators describe independent electrons on a lattice driven out of equilibrium by a time-periodic Hamiltonian, beyond the usual adiabatic approximation. In dimension two, such systems are characterized by integer-valued topological indices associated with the unitary propagator, alternatively in the bulk or at the edge of a sample. In this paper, we give new definitions of the two indices, relying neither on translation invariance nor on averaging, and show that they are equal. In particular, weak disorder and defects are intrinsically taken into account. Finally, indices can be defined when two driven samples are placed next to one another either in space or in time and then shown to be equal. The edge index is interpreted as a quantized pumping occurring at the interface with an effective vacuum. PubDate: 2018-03-01 DOI: 10.1007/s00023-018-0657-7 Issue No:Vol. 19, No. 3 (2018)

Authors:Giulio Bonelli; Alba Grassi; Alessandro Tanzini Pages: 743 - 774 Abstract: We describe the magnetic phase of SU(N) \({\mathcal {N}}=2\) super-Yang–Mills theories in the self-dual \(\Omega \) -background in terms of a new class of multi-cut matrix models. These arise from a non-perturbative completion of topological strings in the dual four-dimensional limit which engineers the gauge theory in the strongly coupled magnetic frame. The corresponding spectral determinants provide natural candidates for the \(\tau \) -functions of isomonodromy problems for flat spectral connections associated with the Seiberg–Witten geometry. PubDate: 2018-03-01 DOI: 10.1007/s00023-017-0643-5 Issue No:Vol. 19, No. 3 (2018)

Authors:Pietro Longhi Pages: 775 - 842 Abstract: A new construction of BPS monodromies for 4d \({\mathcal {N}}=2\) theories of class \({\mathcal {S}}\) is introduced. A novel feature of this construction is its manifest invariance under Kontsevich–Soibelman wall crossing, in the sense that no information on the 4d BPS spectrum is employed. The BPS monodromy is encoded by topological data of a finite graph, embedded into the UV curve C of the theory. The graph arises from a degenerate limit of spectral networks, constructed at maximal intersections of walls of marginal stability in the Coulomb branch of the gauge theory. The topology of the graph, together with a notion of framing, encode equations that determine the monodromy. We develop an algorithmic technique for solving the equations and compute the monodromy in several examples. The graph manifestly encodes the symmetries of the monodromy, providing some support for conjectural relations to specializations of the superconformal index. For \(A_1\) -type theories, the graphs encoding the monodromy are “dessins d’enfants” on C, the corresponding Strebel differentials coincide with the quadratic differentials that characterize the Seiberg–Witten curve. PubDate: 2018-03-01 DOI: 10.1007/s00023-017-0635-5 Issue No:Vol. 19, No. 3 (2018)

Authors:José M. Gracia-Bondía; Jens Mund; Joseph C. Várilly Pages: 843 - 874 Abstract: We show how chirality of the weak interactions stems from string independence in the string-local formalism of quantum field theory. PubDate: 2018-03-01 DOI: 10.1007/s00023-017-0637-3 Issue No:Vol. 19, No. 3 (2018)

Authors:Paweł Duch Pages: 875 - 935 Abstract: We construct the Wightman and Green functions in a large class of models of perturbative QFT in the four-dimensional Minkowski space in the Epstein–Glaser framework. To this end we prove the existence of the weak adiabatic limit, generalizing the results due to Blanchard and Seneor. Our proof is valid under the assumption that the time-ordered products satisfy certain normalization condition. We show that this normalization condition may be imposed in all models with interaction vertices of canonical dimension 4 as well as in all models with interaction vertices of canonical dimension 3 provided each of them contains at least one massive field. Moreover, we prove that it is compatible with all the standard normalization conditions which are usually imposed on the time-ordered products. The result applies, for example, to quantum electrodynamics and non-abelian Yang–Mills theories. PubDate: 2018-03-01 DOI: 10.1007/s00023-018-0652-z Issue No:Vol. 19, No. 3 (2018)

