On the state space of the dipole ghost

A particular representation of SO(4, 2) is identified with the state space of the free dipole ghost. This representation is then given an explicit realization as the solution space of a 4th-order wave equation on a spacetime locally isomorphic to Minkowski space. A discrete basis for this solution space is given, as well as an explicit expression for its SO(4, 2) invariant inner product. The connection between the modes of dipole field and those of the massless scalar field is clarified, and a recent conjecture concerning the restriction of the dipole representation to the Poincaré subgroup is confirmed. A particular coordinate transformation then reveals the theory of the dipole ghost in Minkowski space. Finally, it is shown that the solution space of the dipole equation is not unitarizable in a Poincaré invariant manner.     

原子-辐射场相互作用系统量子统计力学的相干态表述

用Ｈｅｉｓｅｎｂｅｒｇ－ｗｅｙｌｅ（简称Ｈ－Ｗ）群直积ＳＵ（２）群上的相干态表述了原子－辐射场相互作用系统的量子统计力学,引入一个新定义的分布函数,将平衡态下密度矩阵的Ｂｌｏｃｈ方程转换成可分离变量的偏微分方程,给出了方程的形式解,算符和其平均值的表示。     

Degenerate Solutions of the Nonlinear Self-Dual Network Equation

The N-fold Darboux transformation(DT) T_n~([N]) of the nonlinear self-dual network equation is given in terms of the determinant representation. The elements in determinants are composed of the eigenvalues λ_j(j = 1, 2..., N)and the corresponding eigenfunctions of the associated Lax equation. Using this representation, the N-soliton solutions of the nonlinear self-dual network equation are given from the zero "seed" solution by the N-fold DT. A general form of the N-degenerate soliton is constructed from the determinants of N-soliton by a special limit λ_j →λ_1 and by using the higher-order Taylor expansion. For 2-degenerate and 3-degenerate solitons, approximate orbits are given analytically,which provide excellent fit of exact trajectories. These orbits have a time-dependent "phase shift", namely ln(t~2).     

Representation of solutions of the Burgers equation using functional integrals

In the paper, a representation of a solution of the Burgers equation in ℝ ⁿ is obtained by using integrals with respect to the Wiener measure on the space of trajectories in ℝ ⁿ . The Burgers equation is considered in a rigged Hilbert space. It is proved that, in the infinite-dimensional case, there is an analog of the Cole-Hopf transformation relating the Burgers equation and an analog of the heat equation with respect to measures. The Feynman-Kac formula for the heat equation (with potential) with respect to measures in a rigged Hilbert space is obtained.     

原子-辐射场相互作用系统 Heisenberg运动方程的相干态表述

用Heisenberg-Weyle(简称H-W)群直积SU(2)群上的相干态表述了原 子-辐射场相互作用系统的Heisenberg运动方程。引入一个算符振幅函数,将Heisenberg绘 景中算符运动方程转化成可分离变量的偏微分方程,给出了方程的形式解以及时间演化算符 在该表述下的表示。     

A propagator theory applied to wave mechanics in phase space

The fact that the classical Liouville equation can be analyzed as a dynamical equation in Hilbert-Koopman (HK.) space is used in order to develop a perturbative method for the wave mechanics in phase space: an explicit solution of the Liouville equation inqp representation is exhibited. The connection between the solution obtained and the dynamics of correlations is established by computing theqp-kp transformation function in HK space. To elucidate the method, an application is presented and the result compared to that available in the literature.     

Analytic solution of the wave equation for an electron in the field of a molecule with an electric dipole moment

We relax the usual diagonal constraint on the matrix representation of the eigenvalue wave equation by allowing it to be tridiagonal. This results in a larger representation space that incorporates an analytic solution for the non-central electric dipole potential cosθ/r², which was believed not to belong to the class of exactly solvable potentials. Therefore, we were able to obtain a closed form solution of the three-dimensional time-independent Schrödinger equation for a charged particle in the field of a point electric dipole that could carry a nonzero net charge. This problem models the interaction of an electron with a molecule (neutral or ionized) that has a permanent electric dipole moment. The solution is written as a series in a basis composed of special functions that support a tridiagonal matrix representation for the angular and radial components of the wave operator. Moreover, this solution is for all energies, the discrete (for bound states) as well as the continuous (for scattering states). The expansion coefficients of the radial and angular components of the wavefunction are written in terms of orthogonal polynomials satisfying three-term recursion relations. For the Coulomb-free case, where the molecule is neutral, we calculate critical values for its dipole moment below which no electron capture is allowed. These critical values are obtained not only for the ground state, where it agrees with already known results, but also for excited states as well.     

