Five-dimensional axisymmetric spacetime solutions

We give a method by which a class of 5D potential solutions and a class of 5D spacetime solutions with magnetic field can be generalized, and show some examples.     

Time-dependent correlations for a one-component plasma in a uniform magnetic field

We derive various sum rules for the time-displaced structure function of a classical one-component plasma subjected to an external uniform magnetic field. When the plasma has some translational invariance (i.e., homogeneous or translation-invariant along the field), we find that there are long-wavelength oscillations with well-defined frequencies. The results are obtained from linear response and macroscopic electrodynamics, as well as from the microscopic equations of motion (BBGKY hierarchy). In the presence of the magnetic field, the time-displaced structure function has a polynomial decay at large distances, even in the homogeneous case. When the plasma has no translational invariance, examples show a more complicated temporal behaviour in the long-length-scale limit, involving a superposition of oscillations over a continuous range of frequencies.     

Feynman integral relations from parametric annihilators

We study shift relations between Feynman integrals via the Mellin transform through parametric annihilation operators. These contain the momentum space integration by parts relations, which are well known in the physics literature. Applying a result of Loeser and Sabbah, we conclude that the number of master integrals is computed by the Euler characteristic of the Lee–Pomeransky polynomial. We illustrate techniques to compute this Euler characteristic in various examples and compare it with numbers of master integrals obtained in previous works.

     

Polynomial deformations and cohomology of Calabi-Yau manifolds

We study the method of polynomial deformations that is used in the physics literature to determine the Hodge numbers of Calabi-Yau manifolds as well as the related Yukawa couplings. We show that the argument generally presented in the literature in support of these computations is seriously misleading, give a correct proof which applies to all the cases we found in the literature, and present examples which show that the method is not universally valid. We present a general analysis which applies to all Calabi-Yau manifolds embedded as complete intersections in products of complex projective spaces, yields sufficient conditions for the validity of the polynomial deformation method, and provides an alternative computation of all the Hodge numbers in many cases in which the polynomial method fails.     

Non-Weyl resonance asymptotics for quantum graphs in a magnetic field

We study asymptotical behaviour of resonances for a quantum graph consisting of a finite internal part and external leads placed into a magnetic field, in particular, the question whether their number follows the Weyl law. We prove that the presence of a magnetic field cannot change a non-Weyl asymptotics into a Weyl one and vice versa. On the other hand, we present examples demonstrating that for some non-Weyl graphs the “effective size” of the graph, and therefore the resonance asymptotics, can be affected by the magnetic field.     

Conformally Flat Spacetimes and Weyl Frames

We discuss the concepts of Weyl and Riemann frames in the context of metric theories of gravity and state the fact that they are completely equivalent as far as geodesic motion is concerned. We apply this result to conformally flat spacetimes and show that a new picture arises when a Riemannian spacetime is taken by means of geometrical gauge transformations into a Minkowskian flat spacetime. We find out that in the Weyl frame gravity is described by a scalar field. We give some examples of how conformally flat spacetime configurations look when viewed from the standpoint of a Weyl frame. We show that in the non-relativistic and weak field regime the Weyl scalar field may be identified with the Newtonian gravitational potential. We suggest an equation for the scalar field by varying the Einstein-Hilbert action restricted to the class of conformally-flat spacetimes. We revisit Einstein and Fokker’s interpretation of Nordstr?m scalar gravity theory and draw an analogy between this approach and the Weyl gauge formalism. We briefly take a look at two-dimensional gravity as viewed in the Weyl frame and address the question of quantizing a conformally flat spacetime by going to the Weyl frame.     

A motivic approach to phase transitions in Potts models

We describe an approach to the study of phase transitions in Potts models based on an estimate of the complexity of the locus of real zeros of the partition function, computed in terms of the classes in the Grothendieck ring of the affine algebraic varieties defined by the vanishing of the multivariate Tutte polynomial. We give completely explicit calculations for the examples of the chains of linked polygons and of the graphs obtained by replacing the polygons with their dual graphs. These are based on a deletion–contraction formula for the Grothendieck classes and on generating functions for splitting and doubling edges.     

