Photon-photon scattering: a tutorial

Long-established results for the low-energy photon-photon scattering, gamma gamma --> gamma gamma, have recently been questioned. We analyze that claim and demonstrate that it is inconsistent with experience. We demonstrate that the mistake originates from an erroneous manipulation of divergent integrals and discuss the connection with another recent claim about the Higgs decay into two photons. We show a simple way of correctly computing the low-energy gamma gamma scattering.Comment: 9 pages, 2 figures; v2: typographical errors corrected in eqs 1, 20, and 21; ref. [23] adde

Dirac Operators and the Calculation of the Connes Metric on arbitrary (Infinite) Graphs

As an outgrowth of our investigation of non-regular spaces within the context of quantum gravity and non-commutative geometry, we develop a graph Hilbert space framework on arbitrary (infinite) graphs and use it to study spectral properties of graph-Laplacians and graph-Dirac-operators. We define a spectral triplet sharing most of the properties of what Connes calls a spectral triple. With the help of this scheme we derive an explicit expression for the Connes-distance function on general directed or undirected graphs. We derive a series of apriori estimates and calculate it for a variety of examples of graphs. As a possibly interesting aside, we show that the natural setting of approaching such problems may be the framework of (non-)linear programming or optimization. We compare our results (arrived at within our particular framework) with the results of other authors and show that the seeming differences depend on the use of different graph-geometries and/or Dirac operators.Comment: 27 pages, Latex, comlementary to an earlier paper, general treatment of directed and undirected graphs, in section 4 a series of general results and estimates concerning the Connes Distance on graphs together with examples and numerical estimate

Boundary relations and generalized resolvents of symmetric operators

The Kre\u{\i}n-Naimark formula provides a parametrization of all selfadjoint exit space extensions of a, not necessarily densely defined, symmetric operator, in terms of maximal dissipative (in \dC_+) holomorphic linear relations on the parameter space (the so-called Nevanlinna families). The new notion of a boundary relation makes it possible to interpret these parameter families as Weyl families of boundary relations and to establish a simple coupling method to construct the generalized resolvents from the given parameter family. The general version of the coupling method is introduced and the role of boundary relations and their Weyl families for the Kre\u{\i}n-Naimark formula is investigated and explained.Comment: 47 page

Looking for magnetic monopoles at LHC with diphoton events

Magnetic monopoles have been a subject of interest since Dirac established the relation between the existence of monopoles and charge quantization. The intense experimental search carried thus far has not met with success. The Large Hadron Collider is reaching energies never achieved before allowing the search for exotic particles in the TeV mass range. In a continuing effort to discover these rare particles we propose here other ways to detect them. We study the observability of monopoles and monopolium, a monopole-antimonopole bound state, at the Large Hadron Collider in the $\gamma \gamma$ channel for monopole masses in the range 500-1000 GeV. We conclude that LHC is an ideal machine to discover monopoles with masses below 1 TeV at present running energies and with 5 fb$^{-1}$ of integrated luminosity.Comment: This manuscript contains information appeared in Looking for magnetic monopoles at LHC, arXiv:1104.0218 [hep-ph] and Monopolium detection at the LHC.,arXiv:1107.3684 [hep-ph] by the same authors, rewritten for joint publication in The European Physica Journal Plus. 26 pages, 22 figure

Kirchhoff's Rule for Quantum Wires

In this article we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with $n$ open ends and where we define the Hamiltonian to be (minus) the Laplace operator with general boundary conditions at the vertices. This results in a scattering theory with $n$ channels. The corresponding on-shell S-matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy $E>0$ is explicitly given in terms of the boundary conditions and the lengths of the internal lines. It is shown to be unitary, which may be viewed as the quantum version of Kirchhoff's law. We exhibit covariance and symmetry properties. It is symmetric if the boundary conditions are real. Also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory. Finally we provide a composition rule by which the on-shell S-matrix of a graph is factorizable in terms of the S-matrices of its subgraphs. All proofs only use known facts from the theory of self-adjoint extensions, standard linear algebra, complex function theory and elementary arguments from the theory of Hermitean symplectic forms.Comment: 40 page

Generalized boundary triples, I. Some classes of isometric and unitary boundary pairs and realization problems for subclasses of Nevanlinna functions

© 2020 The Authors. Mathematische Nachrichten published by Wiley‐VCH Verlag GmbH & Co. KGaA. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.fi=vertaisarvioitu|en=peerReviewed

Two-dimensional Hamiltonian systems

This survey article contains various aspects of the direct and inverse spectral problem for twodimensional Hamiltonian systems, that is, two dimensional canonical systems of homogeneous differential equations of the form Jy'(x) = -zH(x)y(x); x ∈ [0;L); 0 < L ≤ ∞; z ∈ C; with a real non-negative definite matrix function H ≥ 0 and a signature matrix J, and with a standard boundary condition of the form y1(0+) = 0. Additionally it is assumed that Weyl's limit point case prevails at L. In this case the spectrum of the canonical system is determined by its Titchmarsh-Weyl coefficient Q which is a Nevanlinna function, that is, a function which maps the upper complex half-plane analytically into itself. In this article an outline of the Titchmarsh-Weyl theory for Hamiltonian systems is given and the solution of the direct spectral problem is shown. Moreover, Hamiltonian systems comprehend the class of differential equations of vibrating strings with a non-homogenous mass-distribution function as considered by M.G. Krein. The inverse spectral problem for two{dimensional Hamiltonian systems was solved by L. de Branges by use of his theory of Hilbert spaces of entire functions, showing that each Nevanlinna function is the Titchmarsh-Weyl coefficient of a uniquely determined normed Hamiltonian. More detailed results of this connection for e.g. systems with a semibounded or discrete or finite spectrum are presented, and also some results concerning spectral perturbation, which allow an explicit solution of the inverse spectral problem in many cases

Towards Comprehensive Foundations of Computational Intelligence

Abstract. Although computational intelligence (CI) covers a vast variety of different methods it still lacks an integrative theory. Several proposals for CI foundations are discussed: computing and cognition as compression, meta-learning as search in the space of data models, (dis)similarity based methods providing a framework for such meta-learning, and a more general approach based on chains of transformations. Many useful transformations that extract information from features are discussed. Heterogeneous adaptive systems are presented as particular example of transformation-based systems, and the goal of learning is redefined to facilitate creation of simpler data models. The need to understand data structures leads to techniques for logical and prototype-based rule extraction, and to generation of multiple alternative models, while the need to increase predictive power of adaptive models leads to committees of competent models. Learning from partial observations is a natural extension towards reasoning based on perceptions, and an approach to intuitive solving of such problems is presented. Throughout the paper neurocognitive inspirations are frequently used and are especially important in modeling of the higher cognitive functions. Promising directions such as liquid and laminar computing are identified and many open problems presented.