Volume 38, Issue 8, August 1997
Index of content:

A oscillator Green’s function
View Description Hide DescriptionBy using the generating function formula for the product of two Hermite polynomials,deformation of the Feynman Green’s function for the harmonic oscillator is obtained.

Relativistic classical theory of a free particle
View Description Hide DescriptionBy shifting the emphasis from the concept of trajectory to the concept of probability density it is possible to incorporate the uncertainty principle into classical mechanics. This amendment in the nonrelativistic classical theory is sufficient to derive the Schrödinger equation for a general potential. In order to show that the approach has general validity it is necessary to show that it can be generalized to the classical relativistic dynamics. In this paper it is shown how this generalization is achieved for a free particle, and as a result the Dirac instead of the Klein–Gordon equation is obtained. It is shown that the spin and the magnetic moment of charged particles are classical in character because their correct values are calculated as the averages over the classical (relativistic) phase space density, subject to the constraint imposed by the uncertainty principle. Since the Dirac equation has direct connection to the classical (relativistic) dynamics the problem of the positive and negative energy states is discussed.

Nonperturbative formulation of relativistic twoparticle states in the scalar Yukawa model
View Description Hide DescriptionWe investigate a variational approach to the study of twoparticle bound states in quantum field theory. The scalar Yukawa model, in which scalar “nucleons” interact via “meson” exchange, is considered. A variational trial state that contains one or two scalar nucleons with any number of mediating scalar mesons is used to derive integral equations for the one and twonucleon systems.Numerical solutions of these equations are obtained in dimensions. Comparison is made with perturbative and nonrelativistic approximations that have been used in earlier calculations.

Longrange scattering in the position representation
View Description Hide DescriptionFor a large class of longrange potentials we prove the asymptotic completeness of modified wave operators constructed using solutions of the eikonal equation .

Vector bundles over configuration spaces of nonidentical particles: Topological potentials and internal degrees of freedom
View Description Hide DescriptionWe consider configuration spaces of nonidentical pointlike particles. The physically motivated assumption that any two particles cannot be located at the same point in space–time leads to nontrivial topological structure of the configuration space. For a quantum mechanical description of such a system, we classify complex vector bundles over the configuration space and obtain potentials of topological origin, similar to those that occur in the fiber bundle approach to Dirac’s magnetic monopole or in Yang–Mills theory.

Nöther formalism for conserved quantities in classical gauge field theories. II. The arbitrary Bosonic matter case
View Description Hide DescriptionA formalism suited to deal with Bosonic matter represented by tensor densities coupled with a gauge field and gravitational field is developed. Nöther conserved currents and superpotentials are found and the relation between the canonical and the Hilbert stress tensors are investigated in the general case. The particular cases of gauge theories with scalar matter fields and of purely geometrical theories with arbitrary matter are recovered from this general formalism; as an example of application we consider scalar densities and the vector matter fields.

Negative binomial and multinomial states: Probability distributions and coherent states
View Description Hide DescriptionFollowing the relationship between probability distribution and coherent states, for example the well known Poisson distribution and the ordinary coherent states and relatively less known one of the binomial distribution and the su(2) coherent states, we propose interpretation of su(1,1) and coherent statesin terms of probability theory. They will be called the negative binomial (multinomial) states which correspond to the negative binomial (multinomial) distribution, the noncompact counterpart of the well known binomial (multinomial) distribution. Explicit forms of the negative binomial (multinomial) states are given in terms of various boson representations which are naturally related to the probability theory interpretation. Here, we show fruitful interplay of probability theory,group theory, and quantum theory.

Grassmann manifold bosonization of QCD in two dimensions
View Description Hide DescriptionTwodimensional QCD is bosonized to be an integrably deformed Wess–Zumino–Witten model under proper limit. Fermions are identified having indices of the Grassmann manifold. Conditions for integrability are analyzed and their physical meanings are discussed. We also address the nature of the exactly solvable part of the theory and find the infinitely many conserved quantities.

The Hamiltonian near the critical value
View Description Hide DescriptionWe elucidate the behavior of the operator near the critical value where it ceases to be bounded below, by obtaining a family of operators which is selfadjoint holomorphic in a domain including all real and such that is just the operator or its Friedrich extension, while is another selfadjoint extension. The operators ( real) are shown to be positive, and to have only discrete spectrum below The eigenvalues are then analytic functions of near (and become the eigenvalues of a nonselfadjoint operator when ). We show that these eigenvalues cannot vanish, but that the lowest eigenvalue of goes to zero when The eigenvalues are analytic in α at

A solvable model for excitonic complexes in one dimension
View Description Hide DescriptionIt is known experimentally that stable fewbody clusters containing negativelycharged electrons (e) and positivelycharged holes (h) can exist in lowdimensional semiconductor nanostructures. In addition to the familiar exciton (e+h), threebody “charged excitons” (2e+h and 2h+e) have also been observed. Much less is known about the properties of such charged excitons since threebody problems are generally very difficult to solve, even numerically. Here we introduce a simple model, which can be considered as an extended Calogero model, to calculate analytically the energy spectra for both a charged exciton and a neutral exciton in a onedimensional nanostructure, such as a finitelength quantum wire. Apart from its physical motivation, the model is of mathematical interest in that it can be related to the Heun (or Heine) equation and, as shown explicitly, highly accurate, closed form solutions can be obtained.

