Volume 33, Issue 4, April 1992
Index of content:

E(2)‐symmetric two‐mode sheared states
View Description Hide DescriptionIt is noted that the symmetry of two‐mode squeezed states of light is governed by the group Sp(4) that is locally isomorphic to O(3,2). This group has subgroups that are locally isomorphic to the two‐dimensional Euclidean group. Two‐mode states having the E(2) symmetry are constructed. The translation‐like transformations of this symmetry group shear the Wigner distribution function defined over the four‐dimensional phase space consisting of two pairs of canonical variables. Sheared states are constructed in the Schrödinger picture of quantum mechanics and in the Fock space for photon numbers. It is shown that the Wigner phase‐space picture is a convenient representation of quantum mechanics for calculating measurable quantities of the sheared states.

Canonical representations of sp(2n,R)
View Description Hide DescriptionIn this paper a rather unconventional real basis for the real symplectic algebra sp(2n,R) is studied. This basis is valid for representations carried by homogeneous polynomials of the 2n phase‐space variables. The utility of this basis for practical computations is demonstrated by giving a simple derivation of the second‐ and fourth‐order indices of irreducible representations of sp(2n,R).

Eigenfunction expansions associated with the curl derivatives in cylindrical geometries: Completeness of Chandrasekhar–Kendall eigenfunctions
View Description Hide DescriptionEigenfunctions of the curl derivatives are used in many different fields of theoretical physics. For example, in plasma physics, an eigenfunction of curl represents a certain eigenstate of a plasma that has the force‐free property. The present theory proves that the spectral resolution of the curl operator gives a complete eigenfunction expansion of a solenoidal vector field with homogeneous boundary and flux conditions. Useful decomposition theories and a previous abstract result [Math. Z. 204, 235 (1990)] are concisely summarized. Explicit eigenfunctions have been studied in cylindrical geometries. Chandrasekhar–Kendall functions [Astrophys. J. 126, 457 (1957)], which are eigenfunctions of the curl derivatives in a cylindrical domain, give an orthogonal complete basis of the Hilbert space that is the orthogonal complement of the space of irrotational vector fields.

C‐Integrable nonlinear partial differential equations in N+1 dimensions
View Description Hide DescriptionTwo techniques are introduced, which are suitable to manufacture C‐integrable nonlinear PDEs (i.e., nonlinear PDEs solvable by an appropriate Change of variables) in N+1 dimensions. Several examples are exhibited.

Models with Hopf terms
View Description Hide DescriptionThe geometrical structure of the Hopf term and its relation to the spin of a skyrmion is studied. An ansatz describing the most general one‐skyrmion field configuration in three dimensions is introduced and various properties of the physical Hopf term for it are exhibited. The extension to seven dimensions is also made. Some comments are also made about the quantization of the coefficient of the Hopf term in the action.

Symplectic reduction and topology for applications in classical molecular dynamics
View Description Hide DescriptionThis paper aims to introduce readers with backgrounds in classical molecular dynamics to some ideas in geometric mechanics that may be useful. This is done through some simple but specific examples: (i) the separation of the rotational and internal energies in an arbitrarily floppy N‐body system and (ii) the reduction of the phase space accompanying the change from the laboratory coordinate system to the center of mass coordinate system relevant to molecular collisiondynamics. For the case of two‐body molecular systems constrained to a plane, symplectic reduction is employed to demonstrate explicitly the separation of translational, rotational, and internal energies and the corresponding reductions of the phase space describing the dynamics for Hamiltonian systems with symmetry. Further, by examining the topology of the energy‐momentum map, a unified treatment is presented of the reduction results for the description of (i) the classical dynamics of rotating and vibrating diatomic molecules, which correspond to bound trajectories and (ii) the classical dynamics of atom–atom collisions, which correspond to scattering trajectories. This provides a framework for the treatment of the dynamics of larger N‐body systems, including the dynamics of larger rotating and vibrating polyatomic molecular systems and the dynamics of molecule–molecule collisions.

