Volume 36, Issue 2, February 1995
Index of content:

Operator‐theoretical analysis of a representation of a supersymmetry algebra in Hilbert space
View Description Hide DescriptionOperator‐theoretical analysis is made on (unbounded) representations, in Hilbert spaces, of a supersymmetry(SUSY)algebra coming from a supersymmetric quantum field theory in two‐dimensional space–time. A basic idea for the analysis is to apply the theory of strongly anticommuting self‐adjoint operators. A theorem on integrability of a representation of the SUSYalgebra is established. Moreover, it is shown that strong anticommutativity of self‐adjoint operators is a natural and suitable concept in analyzing representations of the SUSYalgebra in Hilbert space.

Some extensions of semiclassical limit ℏ → 0 for Wigner functions on phase space
View Description Hide DescriptionThe semiclassical asymptotics of the unitary quantum propagator applied to the Wigner functions, which are associated with the initial quantum state belonging to Schwartz class, are considered herein. The limit ℏ → 0 of the semiclassical asymptotics of these Wigner functions is investigated. As a consequence of this result, there exists a simple semiclassical asymptotic, which obeys the unitary conservation law over any finite time interval [0,T].

Classical deformations, Poisson–Lie contractions, and quantization of dual Lie bialgebras
View Description Hide DescriptionA Poisson–Hopf algebra of smooth functions is simultaneously constructed on the two dimensional Euclidean, Poincaré, and Heisenberg groups by using a classical r‐matrix which is invariant under contraction. The quantization for this algebra of functions is developed, and its dual Hopf algebra is also computed. Contractions on these quantum groups are studied. It is shown that, within this setting, classical deformations are transformed into quantum ones by Hopf algebra duality and the quantum Heisenberg algebra is derived by means of a (dual) Poisson–Lie quantization that deforms the standard Moyal–Weyl *_{ℏ}‐product.

On supersymmetry and radial equations for three‐dimensional central problems
View Description Hide DescriptionSupersymmetric three‐dimensional Schrödinger equations describing central problems do not lead, in general, to supersymmetric one‐dimensional radial equations. Such a context is discussed, in connection with the two main supersymmetrization procedures (with or without spin‐orbit coupling terms). The inverse problem is also considered, starting from a supersymmetric radial equation.

Divergences in the moduli space integral and accumulating handles in the infinite‐genus limit
View Description Hide DescriptionThe symmetries associated with the closed bosonic string partition function are examined so that the integration region in Teichmuller space can be determined. The conditions on the period matrix defining the fundamental region can be translated to relations on the parameters of the uniformizing Schottky group. The growth of the lower bound for the regularized partition function is derived through integration over a subset of the fundamental region.

The Hurwitz transformation: Nonbilinear version
View Description Hide DescriptionAn alternative approach to the Hurwitz (H) transformation reducing Euclidean spaceE ^{8} to Euclidean spaceE ^{5} is developed. It is shown how refusal of the bilinearity condition leads to the replacement of the H‐transformation by its modified nonbilinear (left and right) version: H _{ L }‐ and H _{ R }‐transformations. The H _{ L }(H _{ R })‐transformation of the specific type eight‐dimensional hyperspherical coordinates is investigated. The radial coordinate u and the polar angle θ/2, in this approach, transform into u ^{2} and θ, respectively, like in the H‐case. Action of these transformations on the remaining hyperspherical coordinates, unlike in the H‐case, is equivalent to the invariance (shutting) of one angular triplet and the shutting (invariance) of another triplet. The connection of H _{ L } and H _{ R } with H is established and the structure of the H‐transformation itself is revealed on this basis.