The effect of the heat capacity of the liquid phase on the heat of fusion liquidus equation of compound semiconductors

We have critically examined the assumptions involved in the derivation of Vieland's widely used heat of fusion liquidus equation for binary compounds and conclude that the thermodynamic form of this equation ignores the relative partial molar heat capacity of the liquid solution. Taking into account this quantity, we obtain the generalized heat of fusion equation which is exact and show its complete equivalence to its alternative, the heat of formation equation. The generalized result provides a correction term to Vieland's equation which can be expressed as a function of the activity coefficients at the compound composition. Applying the correction term to the activity coefficients derived for a number of useful solution models, we find that the regular solution form of Vieland's equation is exact, as shown previously, if α (interchange energy) is a constant or a linear function of temperature. But when α is expanded as an nth order polynomial in temperature (simple solution), Vieland's equation is inexact for n ? 2. In addition, it is demonstrated that for a regular associated solution and for Darken's quadratic representation, Vieland's thermodynamic equation is exact only with certain restrictions, while for a quasi-chemical solution it is invalid.     

Anisotropic scattering in inhomogeneous media—1. A new representation formula for the solution of the auxiliary integral equation for the source function

A new representation formula for the solution of the auxiliary integral equation for the source function in inhomogeneous, anisotropically scattering media is presented. It involves two new functions Φ and ψ of two variables instead of the original five variables. This generalizes earlier results of Kagiwada et al. (1969) and Sobolev (1972) applicable to homogeneous atmospheres. The corresponding Bellman-Krein formula for the resolvent kernel is also derived. The present representation for the solution of Fredholm integral equations of second kind with unsymmetric kernels provides a new approach to radiative transfer in anisotropic inhomogeneous media.     

Fractional Fokker-Planck Equation with Space and Time Dependent Drift and Diffusion

In this paper, we aim to answer the question proposed by Magdziarz (Stoch. Proc. Appl. 119:3238?C3252, 2009), i.e. we investigate the solution of an anomalous diffusion equation with time and space dependent force and diffusion coefficient. First, we try to find the stochastic representation of this equation, which means the PDF of this stochastic process is rightly the solution of the equation we aim to solve. Then, we also simulate the sample paths of the stochastic process. At last, taking advantage of the stochastic representation method, we employed Monte Carlo method to approximate the solution of the mentioned equation.     

Electron in the field of a molecule with an electric dipole moment

In solving the eigenvalue wave equation, we relax the usual diagonal constraint on its matrix representation by allowing it to be tridiagonal. This results in a larger representation space that incorporates an analytic solution for the noncentral electric dipole pole potential cos theta/r2, which was believed not to belong to the class of exactly solvable potentials. Consequently, we obtain closed form solution of the time-independent Schr?dinger equation for an electron in the field of a molecule treated as a point electric dipole.     

The Chern-Simons theory and knot polynomials

The Chern-Simons gauge theory is studied using a functional integral quantization. This leads to a differential equation for expectations of Wilson lines. The solution of this differential equation is shown to be simply related to the two-variable Jones polynomial of the corresponding link, in the case where the gauge group isSU(N). A similar equation has been used before to get the Jones polynomial from a braid representation of the link. The main novelty of our approach is that we get the Jones polynomial from a plat representation of the link.     

Statistical bootstrap of fireball decay spectra

We propose a generalized statistical bootstrap equation for the generating functional of the fireball decay spectra which includes the bootstrap of the hadronic mass spectrum. In the form of an integral representation a solution is given for some n-particle distributions as well as multiplicity moments in the case of identical particles. Within this formalism we are able to discuss decay-chain end effects and to treat quantum number conservation explicity. The general equation is approximated by a simpler bootstrap equation for the linear decay chain with quantum-number conservation. An asymptotic solution for the single particle distributions according to this equation is discussed.     

A truncated Painlevé expansion associated with the Tzitzéica equation: consistency and general solution

It is shown that the overdetermined system obtained by truncating a Painlevé expansion for solutions of the classical Tzitzéica equation may be reduced to a compatible Frobenius system. Its general solution is expressed in terms of solutions of the standard linear representation of the Tzitzéica equation.     

Construction of effective operators for the time-dependent Schrödinger equation

The contact transformation method is used to solve the time-dependent Schrödinger equation describing the behavior of a multilevel system in a resonance electromagnetic field. A solution not containing secular terms is obtained for the evolution operator in the interaction representation U₀. The derivation of the equation for the probability of the simultaneous absorption of two photons with the dipole interaction is presented as an example of the effectiveness of the obtained solution.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 78–81, August, 1982.     

Quantum mechanical masterequation and Fokker-Planck equation for the damped harmonic oscillator

For the statistical operator of the damped harmonic oscillator a Masterequation is given in operator form describing both inelastic and elastic, purely phase destroying processes. By expressing the statistical operator in the diagonal representation with respect toGlauber's coherent states the Masterequation is transformed into a Fokker-Planck equation forGlauber's quasiprobability distribution function. The general solution of this Fokker-Planck equation is calculated. It is shown how the solution of a Masterequation can be used for calculating correlation functions and expressions are given for the amplitude and intensity correlation functions which are in complete formal agreement with the corresponding classical formulae.     

Effective theory of NN interactions in a separable representation

We consider the effective field theory of the NN system in a separable representation. The pionic part of the effective potential is included nonperturbatively and approximated by a separable potential. The use of a separable representation allows for the explicit solution of the Lippmann-Schwinger equation and a consistent renormalization procedure. The phase shifts in the ¹ S ₀ channel are calculated to subleading order.