A simple proof of geodesical completeness for compact space‐times of zero curvature
View Description Hide DescriptionAny Riemannian metric on a compact manifold is geodesically complete. By contrast, it is widely known that incomplete compact Lorentzian manifolds exist. In this article it will be proven that a compact manifold with a smooth Lorentz metric must be complete if the metric is flat everywhere. This question of completeness; which has significant implications for the classification of (compact) Lorentzian space forms, has remained unresolved until quite recently, when it was finally established by Carrière as a special case of an even more general result. Here, a simple, alternative proof of this special case of Carrière’s theorem is given; the proof requires minimal mathematical machinery but still involves the use of some intriguing connections between the topology and global geometry of a compact flat space‐time.

An introduction to the semiclassical limit of Euclidean quantum mechanics
View Description Hide DescriptionA new class of Gaussian (‘‘Bernstein’’) diffusion processes associated with semiclassical quantum dynamics is introduced and studied according to a new Euclidean strategy for the quantization of systems of particles in a potential (‘‘Euclidean quantum mechanics’’). All the measures known by other means, up to now, are included as particular cases of the present framework. It is shown that the theory may be regarded as a well‐defined probabilistic version of Feynman’s space‐time approach distinct, in particular, from the one due to Kac, and with much closer relations with classical Lagrangian dynamics. Many explicit examples are considered.

Taylor series and δ‐perturbation expansions for a nonlinear semiconductor transport equation
View Description Hide DescriptionA one‐dimensional, nonlinear equation for steady‐state carrier transport in semiconductors is solved by two approaches: a Taylor series–first integral method and the δ‐perturbation method. The square of the carrier speed is found as a function of the distance where collisions are treated by an energy‐independent mean‐free time. The coefficients of the Taylor series at the first maximum are exact expressions in the function determined from the first integral. In the perturbation expansion the function is evaluated to first order in δ. For both methods expressions for the distance from the first maximum to the emitting electrode are found and the distance is evaluated numerically for the Taylor series–first integral method.

Canonical quantization of supersymmetric second‐order systems
View Description Hide DescriptionThe canonical quantization of systems described by Lagrangians containing second‐order time derivatives of both even and odd variables is developed. It is shown that a correct application of Dirac’s method eliminates the ambiguity of different quantized versions associated with the same superclassical dynamics.

Stochastic quantization for quantum mechanics and conditional probabilities
View Description Hide DescriptionThe applicability of the stochastic quantization method to the quantum mechanics is examined. The imaginary character of the action is taken into account by the introduction of a pseudowhite noise defined by an imaginary Gaussian density. The intervening of end points in the paths contributing to the functional integral is treated by introducing Gaussian densities with vanishing variance. The same method allows the calculation of conditional probabilities for the Brownian motion.

The metaplectic representation of su_{ q }(1,1) and the q‐Gegenbauer polynomials
View Description Hide DescriptionThe metaplectic representation of the quantum algebra su_{q}(1,1) is shown to provide a group‐theoretic setting for certain basic orthogonal polynomials generalizing the usual Gegenbauer polynomials.

The Gel’fand–Tsetlin representations of the orthogonal Cayley–Klein algebras
View Description Hide DescriptionThe explicit expressions of the operators of the irreducible representations of the orthogonal Cayley–Klein algebras so_{ n+1}(j) are obtained from the well‐known Gel’fand–Tsetlin representations of so_{ n+1}. The contractions and the analytic continuations of the representations of so_{3}(j _{1},j _{2}),so_{4}(j _{1}, j _{2}, j _{3}) are regarded as examples. This approach gives the representations of the contracted and the analytically continued algebras in different bases, for example, in discrete and continuous ones. Possible contractions of the irreducible representations are discussed.

Contractions of the irreducible representations of the quantum algebras su_{ q }(2) and so_{ q }(3)
View Description Hide DescriptionThe contractions of the irreducible representations of the unitary quantum algebra su_{ q }(2) and the orthogonal quantum algebra so_{ q }(3) in the Gel’fand–Tsetlin basis are regarded in detail with the help of the dual numbers.

