Hyperboloidal layers for hyperbolic equations on unbounded domains

We show how to solve hyperbolic equations numerically on unbounded domains by compactification, thereby avoiding the introduction of an artificial outer boundary. The essential ingredient is a suitable transformation of the time coordinate in combination with spatial compactification. We construct a new layer method based on this idea, called the hyperboloidal layer. The method is demonstrated on numerical tests including the one dimensional Maxwell equations using finite differences and the three dimensional wave equation with and without nonlinear source terms using spectral techniques.Comment: 23 pages, 23 figure

Qubit State Discrimination

We show how one can solve the problem of discriminating between qubit states. We use the quantum state discrimination duality theorem and the Bloch sphere representation of qubits which allows for an easy geometric and analytical representation of the optimal guessing strategies.Comment: 6 pages, 4 figures. v2 has small corrections and changes in reference

Relative-locality geometry for the Snyder model

We investigate the geometry of the energy-momentum space of the Snyder model of noncommmutative geometry and of its generalizations, according to the postulates of relative locality. These relate the geometric structures to the deformed composition law of momenta. It turns out that the Snyder energy-momentum spaces are maximally symmetric, with vanishing torsion and nonmetricity. However, one cannot apply straightforwardly the phenomenological relations between the geometry and the dynamics postulated in the standard prescription of relative locality, because they were obtained assuming that the leading corrections to the composition law of momenta are quadratic, which is not the case with the Snyder model and its generalizationsComment: 13 page

Distributions and Integration in superspace

Distributions in superspace constitute a very useful tool for establishing an integration theory. In particular, distributions have been used to obtain a suitable extension of the Cauchy formula to superspace and to define integration over the superball and the supersphere through the Heaviside and Dirac distributions, respectively. In this paper, we extend the distributional approach to integration over more general domains and surfaces in superspace. The notions of domain and surface in superspace are defined by smooth bosonic phase functions $g$. This allows to define domain integrals and oriented (as well as non-oriented) surface integrals in terms of the Heaviside and Dirac distributions of the superfunction $g$. It will be shown that the presented definition for the integrals does not depend on the choice of the phase function $g$ defining the corresponding domain or surface. In addition, some examples of integration over a super-paraboloid and a super-hyperboloid will be presented. Finally, a new distributional Cauchy-Pompeiu formula will be obtained, which generalizes and unifies the previously known approaches.Comment: 25 page

Spin-dependent Bohm trajectories for hydrogen eigenstates

The Bohm trajectories for several hydrogen atom eigenstates are determined, taking into account the additional momentum term that arises from the Pauli current. Unlike the original Bohmian result, the spin-dependent term yields nonstationary trajectories. The relationship between the trajectories and the standard visualizations of orbitals is discussed. The trajectories for a model problem that simulates a 1s-2p transition in hydrogen are also examined.Comment: 11 pages, 3 figure

An inverse indefinite numerical range problem

https://thekeep.eiu.edu/den_1997_feb/1008/thumbnail.jp

Entanglement, Holography and Causal Diamonds

We argue that the degrees of freedom in a d-dimensional CFT can be re-organized in an insightful way by studying observables on the moduli space of causal diamonds (or equivalently, the space of pairs of timelike separated points). This 2d-dimensional space naturally captures some of the fundamental nonlocality and causal structure inherent in the entanglement of CFT states. For any primary CFT operator, we construct an observable on this space, which is defined by smearing the associated one-point function over causal diamonds. Known examples of such quantities are the entanglement entropy of vacuum excitations and its higher spin generalizations. We show that in holographic CFTs, these observables are given by suitably defined integrals of dual bulk fields over the corresponding Ryu-Takayanagi minimal surfaces. Furthermore, we explain connections to the operator product expansion and the first law of entanglement entropy from this unifying point of view. We demonstrate that for small perturbations of the vacuum, our observables obey linear two-derivative equations of motion on the space of causal diamonds. In two dimensions, the latter is given by a product of two copies of a two-dimensional de Sitter space. For a class of universal states, we show that the entanglement entropy and its spin-three generalization obey nonlinear equations of motion with local interactions on this moduli space, which can be identified with Liouville and Toda equations, respectively. This suggests the possibility of extending the definition of our new observables beyond the linear level more generally and in such a way that they give rise to new dynamically interacting theories on the moduli space of causal diamonds. Various challenges one has to face in order to implement this idea are discussed.Comment: 84 pages, 12 figures; v2: expanded discussion on constraints in section 7, matches published versio