Solving the Bethe–Salpeter Equation in Euclidean Space

Different approaches to solve the spinor–spinor Bethe–Salpeter (BS) equation in Euclidean space are considered. It is argued that the complete set of Dirac matrices is the most appropriate basis to define the partial amplitudes and to solve numerically the resulting system of equations with realistic interaction kernels. Other representations can be obtained by performing proper unitary transformations. A generalization of the iteration method for finding the energy spectrum of the BS equation is discussed and examples of concrete calculations are presented. Comparison of relativistic calculations with available experimental data and with corresponding non relativistic results together with an analysis of the role of Lorentz boost effects and relativistic corrections are presented. A novel method related to the use of hyperspherical harmonics is considered for a representation of the vertex functions suitable for numerical calculations.     

Solutions of the Bethe–Salpeter Equation in Minkowski Space and Applications to Electromagnetic Form Factors

We present a new method for solving the two-body Bethe–Salpeter equation in Minkowski space. It is based on the Nakanishi integral representation of the Bethe–Salpeter amplitude and on subsequent projection of the equation on the light-front plane. The method is valid for any kernel given by the irreducible Feynman graphs and for systems of spinless particles or fermions. The Bethe–Salpeter amplitudes in Minkowski space are obtained. The electromagnetic form factors are computed and compared to the Euclidean results.     

From Ecology to Finance (and Back?): A Review on Entropy-Based Null Models for the Analysis of Bipartite Networks

Journal of Statistical Physics - Bipartite networks provide an insightful representation of many systems, ranging from mutualistic networks of species interactions to investment networks in...     

A Search on the Klein-Gordon Equation

The s-wave Klein-Gordon equation for the bound states is separated in two parts to see clearly the relativistic contributions to the solution in the non-relativistic limit. The reliability of the model is discussed with two examples chosen specifically.     

Review of AdS/CFT Integrability, Chapter III.1: Bethe Ans?tze and the R-Matrix Formalism

The one-dimensional Heisenberg XXX spin chain appears in a special limit of the AdS/CFT integrable system. We review various ways of proving its integrability, and discuss the associated methods of solution. In particular, we outline the coordinate and the algebraic Bethe ansatz, giving reference to literature suitable for learning these techniques. Finally, we speculate which of the methods might lift to the exact solution of the AdS/CFT system, and sketch a promising method for constructing the Baxter Q-operator of the XXX chain. It allows to find the spectrum of the model using certain algebraic techniques, while entirely avoiding Bethe’s ansatz.     

A Harmonic Solution for the Hyperbolic Heat Conduction Equation and Its Relationship to the Guyer–Krumhansl Equation

A particular solution of the hyperbolic heat-conduction equation was constructed using the method of operators. The evolution of a harmonic solution is studied, which simulates the propagation of electric signals in long wire transmission lines. The structures of the solutions of the telegraph equation and of the Guyer–Krumhansl equation are compared. The influence of the phonon heat-transfer mechanism in the environment is considered from the point of view of heat conductivity. The fulfillment of the maximum principle for the obtained solutions is considered. The frequency dependences of heat conductivity in the telegraph equation and in an equation of the Guyer–Krumhansl type are studied and compared with each other. The influence of the Knudsen number on heat conductivity in the model of thin films is studied.     

On the Riemann�CHilbert�CBirkhoff Inverse Monodromy Problem Associated with the Third Painlev�� Equation

A linear system of ordinary differential equations corresponding by the isomonodromy deformation method to the third Painlevé equation is considered. The surjectivity of the monodromy map generated by this system is proven using the Riemann?CHilbert factorization method.     

Braces and the Yang–Baxter Equation

Several aspects of relations between braces and non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation are discussed and many consequences are derived. In particular, for each positive integer n a finite square-free multipermutation solution of the Yang–Baxter equation with multipermutation level n and an abelian involutive Yang–Baxter group is constructed. This answers a problem of Gateva-Ivanova and Cameron. It is proved that finite non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation whose associated involutive Yang–Baxter group is abelian are multipermutation solutions. Earlier the authors proved this with the additional square-free hypothesis on the solutions. It is also proved that finite square-free non-degenerate involutive set-theoretic solutions associated to a left brace are multipermutation solutions.     

Exact Bound Solution of the Klein-Gordon Equation and the Dirac Equation with Rosen-MorseⅡ Potential

In this paper, the relativistic Rosen-Morse II potential is investigated by solving the Klein-Gordon and the Dirac equations with equal attractive scalar s(r) and repulsive vector v(r) potentials. The exact energy equations of the bound state are obtained by the method of supersymmetric and shape invariance. Finally, a kind of special potential about Rosen-MorseⅡ potential is discussed.     

Thermodynamic Gravity and the Schrödinger Equation

We adopt a formulation of the Mach principle that the rest mass of a particle is a measure of it’s long-range collective interactions with all other particles inside the horizon. As a consequence, all particles in the universe form a ‘gravitationally entangled’ statistical ensemble and one can apply the approach of classical statistical mechanics to it. It is shown that both the Schrödinger equation and the Planck constant can be derived within this Machian model of the universe. The appearance of probabilities, complex wave functions, and quantization conditions is related to the discreetness and finiteness of the Machian ensemble.     

A Dirac–Dunkl Equation on S2 and the Bannai–Ito Algebra

The Dirac–Dunkl operator on the two-sphere associated to the \({{\mathbb{Z}_{2}^{3}}}\) reflection group is considered. Its symmetries are found and are shown to generate the Bannai–Ito algebra. Representations of the Bannai–Ito algebra are constructed using ladder operators. Eigenfunctions of the spherical Dirac–Dunkl operator are obtained using a Cauchy–Kovalevskaia extension theorem. These eigenfunctions, which correspond to Dunkl monogenics, are seen to support finite-dimensional irreducible representations of the Bannai–Ito algebra.     

A Note on Symmetries and Generalized W∞ Algebra of the Modified KP Equation

Some remarks on the paper Symmetries and generalized W algebra of the modified KP equation (S. Y. Lou and G. J. Ni, Lett. Math. Phys. 34 (1995), 327–331) are given. It is pointed out that if we consider the inverse operator of a differential operator to be a linear operator, the vector fields nv(f) defined in the above Letter are not certain to be symmetries of the modified KP equation under consideration.     

A Super mKdV Equation: Bosonization,Painlevé Property and Exact Solutions

The symmetry of the fermionic field is obtained by means of the Lax pair of the mKdV equation. A new super mKdV equation is constructed by virtue of the symmetry of the fermionic form. The super mKdV system is changed to a system of coupled bosonic equations with the bosonization approach. The bosonized SmKdV(BSmKdV)equation admits Painlevé property by the standard singularity analysis. The traveling wave solutions of the BSmKdV system are presented by the mapping and deformation method. We also provide other ideas to construct new super integrable systems.