On the Feynman Path Integral for the Dirac Equation in the General Dimensional Spacetime

The Feynman path integral for the Dirac equation in the general dimensional spacetime is determined mathematically in the form of the sum-over-histories satisfying the superposition principle, i.e., the “sum” of the probability amplitudes with a common weight over all possible paths that go in any direction at any speed forward and backward in time. It is noted that our Feynman path integral is determined not in configuration space, but in phase space.     

The (2+1) Dirac Equation with a Delta Potential

In this Letter the bound states of (2+1) Dirac equation with the cylindrically symmetric (r–r ₀) potential are discussed. It is surprisingly found that the relation between the radial functions at two sides of r ₀ can be established by an SO(2) transformation. We obtain a transcendental equation for calculating the energy of the bound state from the matching condition in the configuration space. The condition for existence of bound states is determined by the Sturm-Liouville theorem.     

2+1 Dimensional Noncommutative Dirac Oscillator and (Anti)-Jaynes-Cummings Models

We study Dirac oscillator in 2+1 dimensional noncommutative space. The model is solved exactly and the relationship with Jaynes-Cummings (JC) or anti-Jaynes-Cummings (AJC) models are investigated. We find that for a positive noncommutative parameter, there is an exact map from the 2+1 dimensional noncommutative Dirac oscillator to AJC model. However, for a negative noncommutative parameter, the noncommutative planar Dirac oscillator contains both AJC and JC terms simultaneously. Our investigation may afford a new way to study relativistic quantum mechanics models in noncommutative space by means of quantum optics method, and vice verse.     

The Darboux Transformation for the One-Dimensional Stationary Dirac Equation with a Pseudoscalar Potential

This paper consider the Darboux transform for the Dirac equation with a pseudoscalar-type potential. Formulas for the potential difference and for the solutions of the transformed equation are derived. The relationship between the Darboux transforms for Dirac and Schrödinger equations is analyzed. New potentials with the spectrum of a relativistic harmonic oscillator are obtained as examples.     

Path Integral Discussion for Smorodinsky-Winternitz Potentials: I. Two- and Three Dimensional Euclidean Space

Path integral formulations for the Smorodinsky-Winternitz potentials in two- and three-dimensional Euclidean space are presented. We mention all coordinate systems which separate the Smorodinsky-Winternitz potentials and state the corresponding path integral formulations. Whereas in many coordinate systems an explicit path integral formulation is not possible, we list in all soluble cases the path integral evaluations explicitly in terms of the propagators and the spectral expansions into the wave-functions.     

Dirac Equation and Ground State of Solvable Potentials: Supersymmetry Method

The supersymmetry in non-relativistic quantum mechanics is applied as an algebraic method to obtain the solutions of the Dirac equation with spherical symmetry electromagnetic potentials. We show that some of the superpotentials related to ground state of the solvable potentials in non-relativistic quantum mechanics can be used for studying of the Dirac equation.     

Invariance Analysis of the (2+1) Dimensional Long Dispersive Wave Equation

Abstract

In this paper, we bring out the Lie symmetries and associated similarity reductions of the recently proposed (2+1) dimensional long dispersive wave equation. We point out that the integrable system admits an infinite-dimensional symmetry algebra along with Kac-Moody-Virasoro-type subalgebras. We also bring out certain physically interesting solutions.     

Path Integral Solution for a Dirac Particle in Non-Abelian SU(N) Gauge Field

The propagator of a Dirac particle in interaction with a non-Abelian SU(N) gauge field is determined according to the path integral formalism of Alexandrou et al. by using the representation so called “local projection” and the wave functions are extracted. Furthermore, it is shown that certain selected equations obtained during the integrations can also be classically derived.     

Exact Solutions of Dirac Equation and Particle Creation in (1+3)-Dimensional Robertson-Walker Spacetime

The exact solutions of the Dirac equation are discussed for a Robertson-Walker spacetime with asymptotically Minkowskian in and out regions. We obtain the mode solutions which reduce to positive and negative Minkowskian spinors in asymptotically regions. Using the obtained solutions we compute the density of created particles.     

Solution of Spin and Pseudo-Spin Symmetric Dirac Equation in (1+1) Space-Time Using Tridiagonal Representation Approach

The aim of this work is to find exact solutions of the Dirac equation in(1+1) space-time beyond the already known class.We consider exact spin(and pseudo-spin) symmetric Dirac equations where the scalar potential is equal to plus(and minus) the vector potential.We also include pseudo-scalar potentials in the interaction.The spinor wavefunction is written as a bounded sum in a complete set of square integrable basis,which is chosen such that the matrix representation of the Dirac wave operator is tridiagonal and symmetric.This makes the matrix wave equation a symmetric three-term recursion relation for the expansion coefficients of the wavefunction.We solve the recursion relation exactly in terms of orthogonal polynomials and obtain the state functions and corresponding relativistic energy spectrum and phase shift.     

Exact Solutions of Dirac Equation on (1+1)-Dimensional Spacetime Coupled to a Static Scalar Field

We use a generalized scheme of supersymmetric quantum mechanics to obtain the energy spectrum and wave function for Dirac equation in (1+1)-dimensional spacetime coupled to a static scalar field.     

Dirac equation in (1+1)-dimensional curved space-time

The Dirac equation in (1+1)-dimensional curved space-time is solved explicitly for the spatially flat Robertson-Walker space-time and the cigar metric considered by Witten.     

On Peakon and Kink-peakon Solutions to a (2 + 1) Dimensional Generalized Camassa-Holm Equation

In this paper, we study a (2 + 1)-dimensional generalized Camassa-Holm (2dgCH) equation with both quadratic and cubic nonlinearity. We derive a peaked soliton (peakon) solution, double-peakon solutions, and kink-peakon solutions. In particular, weak kink - peakon solution is the first time to address in the 2 + 1-dimensional integrable system.     

Exact Periodic-Wave Solutions for (2+1)-Dimensional Boussinesq Equation and (3+1)-Dimensional KP Equation

The (2 1)-dimensional Boussinesq equation and (3 1)-dimensional KP equation are studied by using the extended Jacobi elliptic-function method. The exact periodic-wave solutions for the two equations are obtained.     

Exact Periodic-Wave Solutions for （2＋1）-Dimensional Boussinesq Equation and（3＋1）-Dimensional KP Equation

The (2 1)-dimensional Boussinesq equation and (3 1)-dimensional KP equation are studied by using the extended Jacobi elliptic-function method. The exact periodic-wave solutions for the two equations are obtained.     

1+1维含噪声Kuramoto-Sivashinsky方程表面宽度分布率的数值计算

通过对1+1维含噪声Kuramoto-Sivashinsky(KS)方程进行数值计算,得到其在饱和状态下的表面宽度分布率并与Kardar-Parisi-Zhang(KPZ)方程进行比较.结果表明,1+1维含噪声KS方程的表面宽度分布率标度函数受有限尺寸效应影响较小,并与KPZ方程具有相近的表面宽度分布率标度函数.     

Most General Form of the Lagrange Function for Classical Two Particle Equation of Motion Covariant under Galilei Transformation in (1 + 1) and (3 + 1) Dimensional Spaces

In the present note we give the most general form of a non-relativistic Lagrange function for two point particles moving according to some classical equations of motion. We do not specify the form of these equations. We require only that these equations should be form invariant with respect to the Galilei transformations and should be derivable from a Lagrange function as its Euler-Lagrange Equations. We present also some of the generators of the Galilei group and their Lie-Cartan commutation relations.