Index of content:
Volume 35, Issue 7, July 1994

A simple difference realization of the Heisenberg q‐algebra
View Description Hide DescriptionA realization of the Heisenberg q‐algebra whose generators are first‐order difference operators on the full real line is discussed herein. The eigenfunctions of the corresponding q‐oscillator Hamiltonian are given explicitly in terms of the q ^{−1}‐Hermite polynomials. The nonuniqueness of the measure for these q‐oscillator states is also studied.

On integrable models related to the osp(1,2) Gaudin algebra
View Description Hide DescriptionThe osp(1,2) Gaudin algebra is defined and integrable models described by it are considered. The models include the osp(1,2) Gaudin magnet and the Dicke model related to it. Detailed discussion of the simplest cases of these models is presented. The effect of the presence of fermions on the separation of variables is indicated.

q‐Deformation of the Coulomb problem
View Description Hide DescriptionThe hydrogenic Schrödinger equation may be transformed into an integral equation on the group space of SO(3). The solutions of this integral equation are the Wigner functions D ^{ j } _{ mn }, the matrix elements of the irreducible representations of SO(3). In this article we examine a q‐deformation of the Coulomb problem for which the new wave functions are the matrix elements of the irreducible representations of the quantum groupSU_{ q }(2). The new integral equation is formulated in terms of the Woronowicz integral. The eigenvalues are given by a modified Balmer formula which lifts the Coulomb degeneracy and replaces n by [n].

Continuity of the S matrix for the perturbed Hill’s equation
View Description Hide DescriptionThe behavior of the scattering matrix associated with the perturbed Hill’s equation as the spectral parameter approaches an endpoint of a spectral band is studied. In particular, the continuity of the scattering matrix at the band edges is proven and explicit expressions for the transmission and reflection coefficients at those points are derived. All possible cases are discussed and our fall‐off assumptions on the perturbation are weaker than those made by other authors.

Algebraic solution for the Natanzon hypergeometric potentials
View Description Hide DescriptionAn algebraic method—based on a strategy that makes use of a realization of the algebra SO(2,1)—in terms of differential operators is used to solve the bound state problem for the most general Natanzon potentials for which the Schrödinger equation can be reduced to hypergeometric form (hence, hypergeometric potentials).

The vacuum energy density for spherical and cylindrical universes
View Description Hide DescriptionThe vacuum energy density (Casimir energy) corresponding to a massless scalar quantum field living in different universes (mainly no‐boundary ones), in several dimensions, is calculated. Hawking’s zeta function regularization procedure supplemented with binomial expansion is shown to be a rigorous and well suited method for performing the analysis. It is compared with other more involved techniques. The principal‐part prescription is used to deal with the poles that eventually appear. Results of the analysis are the absence of poles at four dimensions (4D) (for a 4D Riemann sphere and for a 4D cylinder of 3D Riemann spherical section), the total coincidence of the results corresponding to a 3D and a 4D cylinder (the first after pole subtraction), and the fact that the vacuum energy density for cylinders is (in absolute value) over an order of magnitude smaller than for spheres of the same dimension.

Nonlinear Schrödinger equations and the separation property
View Description Hide DescriptionHierarchies of nonlinear Schrödinger equations were investigated for multiparticle systems, satisfying the separation property, i.e., where product wave functions evolve by the separate evolution of each factor. Such a hierarchy defines a nonlinear derivation on tensor products of the single‐particle wave‐function space, and satisfies a certain homogeneity property characterized by two new universal physical constants. A canonical construction of hierarchies is derived that allows the introduction, at any particular ‘‘threshold’’ number of particles, of truly new physical effects absent in systems having fewer particles. In particular, if single quantum particles satisfy the usual (linear) Schrödinger equation, a system of two particles can evolve by means of a fairly simple nonlinear Schrödinger equation without violating the separation property. Examples of Galilean‐invariant hierarchies are given.

Mass spectra from field equations
View Description Hide DescriptionA type of field quantization is developed which, in some cases, leads directly from a field equation to a mass spectrum. The starting point is a field‐Hamiltonian setup of the field equations. A Schrödinger‐type wave function, a function of the field variables, is postulated and a Hamiltonian operator is inferred from the Hamiltonian function. A wave equation is suggested which has particle‐type solutions only for special values of the rest mass. As examples, the spectra for the linear wave equation and for an equation with a sharply limited range of the field variable are discussed.

Quantum electrodynamics in external fields from the spin representation
View Description Hide DescriptionSystematic use of the infinite‐dimensional spin representation simplifies and rigorizes several questions in quantum field theory. This representation permutes ‘‘Gaussian’’ elements in the fermion Fock space, and is necessarily projective: its cocycle at the group level is computed, and Schwinger terms and anomalies from infinitesimal versions of this cocycle are obtained. Quantization, in this framework, depends on the choice of the ‘‘right’’ complex structure on the space of solutions of the Dirac equation. We show how the spin representation allows one to compute exactly the S‐matrix for fermions in an external field; the cocycle yields a causality condition needed to determine the phase.