Survival,Extinction and Approximation of Discrete-time Branching Random Walks

We consider a general discrete-time branching random walk on a countable set X. We relate local, strong local and global survival with suitable inequalities involving the first-moment matrix M of the process. In particular we prove that, while the local behavior is characterized by M, the global behavior cannot be completely described in terms of properties involving M alone. Moreover we show that locally surviving branching random walks can be approximated by sequences of spatially confined and stochastically dominated branching random walks which eventually survive locally if the (possibly finite) state space is large enough. An analogous result can be achieved by approximating a branching random walk by a sequence of multitype contact processes and allowing a sufficiently large number of particles per site. We compare these results with the ones obtained in the continuous-time case and we give some examples and counterexamples.     

On the semiclassical analysis of the ground state energy of the Dirichlet Pauli operator III: magnetic fields that change sign

We consider the semiclassical Dirichlet Pauli operator in bounded connected domains in the plane. Rather optimal results have been obtained in previous papers by Ekholm–Kova?ík–Portmann and Helffer–Sundqvist for the asymptotics of the ground state energy in the semiclassical limit when the magnetic field has constant sign. In this paper, we focus on the case when the magnetic field changes sign. We show, in particular, that the ground state energy of this Pauli operator will be exponentially small as the semiclassical parameter tends to zero and give lower bounds and upper bounds for this decay rate. Concrete examples of magnetic fields changing sign on the unit disk are discussed. Various natural conjectures are disproved, and this leaves the research of an optimal result in the general case still open.

     

On the non-integrability of the spherically-symmetric nonlinear Schrödinger equation in the Langmuir collapse problem

In the present paper we give some analytic considerations of the spherically-symmetric nonlinear Schrödinger equation arising in the Langmuir collapse problem. We will make a systematic exploration of the various group symmetries of thie equation and show that the latter possesses only a three-parameter symmetry group. We will then give a variational formulation of this equation and use the three-parameter symmetry group to show that the equation in question possesses apparently only two polynomial conservation laws. Finally, we will make a study of the singularity structure of the present equation and show that it does not seem to possess the Painlevé property. The conclusion is that the spherically-symmetric nonlinear Schrödinger equation in question is apparently not integrable.     

The Ponzano–Regge Model and Parametric Representation

We give a parametric representation of the effective noncommutative field theory derived from a ${\kappa}$ -deformation of the Ponzano–Regge model and define a generalized Kirchhoff polynomial with ${\kappa}$ -correction terms, obtained in a ${\kappa}$ -linear approximation. We then consider the corresponding graph hypersurfaces and the question of how the presence of the correction term affects their motivic nature. We look in particular at the tetrahedron graph, which is the basic case of relevance to quantum gravity. With the help of computer calculations, we verify that the number of points over finite fields of the corresponding hypersurface does not fit polynomials with integer coefficients, hence the hypersurface of the tetrahedron is not polynomially countable. This shows that the correction term can change significantly the motivic properties of the hypersurfaces, with respect to the classical case.     

Comments, with reply , on `An MHD model for the Earth's magneticfield with spatially dependent electrical conductivity' by I. Alexeffand J.R. Roth

It is shown that the MHD model for the Earth's magnetic field recently proposed by Alexeff and Roth (ibid., vol.PS-14, no.6 pp.862-4, Dec. 1986) does not work as a self-excited dynamo but has to be considered as an amplification mechanism for a field due to causes different from fluid motions. The authors reply that they agree with D. Radler's comments, and that they had explicitly stated the point in question in the latter part of their paper. They also give two additional points to consider     