Authors:Vincenzo Morinelli Pages: 937 - 958 Abstract: We discuss the Bisognano–Wichmann property for localPoincaré covariant nets of standard subspaces. We provide a sufficient algebraic condition on the covariant representation ensuring the Bisognano–Wichmann and the duality properties without further assumptions on the net. We call it modularity condition. It holds for direct integrals of scalar massive and massless representations. We present a class of massive modular covariant nets not satisfying the Bisognano–Wichmann property. Furthermore, we give an outlook on the relation between the Bisognano–Wichmann property and the split property in the standard subspace setting. PubDate: 2018-03-01 DOI: 10.1007/s00023-017-0636-4 Issue No:Vol. 19, No. 3 (2018)

Authors:Tobias Kuna; Dimitrios Tsagkarogiannis Abstract: We prove absolute convergence of the multi-body correlation functions as a power series in the density uniformly in their arguments. This is done by working in the context of the cluster expansion in the canonical ensemble and by expressing the correlation functions as the derivative of the logarithm of an appropriately extended partition function. In the thermodynamic limit, due to combinatorial cancellations, we show that the coefficients of the above series are expressed by sums over some class of two-connected graphs. Furthermore, we prove the convergence of the density expansion of the “direct correlation function” which is based on a completely different approach and it is valid only for some integral norm. Precisely, this integral norm is suitable to derive the Ornstein–Zernike equation. As a further outcome, we obtain a rigorous quantification of the error in the Percus–Yevick approximation. PubDate: 2018-02-24 DOI: 10.1007/s00023-018-0655-9

Authors:P. J. Forrester; J. R. Ipsen; Dang-Zheng Liu Abstract: We investigate spectral properties of a Hermitised random matrix product which, contrary to previous product ensembles, allows for eigenvalues on the full real line. We prove that the eigenvalues form a bi-orthogonal ensemble, which reduces asymptotically to the Hermite Muttalib–Borodin ensemble. Explicit expressions for the bi-orthogonal functions as well as the correlation kernel are provided. Scaling the latter near the origin gives a limiting kernel involving Meijer G-functions, and the functional form of the global density is calculated. As a part of this study, we introduce a new matrix transformation which maps the space of polynomial ensembles onto itself. This matrix transformation is closely related to the so-called hyperbolic Harish-Chandra–Itzykson–Zuber integral. PubDate: 2018-02-24 DOI: 10.1007/s00023-018-0654-x

Authors:Jan Felipe van Diejen; Erdal Emsiz; Ignacio Nahuel Zurrián Abstract: We employ a discrete integral-reflection representation of the double affine Hecke algebra of type \(C^\vee C\) at the critical level \(\text {q}=1\) , to endow the open finite q-boson system with integrable boundary interactions at the lattice ends. It is shown that the Bethe Ansatz entails a complete basis of eigenfunctions for the commuting quantum integrals in terms of Macdonald’s three-parameter hyperoctahedral Hall–Littlewood polynomials. PubDate: 2018-02-23 DOI: 10.1007/s00023-018-0658-6

Authors:L. Castellani; R. Catenacci; P. A. Grassi Abstract: We reformulate super-quantum mechanics in the context of integral forms. This framework allows to interpolate between different actions for the same theory, connected by different choices of picture changing operators (PCO). In this way we retrieve component and superspace actions and prove their equivalence. The PCO are closed integral forms and can be interpreted as super-Poincaré duals of bosonic submanifolds embedded into a supermanifold. We use them to construct Lagrangians that are top integral forms, and therefore can be integrated on the whole supermanifold. The \(D=1, N=1\) and the \(D=1, N=2\) cases are studied, in a flat and in a curved supermanifold. In this formalism, we also consider coupling with gauge fields, Hilbert space of quantum states, and observables. PubDate: 2018-02-23 DOI: 10.1007/s00023-018-0653-y

Authors:Dmitry Ostrovsky Abstract: A theory of intermittency differentiation is developed for a general class of 1D infinitely divisible multiplicative chaos measures. The intermittency invariance of the underlying infinitely divisible field is established and utilized to derive a Feynman–Kac equation for the distribution of the total mass of the limit measure by considering a stochastic flow in intermittency. The resulting equation prescribes the rule of intermittency differentiation for a general functional of the total mass and determines the distribution of the total mass and its dependence structure to the first order in intermittency. A class of non-local functionals of the limit measure extending the total mass is introduced and shown to be invariant under intermittency differentiation making the computation of the full high-temperature expansion of the total mass distribution possible in principle. For application, positive integer moments and covariance structure of the total mass are considered in detail. PubDate: 2018-02-19 DOI: 10.1007/s00023-018-0656-8