Towards a classification of su(2)⊕...⊕su(2) modular invariant partition functions
View Description Hide DescriptionThe complete classification of Wess–Zumino–Novikov–Witten modular invariant partition functions is known for very few affine algebras and levels, the most significant being all levels of A _{1} and A _{2} and level 1 of all simple algebras. Here, the classification problem is addressed for the nicest high rank semisimple affine algebras: (A _{1} ^{(1)})^{⊕ r }. Among other things, all automorphism invariants are found explicitly for all levels k=(k _{1},...,k _{ r }), and the classification for A ^{(1)} _{1}⊕A ^{(1)} _{1} is completed for all levels k _{1}, k _{2}. The classification problem for (A _{1} ^{(1)})^{⊕ r } is also solved for any levels k _{ i } with the property that for i ≠ j each gcd(k _{ i }+2,k _{ j }+2)≤3. In addition, some physical invariants are found which seem to be new. Together with some recent work by Stanev, the classification for all (A _{1} ^{(1)})_{ k } ^{⊕ r } could now be within sight.

Mass spectra from field equations. II. The electromagnetic contribution
View Description Hide DescriptionIn the first article a method for finding a mass spectrum from a given field equation was developed. The method is extended here to provide for the self‐electromagnetic effect on the rest mass. The process is described by example: the method is applied here for the equations of a complex scalar fieldinteracting with the electromagnetic field. A spectrum of charged‐particle masses is obtained. A quantum number for the charges of the mass levels appears. After some special assumptions about the dependence on the vector potential are made, there are finite electromagnetic contributions to the rest masses of the particle states.

Asymptotic expansion of the Dirac–Coulomb radial Green’s function
View Description Hide DescriptionAn asymptotic expansion of the Whittaker function M _{α,β}(x), for β→∞, with α and x fixed, is employed to obtain the asymptotic expansion of the Dirac–Coulomb radial Green’s functionG _{κ}(x _{2},x _{1},z) when the magnitude of the angular momentum quantum number κ is large. This result has application in numerical calculations of quantum electrodynamic effects in an external Coulomb field.

A Fock space representation for the quantum Lorentz gas
View Description Hide DescriptionA Fock space representation is given for the quantum Lorentz gas, i.e., for random Schrödinger operators of the form H(ω)=p ^{2}+V _{ω}=p ^{2}+∑ φ(x−x _{ j }(ω)), acting in H=L ^{2}(R ^{ d }), with Poisson distributed x _{ j }s. An operator H is defined in K=H⊗P=H⊗L ^{2} (Ω,P(dω))=L ^{2} (Ω,P(dω);H) by the action of H(ω) on its fibers in a direct integral decomposition. The stationarity of the Poisson process allows a unitarily equivalent description in terms of a new family {H(k)‖k∈R ^{ d }}, where each H(k) acts in P [A. Tip, J. Math. Phys. 35, 113 (1994)]. The space P is then unitarily mapped upon the symmetric Fock space over L ^{2}(R ^{ d },ρd x), with ρ the intensity of the Poisson process (the average number of points x _{ j } per unit volume; the scatterer density), and the equivalent of H(k) is determined. Averages now become vacuum expectation values and a further unitary transformation (removing ρ in ρd x) is made which leaves the former invariant. The resulting operator H _{F}(k) has an interesting structure: On the nth Fock layer we encounter a single particle moving in the field of nscatterers and the randomness now appears in the coefficient √ρ in a coupling term connecting neighboring Fock layers. We also give a simple direct self‐adjointness proof for H _{F}(k), based upon Nelson’s commutator theorem. Restriction to a finite number of layers (a kind of low scatterer density approximation) still gives nontrivial results, as is demonstrated by considering an example.

Generalized magnetic monopoles over contact manifolds
View Description Hide DescriptionA generalization of magnetic monopoles is given over an odd dimensional contact manifold and we discuss whether the Yang–Mills–Higgs functional attains at generalized monopoles the absolute minimal value, the topological invariant.

