Solving the Dyson–Schwinger Equation at Zero and Finite Temperatures

The properties of the solutions of the truncated Dyson–Schwinger equation for the quark propagator at finite temperatures within the rainbow-ladder approximation are analysed in some detail.     

Solutions of the Classical Yang–Baxter Equation and Noncommutative Deformations

We show that given a finite-dimensional real Lie algebra acting on a smooth manifold P then, for any solution of the classical Yang–Baxter equation on , there is a canonical Poisson tensor on P and an associated canonical torsion-free and flat contravariant connection. Moreover, we prove that the metacurvature of this contravariant connection vanishes if the isotropy Lie subalgebras of the action are trivial. Those results permit to get a large class of smooth manifolds satisfying the necessary conditions, introduced by Eli Hawkins, to the existence of noncommutative deformations. Recherche menée dans le cadre du Programme Thématique d’Appui à la Recherche Scientifique PROTARS III.     

On the Klein–Gordon Equation in Higher Dimensions: Are Particle Masses Variable?

We consider ways to generalize the 4D Klein–Gordon equation of particle physics to higher dimensions. The most promising approach implies that the mass which appears in the 4D relation is a term in the source-free 5D relation. We check this explicitly for the case of exact solitonic and cosmological solutions of the Kaluza–Klein equations. In general, particle masses are variable; but are constant for the Schwarzschild and late-universe cases, in agreement with data from the solar system and astrophysics. Our results have significant implications for cosmology, and can easily be extended to 10D superstrings, 11D supergravity and higher dimensions.     

Integration of the Hamilton–Jacobi and Maxwell Equations in Diagonal Metrics of Stäckel Spaces

Russian Physics Journal - A classification of metrics and electromagnetic potentials is carried out for the case when the Hamilton–Jacobi equation admits a complete separation of variables in...     

Hamilton–Jacobi equation,reaction action surface and the emergence of the force concept in chemical reaction dynamics

A combination of Hamilton–Jacobi equation and fast marching algorithm can be used to study reaction dynamics by converting the potential energy surface to a reaction action surface. The reaction action surface has been found to be an important tool in theoretical chemistry, allowing us to provide a different force-based perspective of chemical reactions. Several properties such as reaction force, reaction force surface, reaction path force and reaction path force constant have been defined and calculated by using the reaction action surface. This paper investigates these newly defined properties in order to understand the role they play in chemical reaction with reference to a model 4-well potential energy surface.     

Null geodesics and wave front singularities in the Gödel space–time

We explore wave fronts of null geodesics in the Gödel metric emitted from point sources both at, and away from, the origin. For constant time wave fronts emitted by sources away from the origin, we find cusp ridges as well as blue sky metamorphoses where spatially disconnected portions of the wave front appear, connect to the main wave front, and then later break free and vanish. These blue sky metamorphoses in the constant time wave fronts highlight the non-causal features of the Gödel metric. We introduce a concept of physical distance along the null geodesics, and show that for wave fronts of constant physical distance, the reorganization of the points making up the wave front leads to the removal of cusp ridges.     

Phase Turbulence in the Complex Ginzburg-Landau Equation via Kuramoto–Sivashinsky Phase Dynamics

We study the Complex Ginzburg-Landau initial value problem for a complex field uC, with ,R. We consider the Benjamin–Feir linear instability region We show that for all and for all initial data u₀ sufficiently close to 1 (up to a global phase factor eⁱ₀,₀R) in the appropriate space, there exists a unique (spatially) periodic solution of space period L₀. These solutions are small even perturbations of the traveling wave solution, and s, have bounded norms in various L^p and Sobolev spaces. We prove that apart from corrections whenever the initial data satisfy this condition, and that in the linear instability range the dynamics is essentially determined by the motion of the phase alone, and so exhibits phase turbulence. Indeed, we prove that the phase satisfies the Kuramoto–Sivashinsky equation for times while the amplitude 1+² s is essentially constant.Supported in part by the Fonds National Suisse.     

Klein-Gordon and Weyl equations in the Gödel universe

Klein-Gordon and Weyl equations are investigated in the Gödel universe as the background metric. Both equations are solved, the Klein-Gordon equation for arbitrary coupling to the gravitational field.     

Hamilton–Jacobi Formulation for Systems in Terms of?Riesz’s Fractional Derivatives

The paper presents fractional Hamilton–Jacobi formulations for systems containing Riesz fractional derivatives (RFD’s). The Hamilton–Jacobi equations of motion are obtained. An illustrative example for simple harmonic oscillator (SHO) has been discussed. It was observed that the classical results are recovered for integer order derivatives.     

