Shape-invariant hypergeometric type operators with application to quantum mechanics

A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. The associated special functions are eigenfunctions of some shape-invariant operators. These operators can be analysed together and the mathematical formalism we use can be extended in order to define other shape-invariant operators. All the shape-invariant operators considered are directly related to Schrödinger-type equations.     

Operator product expansion in two-dimensional quantum gravity

We consider correlation functions of operators and the operator product expansion in two-dimensional quantum gravity. First we introduce correlation functions with geodesic distances between operators kept fixed. Next, by making two of the operators closer, we examine whether there exists an analog of the operator product expansion in ordinary field theories. Our results suggest that the operator product expansion holds in quantum gravity as well, though special care should be taken regarding the physical meaning of fixing geodesic distances on a fluctuating geometry.     

Multi-variable special polynomials using an operator ordering method

Using an operator ordering method for some commutative superposition operators, we introduce two new multi-variable special polynomials and their generating functions, and present some new operator identities and integral formulas involving the two special polynomials. Instead of calculating complicated partial differential, we use the special polynomials and their generating functions to concisely address the normalization, photocount distributions and Wigner distributions of several quantum states that can be realized physically, the results of which provide real convenience for further investigating the properties and applications of these states.     

光场位相算符和逆算符的Weyl编序展开

用有序算符内的积分技术, 推导了光场位相算符和逆算符的Weyl编序展开形式, 并利用该结果获得了相算符的经典对应以及某些新的特殊函数的生成函数和新的积分公式, 尤其是导出了带负次幂的复高斯积分的积分公式. 关键词： Weyl编序 位相算符 有序算符内的积分     

Source Identities and Kernel Functions for Deformed (Quantum) Ruijsenaars Models

We consider the relativistic generalization of the quantum A _N-1 Calogero–Sutherland models due to Ruijsenaars, comprising the rational, hyperbolic, trigonometric and elliptic cases. For each of these cases, we find an exact common eigenfunction for a generalization of Ruijsenaars analytic difference operators that gives, as special cases, many different kernel functions; in particular, we find kernel functions for Chalykh–Feigin–Veselov–Sergeev-type deformations of such difference operators which generalize known kernel functions for the Ruijsenaars models. We also discuss possible applications of our results.     

Off-diagonal quark distributions in pions in the effective single-instanton approximation

Nonperturbative functions that parametrize off-diagonal hadronic matrix elements of the light-cone leading-twist quark operators are considered. These functions are calculated within the proposed relativistic quark model allowing for the nontrivial structure of the QCD vacuum, special attention being given to gauge invariance. Hadrons are treated as bound states of quarks; strong-interaction quark-pion vertices are described by effective interaction Lagrangians generated by instantons. The parameters of the instanton vacuum, such as the effective radius of the instanton and the quark mass, are related to the vacuum expectation values of the quark-gluon operators of the lowest dimension and to low-energy pion observables.     

利用特殊函数和类比法有序化排列正负指数幂算符

研究算符函数的有序化排列是一项重要的数理任务.本文利用特殊函数和正规乘积排序与反正规乘积排序间的互换法则法导出了幂算符(αα^+)^±n和(α^+α)^±n的正规与反正规乘积排序.进一步,利用类比法得到了算符(XP)^±n和(PX)^±n的坐标-动量排序与动量-坐标排序式.最后,对新得到的这些算符结果的应用进行一些讨论.     

Autonomous models solvable through the full interval method

The most general exclusion single species one dimensional reaction-diffusion models with nearest-neighbor interactions which are both autonomous and can be solved exactly through full interval method are introduced. Using a generating function method, the general solution for F _n , the probability that n consecutive sites be full, is obtained. Some other correlation functions of number operators at nonadjacent sites are also explicitly obtained. It is shown that for a special choice of initial conditions some correlation functions of number operators called full intervals remain uncorrelated.     

Systems of orthogonal polynomials defined by hypergeometric type equations with application to quantum mechanics

A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. We present in a unified and explicit way all these systems of orthogonal polynomials, the associated special functions and the corresponding raising/lowering operators. The equations considered are directly related to some Schrödinger type equations (Pöschl-Teller, Scarf, Morse, etc), and the special functions defined are related to the corresponding bound-state eigenfunctions.     

Generalization of Drinfeld Quantum Affine Algebras

Drinfeld gave a current realization of the quantum affine algebras as a Hopf algebra with a simple comultiplication for the quantum current operators. In this Letter, we will present a generalization of such a realization of quantum Hopf algebras. As a special case, we will choose the structure functions for this algebra to be elliptic functions to derive certain elliptic quantum groups as a Hopf algebra, which degenerates into quantum affine algebras if we take certain degeneration of the structure functions.     

Disk level S-matrix elements at eikonal Regge limit

We examine the calculation of the color-ordered disk level S-matrix element of massless scalar vertex operators for the special case that some of the Mandelstam variables for which there are no open string channel in the amplitude, are set to zero. By explicit calculation we show that the string form factors in the 2n-point functions reduce to one at the eikonal Regge limit.     