Asymptotic of the density of states for the Schrödinger operator with periodic electromagnetic potential
View Description Hide DescriptionFor the Schrödinger operator in with periodic electromagnetic potential, we give an asymptotic formula of the integrate density of states of the form When this estimate enables us to prove the finiteness of gaps in the spectrum.

Path space measures for Dirac and Schrödinger equations: Nonstandard analytical approach
View Description Hide DescriptionA nonstandard path space * measure is constructed to justify the path integral formula for the Dirac equation in twodimensional space–time. A standard measure as well as a standard path integral is obtained from it. We also show that, even for the Schrödinger equation, for which there is no standard measure appropriate for a path integral, there exists a nonstandard measure to define a * path integral whose standard part agrees with the ordinary path integral as defined by a limit from timeslice approximant.

Nonlinear quantization of integrable classical systems
View Description Hide DescriptionIt is demonstrated that the socalled “unavoidable quantum anomalies” can be avoided in the framework of a special nonlinear quantization scheme. In this scheme, the quantized Hamiltonians are represented by nonlinear but homogeneous operators in Hilbert space. The nonlinear terms are of the same order as quantum anomalies, and their role is to cancel anomalies. The quantization method proposed is applicable to integrable classical dynamical systems and the result of quantization is again an integrable (but, generally, nonlinear) “quantum” system. A simple example is discussed in detail. Irrespective of the existence of possible physical applications, the method provides a constructive way for extending the notion of quantum integrability to nonlinear spectral problems and gives a practical tool for building completely integrable nonlinear spectral equations in Hilbert space.

Spinning particle in six dimensions
View Description Hide DescriptionMassive spinning particle in Minkowski space is described as a mechanical system with the configuration space The action functional of the model is unambiguously determined by the requirement of identical (offshell) conservation for the phasespace counterparts of three Casimir operators of the Poincaré group. The model proves to be completely solvable. Its generalization to the constant curvature background is presented. Canonical quantization of the theory leads to the relativistic wave equations for the irreducible fields.

Generalization of Shannon’s theorem for Tsallis entropy
View Description Hide DescriptionBy using the assumptions that the entropy must (i) be a continuous function of the probabilities only; (ii) be a monotonic increasing function of the number of states in the case of equiprobability; (iii) satisfy the pseudoadditivity relation ( and being two independent systems, and a positive constant), and (iv) satisfy the relation where and ), we prove, along Shannon’s lines, that the unique function that satisfies all these properties is the generalized Tsallis entropy

Toda solitons and the dressing symmetry
View Description Hide DescriptionWe present an elementary derivation of the solitonlike solutions in the Toda models which is an alternative to the previously used Hirota method. The solutions of the underlying linear problem corresponding to the solitons are calculated. This enables us to obtain explicit expression for the element which, by dressing group action, produces a generic soliton solution. In the particular example of monosolitons we suggest a relation to the vertex operator formalism, previously used by Olive, Turok, and Underwood. Our results can also be considered as generalization of the approach to the sine–Gordon solitons, proposed by Babelon and Bernard.

Solving the constrained KP hierarchy by gauge transformations
View Description Hide DescriptionWe solve the constrained Kadomtsev–Petviashvili (cKP) hierarchy by using the gauge transformation technique. We show that there are two kinds of gauge transformations which preserve the form of the Lax operator of the cKP hierarchy. One of them is differential type and the other is integral type. Through two such gauge transformations we obtain not only the Wronskiantype functions for the cKP hierarchy, but also the binarytype functions which have not been obtained before.

Spectral transform and solitons for the threewave coupling model with nontrivial boundary conditions
View Description Hide DescriptionA nonlinear threewave coupling model with nontrivial boundary conditions is analyzed in the framework of the inverse spectral transform. Spectral data are determined and their evolution is derived. It is shown that there exists mutual cancellation of divergences of the scattering matrix elements in branching points on the plane of a spectral parameter. A regular darksolitontype solution is obtained.

Universal and integrable nonlinear evolution systems of equations in dimensions
View Description Hide DescriptionIntegrable systems of nonlinear partial differential equations(PDEs) are obtained from integrable equations in dimensions, by means of a reduction method of broad applicability based on Fourier expansion and spatio–temporal rescalings, which is asymptotically exact in the limit of weak nonlinearity. The integrability by the spectral transform is explicitly demonstrated, because the corresponding Lax pairs have been derived, applying the same reduction method to the Lax pair of the initial equation. These systems of nonlinear PDEs are likely to be of applicative relevance and have a “universal” character, inasmuch as they may be derived from a very large class of nonlinear evolution equations with a linear dispersive part.

Superintegrable systems, quadratic constants of motion, and potentials of Drach
View Description Hide DescriptionThe properties of superintegrable systems in two degrees of freedom, possessing three independent globally defined constants of motion, are studied using as an approach, the existence of hidden symmetries and the generalized Noether’s theorem. The potentials are obtained as solution of a system of two partial differential equations. First the case of standard Lagrangians is studied and then the method is applied to the case of Lagrangians with a pseudoEuclidean kinetic term. Finally, the results are related with other approaches and with a family of potentials admitting a second integral of motion cubic in the velocities obtained by Drach.