How to Find Separation Coordinates for the Hamilton–Jacobi Equation: A Criterion of Separability for Natural Hamiltonian Systems

The method of separation of variables applied to the natural Hamilton–Jacobi equation (u/q _i )²+V(q)=E consists of finding new curvilinear coordinates x _i (q) in which the transformed equation admits a complete separated solution u(x)=u _(i)(x _i ;). For a potential V(q) given in Cartesian coordinates, the main difficulty is to decide if such a transformation x(q) exists and to determine it explicitly. Surprisingly, this nonlinear problem has a complete algorithmic solution, which we present here. It is based on recursive use of the Bertrand–Darboux equations, which are linear second order partial differential equations with undetermined coefficients. The result applies to the Helmholtz (stationary Schrödinger) equation as well.     

Born–Infeld Action in (n+2)-Dimension,the Field Equation and a Soliton Solution

In this Letter we introduce the (n+2)-dimensional Born–Infeld action with a dual field strength . We compute the field equation by using Schur polynomials and give a soliton solution.     

Spin 3/2 Field Equation: Separation and Solution in a Class of Lema?tre–Tolman–Bondi Cosmologies

The spin 3/2 field equation is studied in the general Lema?tre–Tolman–Bondi (LTB) space-time. The equation is separated by variable separation. The angular dependence factors out at the level of the general LTB metric. Due to spherical symmetry the separated angular equations coincide with those, previously integrated, relative to the Robertson–Walker and Schwarzschild metric. Separation of time and radial dependence is possible within a class of LTB cosmological models for which the physical radius is a product of a time and a radial function, the last one being further selected by the consistency condition of the radial equations. The separated time dependence, that can be integrated by series, results essentially unique. Instead the radial dependence can be reduced to two independent second order ordinary differential equations that still depend on an arbitrary radial function that is an integration function of the cosmological model. The generalization of the scheme to arbitrary spin field equation is suggested.     

Determination of the Spin–Spin Coupling Constant in Hydrogen Deuteride HD and Estimates of the Manifestation of Axion-Like Particles

An experimental value of the spin–spin coupling constant in deuterated molecular hydrogen HD has been obtained, J _pd = (43.112 ± 0.005) Hz (300 K), while investigating two gaseous samples at pressures of 95 and 155 atm. The experimental result does not coincide with Jpd = (43.31 ± 0.05) Hz that was calculated theoretically by Helkager et al. The observed discrepancy ΔJ _pd ≈ (0.20 ± 0.05 Hz) may point to a manifestation of the involvement of light pseudo-scalar (axion-like) bosons with a mass m _a ≈ 1 keV/c² in the spin–spin coupling of the HD proton and deuteron.     

Decay of the Bethe–Salpeter Kernel and Bound States for Lattice Classical Ferromagnetic Spin Systems at High Temperature

We consider lattice classical ferromagnetic spin systems at high temperature (1) with nearest neighbor interactions and even single-spin distributions (ssd). Associated with each system is an imaginary time lattice quantum field theory. It is known that there is a particle of mass m–ln in the energy-momentum spectrum. If s ⁴–3s ²²<0, where s ^k is the kth moment of the ssd, and is sufficiently small, we show that in the two-particle subspace there is no mass spectrum up to 2m. For >0 we show that the only mass spectrum in (m, 2m) is a bound state of mass m _b=2m+ln(1–)+O(), where =(+2s ²²)^–1. A bound on the decay of the kernel of a Bethe–Salpeter equation is obtained and used to prove these results.     

The Decay of Massive Scalar Field in Non-Static Gödel Type Universe with Viscous Fluid and Heat Flow

In this study, we have investigated the dynamics of non-static Gödel type rotating universe with massive scalar field, viscous fluid and heat flow in the presence of cosmological constant. For various cosmic matter forms, the behavior of the cosmological constant (Λ), shear (η) and bulk (ξ) viscosity coefficients and other kinematic quantities have studied in the early universe. We have showed the decay of massive scalar field in the non-static rotating Gödel type universe and we have obtained constant scalar field with and without source density. Also, we have investigated the effects of massive scalar field on the matter density and pressure. From solutions of the field equations, we have a cosmological model with non-zero expansion, shear, heat flux and rotation. Also some physical and geometrical aspects of the model discussed.     

Erratum: Mayer Expansions and the Hamiltonian–Jacobi Equation. II. Fermions,Dimensional Reduction Formulas

The authors apologize to readers for an egregious error in the proof of Theorem 2.3: contrary to our claim, (2.12) does not follow from Proposition 2.2 and Theorem 2.3 is almost certainly false. We are indebted to Manfred Salmhofer and Christian Wieczerkowski for discovering the error. They are preparing a paper with a corrected theorem proven by different methods. The work in Section 3 is independent of Section 2 and Theorem 3.1 is unaffected.