EXACT SOLUTIONS OF CLASSICAL SCALAR FIELD EQUATIONS

We give a class of exact solutions of quartic scalar field theories. These solutions prove to be interesting as are characterized by the production of mass contributions arising from the nonlinear terms while maintaining a wave-like behavior. So, a quartic massless equation has a nonlinear wave solution with a dispersion relation of a massive wave and a quartic scalar theory gets its mass term renormalized in the dispersion relation through a term depending on the coupling and an integration constant. When spontaneous breaking of symmetry is considered, such wave-like solutions show how a mass term with the wrong sign and the nonlinearity give rise to a proper dispersion relation. These latter solutions do not change the sign maintaining the property of the selected value of the equilibrium state. Then, we use these solutions to obtain a quantum field theory for the case of a quartic massless field. We get the propagator from a first-order correction showing that is consistent in the limit of a very large coupling. The spectrum of a massless quartic scalar field theory is then provided. From this we can conclude that, for an infinite countable number of exact classical solutions, there exist an infinite number of equivalent quantum field theories that are trivial in the limit of the coupling going to infinity.     

Itinerant electron surface magnetism

Although many experiments have been applied to study the surface magnetism, only a few give a direct and unambiguous answer to a so simple question as “what is the value of the surface magnetic moment”. Some examples of all these experiments are reviewed to show the main differences between two- and three-dimensional magnetism. Then we show that the magnetic moment can be enhanced near the surface in itinerant electron magnetism. This may be the case of He₃ atoms in confined geometries, or of transition metal surfaces or more generally when the local surface density of states at the Fermi level is much larger than the bulk one.     

Two- and Four-Level Systems in Magnetic Fields Restricted in Time

We describe some new exact solutions for two- and four-level systems. In all the cases, external fields have a restricted behavior in time. First, we consider a method to construct new solutions for one-spin equation and give some explicit examples: One of them is in a external magnetic field that acts during a finite time interval. Then we show how these solutions can be used to solve the two-spin equation problem. A solution for two interacting spins is analyzed in the case when the field difference between the external fields in each spin varies adiabatically, vanishing on the time infinity. The latter system can be identified with a quantum gate realized by two coupled quantum dots. The probability of the Swap operation for such a gate can be explicitly expressed in terms of special functions. Using the obtained expressions, we construct plots for the Swap operation for some parameters of the external magnetic field and interaction function.     

Thermal entanglement in a two-spin-qutrit system under a nonuniform external magnetic field

The thermal entanglement in a two-spin-qutrit system with two spins coupled by exchange interaction under a magnetic field in an arbitrary direction is investigated. Negativity, the measurement of entanglement, is calculated. We find that for any temperature the evolvement of negativity is symmetric with respect to magnetic field. The behavior of negativity is presented for four different cases. The results show that for different temperature, different magnetic field give maximum entanglement. Both the parallel and antiparallel magnetic field cases are investigated qualitatively (not quantitatively) in detail, we find that the entanglement may be enhanced under an antiparallel magnetic field.     

Intrinsic pinning of vortices as a direct probe of the nonuniform Larkin-Ovchinnikov-Fulde-Ferrell state in layered superconductors

Previously the search for the modulated superconducting Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) state was performed by means of measurements which do not give direct information on spatial modulation of the superconducting state. We propose to measure interlayer conductivity in Josephson-coupled layered superconductors as a function of the strength and the orientation of the parallel magnetic field. We show that interlayer critical current and the conductivity have peaks when the magnetic field is perpendicular to the in-plane wave vector of the LOFF state and when the period of the Josephson vortex lattice induced by the magnetic field is commensurate with the LOFF period.     

Non-Abelian gauge potentials for ultracold atoms with degenerate dark states

We show that the adiabatic motion of ultracold, multilevel atoms in spatially varying laser fields can give rise to effective non-Abelian gauge fields if degenerate adiabatic eigenstates of the atom-laser interaction exist. A pair of such degenerate dark states emerges, e.g., if laser fields couple three internal states of an atom to a fourth common one under pairwise two-photon-resonance conditions. For this so-called tripod scheme we derive general conditions for truly non-Abelian gauge potentials and discuss special examples. In particular we show that using orthogonal laser beams with orbital angular momentum an effective magnetic field can be generated that has a monopole component.