Number density, entropy density, and energy density of statistical mechanics for D‐dimensional grand canonical ensembles
View Description Hide DescriptionA partial differential equation for the number density of a D‐dimensional grand canonical ensemble is used to derive an explicit expression for the semiclassical number density. This is accomplished by first using the Bloch equation and its solutions to derive the classical number density and then combining the classical result with a Planck’s constant expansion of the partial differential equation to obtain the semiclassical formula. Similar results are also derived for the entropy density, energy density, and grand potential density of a D‐dimensional grand canonical ensemble.

Nonexistence of global solutions of the initial‐boundary value problem for the nonlinear Klein–Gordon equation
View Description Hide DescriptionSufficient conditions are given so that the solutions of the initial‐boundary value problem for the nonlinear Klein–Gordon equation do not exist for all t≳0.

Some reductions of the self‐dual Yang–Mills equations to integrable systems in 2+1 dimensions
View Description Hide DescriptionA reduction of the self‐dual Yang–Mills (SDYM) equations is studied by imposing two space–time symmetries and by requiring that the connection one‐form belongs to a Lie algebra of formal matrix‐valued differential operators in an auxiliary variable. In this article, the scalar case and the canonical cases for 2×2 matrices are examined. In the scalar case, it is shown that the field equations can be reduced to the forced Burgers equation. In the matrix case, several well‐known 2+1 integrable equations are obtained. Also examined are certain transformation properties between the solutions of some of these 2+1 equations.

Knotted periodic orbits in Rössler’s equations
View Description Hide DescriptionKnots corresponding to the Rössler chaotic model are defined. A systematic method to determine the knots for any order of the chaotic trajectory is given. For the m chaotic path R _{ m }, a minimal (2m−1)‐braid representative is given. Hence the order of the chaotic trajectory is knot invariant. An algorithm for constructing any Rössler knot, using a figure eight knot, is given.

Fluids with spin and twist
View Description Hide DescriptionFluids with persistentvortices that exhibit shear plus expansion (or contraction) in noninertial frames are common physical phenomena. The concept of intrinsic rotation is commonly referred to as spin; the equivalent concept for shear would be shear momenta, referred to as twist in this work. The motion of the Earth’s atmosphere is a prime example of such motion in which the driving engine is the rotation of the Earth plus solar radiation. The general analytical features of persistent vortices that exhibit shear plus expansion and contraction are introduced using the methods of affine geometry. The same theoretical considerations can also be applied to astrophysical examples.

Low‐energy dynamics of a C P ^{1} lump on the sphere
View Description Hide DescriptionLow‐energy dynamics in the unit‐charge sector of the C P ^{1} model on spherical space (space‐time S ^{2}×R) is treated in the approximation of geodesic motion on the moduli space of static solutions, a six‐dimensional manifold with nontrivial topology and metric. The structure of the induced metric is restricted by consideration of the isometry group inherited from global symmetries of the full field theory. Evaluation of the metric is then reduced to finding five functions of one coordinate, which may be done explicitly. Some totally geodesic submanifolds are found, and the qualitative features of motion on these described.

Hierarchies of Korteweg–de Vries type equations
View Description Hide DescriptionAn eigenvalue problem is considered. New hierarchies of bi‐Hamiltonian systems are constructed. Some examples of these systems and their reductions are presented.

Particle motion in the gravitational field of a charged, radiating body
View Description Hide DescriptionThe motion of test bodies is examined in the framework of the Vaidya–Bonner model of a charged, radiating mass. The null and time‐like geodesic equations are obtained exactly. Similar equations are obtained exactly for the case of charged test particles. It is demonstrated that the luminosity of the central body induces a relative motion between positive and negative test charges, leading to accretion disk temperatures for luminous bodies which are higher than those of non‐radiating bodies.

A symplectic interpretation of some wave fronts in general relativity
View Description Hide DescriptionIn order to study the symplectic structures of the elementary relativistic free particles in general relativity, a nonvectorial associated fiber bundle is constructed over the space‐time manifold. A foliation in the space‐time is achieved which gives rise to the geodesics principle for massive particles and wave fronts for massless particles.