Authors:Péter Bálint; Péter Nándori; Domokos Szász; Imre Péter Tóth Abstract: We prove exponential correlation decay in dispersing billiard flows on the 2-torus assuming finite horizon and lack of corner points. With applications aimed at describing heat conduction, the highly singular initial measures are concentrated here on 1-dimensional submanifolds (given by standard pairs) and the observables are supposed to satisfy a generalized Hölder continuity property. The result is based on the exponential correlation decay bound of Baladi et al. (Invent Math, 211:39–117, 2018. https://doi.org/10.1007/s00222-017-0745-1) obtained for Hölder continuous observables in these billiards. The model dependence of the bounds is also discussed. PubDate: 2018-02-17 DOI: 10.1007/s00023-018-0648-8

Authors:Laurent Charles; Leonid Polterovich Abstract: We discuss a link between symplectic displacement energy, a fundamental notion of symplectic topology, and the quantum speed limit, a universal constraint on the speed of quantum-mechanical processes. The link is provided by the quantum-classical correspondence formalized within the framework of the Berezin–Toeplitz quantization. PubDate: 2018-02-16 DOI: 10.1007/s00023-018-0649-7

Authors:András Vasy; Michał Wrochna Abstract: We consider the wave equation on asymptotically Minkowski spacetimes and the Klein–Gordon equation on even asymptotically de Sitter spaces. In both cases, we show that the extreme difference of propagators (i.e., retarded propagator minus advanced, or Feynman minus anti-Feynman), defined as Fredholm inverses, induces a symplectic form on the space of solutions with wave front set confined to the radial sets. Furthermore, we construct isomorphisms between the solution spaces and symplectic spaces of asymptotic data. As an application of this result, we obtain distinguished Hadamard two-point functions from asymptotic data. Ultimately, we prove that non-interacting Quantum Field Theory on asymptotically de Sitter spacetimes extends across the future and past conformal boundary, i.e., to a region represented by two even asymptotically hyperbolic spaces. Specifically, we show this to be true both at the level of symplectic spaces of solutions and at the level of Hadamard two-point functions. PubDate: 2018-02-16 DOI: 10.1007/s00023-018-0650-1

Authors:Wei-Xi Li Abstract: In this paper, we consider the Witten Laplacian on 0-forms and give sufficient conditions under which the Witten Laplacian admits a compact resolvent. These conditions are imposed on the potential itself, involving the control of high-order derivatives by lower ones, as well as the control of the positive eigenvalues of the Hessian matrix. This compactness criterion for resolvent is inspired by the one for the Fokker–Planck operator. Our method relies on the nilpotent group techniques developed by Helffer–Nourrigat (Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs, Birkhäuser Boston Inc., Boston, 1985). PubDate: 2018-02-16 DOI: 10.1007/s00023-018-0659-5

Authors:Jeremiah Birrell; Jan Wehr Abstract: We study the dynamics of an inertial particle coupled to forcing, dissipation, and noise in the small mass limit. We derive an expression for the limiting (homogenized) joint distribution of the position and (scaled) velocity degrees of freedom. In particular, weak convergence of the joint distributions is established, along with a bound on the convergence rate for a wide class of expected values. PubDate: 2018-02-06 DOI: 10.1007/s00023-018-0646-x

Authors:Christian D. Jäkel; Jens Mund Abstract: We establish the Haag–Kastler axioms for a class of interacting quantum field theories on the two-dimensional de Sitter space, which satisfy finite speed of light. The \({\mathscr {P}} (\varphi )_2\) model constructed in [3], describing massive scalar bosons with polynomial interactions, provides an example. PubDate: 2018-01-20 DOI: 10.1007/s00023-018-0647-9

Authors:Peter Hintz Abstract: We show that a stationary solution of the Einstein–Maxwell equations which is close to a non-degenerate Reissner–Nordström–de Sitter solution is in fact equal to a slowly rotating Kerr–Newman–de Sitter solution. The proof uses the nonlinear stability of the Kerr–Newman–de Sitter family of black holes with small angular momenta, recently established by the author, together with an extension argument for Killing vector fields. Our black hole uniqueness result only requires the solution to have high but finite regularity; in particular, we do not make any analyticity assumptions. PubDate: 2017-12-15 DOI: 10.1007/s00023-017-0633-7