Algebraic structures of quantum projective field theory related to fusion and braiding. Hidden additive weight
View Description Hide DescriptionThe interaction of various algebraic structures describing fusion, braiding, and group symmetries in quantum projective field theory is the object of investigation in this article. Structures of projective Zamolodchikov algebras, their representations, spherical correlation functions,correlation characters and enveloping quantum projective field theory (QPFT)‐operator algebras, projective Ẅ‐algebras, shift algebras, infinite‐dimensional R matrices R _{proj}(u) and R _{proj} ^{*}(u) of the QPFT, braiding admissible QPFT‐operator algebras, and projective G hypermultiplets are explored. It is proven (in the formalism of shift algebras) that sl(2,C)‐primary fields are characterized by their projective weights and by the hidden additive weight, a hidden quantum number discovered in this article. Special attention is paid to various constructions of projective G hypermultiplets (QPFT‐operator algebras with G symmetries).

Quantization of generalized spinning particles: New derivation of Dirac’s equation
View Description Hide DescriptionQuantization of generalized Lagrangian systems suggests that wave functions for elementary particles must be defined on the kinematical space rather than on configuration space. For spinning particles the center of mass and center of charge are different points. Their separation is of the order of the Compton wavelength. Spin‐1/2 particles arise if the classical model rotates but no half integer spins are obtained for systems with spin of orbital nature. Dirac’s equation is obtained when quantizing the classical relativistic spinning particles whose center of charge is circling around its center of mass at the speed c. Internal orientation of the electron completely characterizes its Dirac’s algebra.

The classical limit of ultralocal scalar fields
View Description Hide DescriptionThe classical limit of the quantum theory of ultralocal scalar fields is approached with the help of coherent state techniques. In the limit ℏ→0, it is demonstrated that the coherent state expectation value of the Heisenberg field operators coincide with the solutions of the corresponding classical canonical equations of motion.

The constraint algorithm for time‐dependent Lagrangians
View Description Hide DescriptionThe aim of this paper is to develop a constraint algorithm for time‐dependent Lagrangian systems which permit us to solve the motion equations. This algorithm extends the Gotay and Nester algorithm for autonomous Lagrangians which is, in fact, a particular case. To do this the almost stable tangent geometry of the evolution space and the notion of cosymplectic structure are used.

On the Kolmogorov’s condition for a special case of the Kirchhoff top
View Description Hide DescriptionIn this article the classical Kirchhoff case of motion of a rigid body in an infinite ideal fluid is considered. Then for the corresponding Hamiltonian system on the zero integral level, the Kolmogorov’s condition which is important for Kolmogorov–Arnold–Moser theory is checked. In contrast to known similar results, there exists a curve in the bifurcation diagram along which the Kolmogorov’s condition vanishes for certain values of the parameters.

On acoustic Helmholtz resonator and on its electromagnetic analog
View Description Hide DescriptionA study of the acoustic Helmholtz resonator and of its electromagnetic analog (a cylinder with a narrow slit) is performed. For the associated Green functions, power series asymptotics with respect to a small parameter ε (‘‘radius’’ of the resonator hole or the width of the slit), and ln ε of poles τ_{ε} with small imaginary parts are obtained by using the method of matching asymptotic expansions. The principal terms of solution asymptotics for corresponding boundary value problems are given.

The Lorentz–Dirac equation in Minkowski space
View Description Hide DescriptionThe Lorentz–Dirac equation is obtained by means of a modification imposed on Maxwell’stensor for the Liénard–Wiechert field. This procedure avoids the mass renormalization of the point charge.

On hearing the shape of rectilinear regions
View Description Hide DescriptionFrom complete knowledge of the eigenvalues of the negative Laplacian on a bounded domain, one may extract information on the geometry and the boundary conditions by analyzing the asymptotic expansion of a spectral function. Explicit calculations are performed for an equilateral triangular domain with Dirichlet or Neumann boundary conditions, yielding in particular the corner angle terms. In three dimensions, some applications to eigenvalue problems for an equilateral triangular prism are dealt with, including the solid vertex terms.

Symplectic analysis of a Dirac constrained theory
View Description Hide DescriptionThe symplectic formalism is applied to a system recently analyzed by the Dirac method. It is shown that this procedure is quite straightforward and elegant for the two versions of the model.

Nonlinear systems related to an arbitrary space–time dependence of the spectral transform
View Description Hide DescriptionA general algebraic analytic scheme for the spectral transform of solutions of nonlinear evolution equations is proposed. This allows one to give the general nonlinear evolution corresponding to an arbitrary time and space dependence of the spectral transform (in general nonlinear and with nonanalytic dispersion relations). The main theorem is that the compatibility conditions always give a true nonlinear evolution because it can always be written as an identity between polynomials in the spectral variable k. This general result is then used to obtain first a method to generate a new class of solutions to the nonlinear Schrödinger equation, and second to construct the spectral transform theory for solving initial‐boundary value problems for resonant wave‐coupling processes (like self‐induced transparency in two‐level media, or stimulated Brillouin scattering of plasma waves, or else stimulated Raman scattering in nonlinear optics, etc.).

The basis of nonlocal curvature invariants in quantum gravity theory. Third order
View Description Hide DescriptionA complete basis of nonlocal invariants in quantum gravitytheory is built to third order in space–time curvature and matter‐field strengths. The nonlocal identities are obtained which reduce this basis for manifolds with dimensionality 2ω<6. The present results are used in heat‐kernel theory,theory of gauge fields and serve as a basis for the model‐independent approach to quantum gravity and, in particular, for the study of nonlocal vacuumeffects in the gravitational collapse problem.