The molecular vibration-rotation Hamiltonian in the Bargmann-Hilbert space

The molecular vibration-rotation Hamiltonian of polyatomic molecules is developed in the Bargmann-Hilbert space of entire functions. The vibration-rotation eigenvectors of polyatomic molecules are described as polynomials in complex variables where the number of complex variables is equal to the number of degrees of freedom of the molecule. Simple differential operator representations are found for all rotational and vibrational operators. A special procedure is developed, in the new representation, for describing the direction cosine operators of a rigid symmetric rotor.The new basis of molecular states is shown to be very convenient for calculating energy or intensity perturbations of molecules with axial symmetry. The theoretical methods include a treatment of doubly degenerate modes of vibration.     

Correlation operators in many-particle physics

Correlation functions and thermodynamic Green's functions are expressed as matrix elements of corresponding superoperators. A connection between the super-operator formalism, which has been used extensively in relaxation problems and in magnetism in recent years, and the method of thermodynamic Green's functions is thus established. The introduction of correlation (super-) operators describing dynamical as well as statistical aspects of a physical system allows for an abstract formulation of the theory independent of any special representation which results in considerable formal simplifications. The investigation of the Laplace-transformed correlation operator leads to Dyson's equation with the self-energy explicitly expressed as a superoperator matrix element.     

Constraints for anomalous dimensions of local light-cone operators in [ϕ 3]6 theory

Using the MS scheme, we derive in [? ³]₆ theory the collinear conformal Ward identity for the Green's functions of local light-cone operators of leading twist. The Ward identity for special collinear conformal transformations and renormalization group invariance give constraints for the off-diagonal part of the anomalous dimension matrix for the general case of β#0. We compute the anomaly of special conformal transformation in lowest loop order and obtain from the constraints the off-diagonal part of the anomalous dimension in 2-loop order.     

Bivariate averaging and the Wigner distribution function

In order to gain insight into the nature of the Wigner and related distribution functions, bivariate averaging functions of real unbounded variables with absolutely continuous marginals that are ordinary probabilities are considered. Accordingly variables are chosen to be phase space variables that are respectively eigenvalues of position and momentum operators. The impact of the condition that the marginals are squared magnitudes of amplitudes that are Fourier transforms of one another is emphasized by the delay of the introduction of this Fourier transform condition until after the form for a bivariate distribution with the given marginals is obtained. When the respective amplitudes are fourier transforms of one another, special cases of the bivariate averaging function correspond to generalized Wigner functions characterized by a parameterα. Such anα-Wigner function can be used as the basis of a consistent averaging procedure if an appropriate corresponding representation for underlying operators to be averaged is specified. Properties of theα-Wigner functions are summarized.     

Flat limit of the de Sitter QFT in the rest frame vacuum

The problem of the flat limits of the scalar and spinor fields on the de Sitter expanding universe is considered in the traditional adiabatic vacuum and in the new rest frame vacuum we proposed recently,in which the frequencies are separated in the rest frames as in special relativity.It is shown that only in the rest frame vacuum can the Minkowskian flat limit be reached naturally fbr any momentum,whereas in the adiabatic vacuum,this limit remains undefined in rest frames in which the momentum vanishes.An important role is played by the phases of the fundamental solutions in the rest frame vacuum,which must be regularized to obtain the desired Minkowskian flat limits.This procedure fixes the phases of the scalar mode functions and Dirac spinors,resulting in their definitive expressions derived here.The physical consequenee is that,in the rest frame vacuum,the flat limits of the oneparticle operators are simply the corresponding operators of special relativity.     

Approximation methods in viscoelasticity theory

According to Volterra’s principle, the solution of the quasistatic problem in linear elasticity theory for an isotropic medium can be obtained from a solution of the corresponding problem of elasticity theory by substituting an integral operator with respect to time for the Poisson ratio and the subsequent interpretation of the constitutive operator function. The method of approximation uses an expansion of constitutive functions into a sum of rational operators. An evaluation is made for the accuracy of the solution, which is related to the quality of approximation. For an anisotropic medium, the constitutive function depends on several integral operators. In this case, a special method of approximation of the constitutive function is suggested which uses the introduction of “canonical” operators. A priori and a posteriori evaluations of the solution are given. Generalizations to nonhomogeneous (composite) media and nonlinear cases are indicated.     

Phenomenological theory of multicritical points splitting in the phase diagrams. The case of Landau’s potentials with ℒ = 3m

A general technique of exact calculation of any correlation functions for the special class of one-dimensional spin models containing small clusters of quantum spins assembled to a chain by alternating with the single Ising spins is proposed. The technique is a natural generalization of that in the models solved by a classical transfer matrix. The general expressions for corresponding matrix operators which are the key components of the technique are obtained. As it is clear from the general principles, the decay of the correlation functions of various types is explicitly shown to be governed by a single correlation length. The technique is illustrated by two examples: symmetric diamond chain and asymmetric sawtooth chain.