U _{ q } covariant oscillators and vertex operators
View Description Hide DescriptionU _{ q } (sl(2)) covariant Bose and Fermi oscillators are studied systematically and applied to constructing vertex operators and U _{ q } invariant blocks. For q ^{ p }=−1 (p=2,3,...) a p ^{2}‐dimensional unitary Fock space representation of the Bose oscillatoralgebraA_{−}(q) is constructed. This representation does not allow, however, for the construction of all U _{ q } invariant blocks associated with an indecomposable representation of U _{ q }. An extension of A_{−} is needed in this case, which involves operators creating states of q spin projection ±p/2 from the Fock vacuum. The extended algebra only has infinite‐dimensional indefinite metric space representations. An algebraic notion of trace is introduced (provided q ^{2}≠1 in the Bose case). This allows the introduction of a family of (in general, mixed) U _{ q } invariant ‘‘temperature’’ states, such that the vacuum expectation value appears in the zero temperature limit.

Inverse scattering in 1‐D nonhomogeneous media and recovery of the wave speed
View Description Hide DescriptionThe inverse scattering problem for the 1‐D Schrödinger equation d ^{2}ψ/dx ^{2} + k ^{2}ψ= k ^{2} P(x)ψ + Q(x)ψ is studied. This equation is equivalent to the 1‐D wave equation with speed 1/√1−P(x) in a nonhomogeneous medium where Q(x) acts as a restoring force. When Q(x) is integrable with a finite first moment, P(x)<1 and bounded below and satisfies two integrability conditions, P(x) is recovered uniquely when the scattering data and Q(x) are known. Some explicitly solved examples are provided.

On integral representations of the Coulomb functions
View Description Hide DescriptionBy dealing with a new integral representation for the irregular s‐wave Coulomb function, exact analytical expressions for the corresponding Jost states in momentum space are constructed, and their usefulness in studying the Coulomb half‐shell scattering is demonstrated. A supersymmetry‐inspired ladder‐operator relation is employed to derive higher partial‐wave generalization of the results presented. Some applications of the integral representation for the associated regular solution are also sought.

Sectional curvature, symmetries, and conformally flat plane waves
View Description Hide DescriptionThis paper is concerned with the sectional curvature function on space‐time and discusses vector fields that generate symmetries of this function. It is shown that such vector fields form a Lie algebra and its general structure and the possibilities for its dimension are given. A consequence of the paper is a complete discussion of the Lie algebra of conformal vector fields for conformally flat plane waves.

Kantowski–Sachs metrics with source: A massless scalar field
View Description Hide DescriptionSolutions of the coupled Einstein massless scalar field equation are discussed under the assumption that the metric belongs to the ‘‘Kantowski–Sachs’’ class of metrics. More precisely, the general solution of the coupled Einstein massless scalar field equations is derived under the following assumptions: (a) The scalar field is massless minimally coupled to gravity and obeys the Klein–Gordon equation; and (b) the metric belongs to the spherical Kantowski–Sachs class of metrics. The main feature of the results is a tractable, very simple, compact form of the space‐time metric. This is accomplished by expressing the solution in a special coordinate gauge. An analysis of the geometry shows the presence of curvature singularities which consist of a spacelike part and a null part. The spacelike singularity appears to ‘‘squeeze’’ the SO(3) orbits to zero proper area while simultaneously stretching their proper separation to infinity length. The null part appears to squeeze to zero values both SO(3) orbits and their proper separation as well. There also exists a subclass of metrics for which the singularity structure is different. The null singularity is replaced by a spacelike part. Both ‘‘initial’’ and ‘‘final’’ singularities exhibit the same qualitative features and they are of finite proper time away from each other.

Kantowski–Sachs metrics with source: A conformally invariant scalar field
View Description Hide DescriptionSolutions of the coupled Einstein conformally invariant massless scalar field equations are derived under the assumption that the metric admits a four‐parameter group of isometries with spacelike generators, where further, the three‐parameter subgroup of isometries acts multiply transitive on two‐dimensional surfaces. The general solution of the system is obtained and some properties concerning the singularity structure are briefly discussed. Results for all three cases are presented, i.e., for the case where the three‐dimensional group of isometries acts on two‐dimensional orbits of positive, negative, or zero curvature. The first two classes belong to the Kantowski–Sachs class of metrics, while the third one is of Bianchi type I with additional rotational symmetry.