Volume 38, Issue 7, July 1997
Index of content:

Novel generalization of threedimensional Yang–Mills theory
View Description Hide DescriptionA class of new nonAbelian gauge theories for vector fields on three manifolds is presented. The theories describe a generalization of threedimensional Yang–Mills theory featuring a novel nonlinear gauge symmetry and field equations for Liealgebravalued vector potential fields. The nonlinear form of the gauge symmetry and field equations relies on the vector crossproduct and vector curl operator available only in three dimensions, and makes use of an auxiliary Lie bracket together with the Lie bracket used in Yang–Mills theory. A gauge covariant formulation of the new theories is given which utilizes the covariant derivative and curvature from the geometrical formulation of Yang–Mills theory. Further features of the new theories are discussed.

Symmetries of preon interactions modeled as a finite group
View Description Hide DescriptionI model preon interactions as a finite group. Treating the elements of the group as the bases of a vector space, I examine those linear mappings under which the transformed bases may be treated as members of a group isomorphic to the original. In some cases these mappings are continuous Lie groups.

Linear canonical transformations in quantum mechanics
View Description Hide DescriptionWe find explicit unitary operators that implement linear canonical transformations of the quantum mechanical operators for a system with degrees of freedom. We then relate the operators effecting this transformation to the previous formulation of quantum canonical transformations in terms of an effective generating function introduced by Ghandour.

Algebraic proof of a sum rule occurring in Stark broadening of hydrogen lines
View Description Hide DescriptionWe present the algebraic proof of a sum rule that is relevant in the theory of Stark broadening of hydrogen and hydrogenic lines. This is accomplished by applying Gauss’ recursion formulas for the hypergeometric function to the analytical expression of the dipole matrix elements involved in the summation.

Properties of eigenstates of the sixvertex model with twisted and open boundary conditions
View Description Hide DescriptionWe use the method proposed by Izergin and Korepin to discuss the Bethe ansatz equations of the sixvertex model with twisted and open boundary conditions. Let two parameters of the Bethe wave functions be equal, then an additional Bethe ansatz equation arises. For the open boundary condition case, we also have discussed the dual pseudovacuum state and have written out the simplest scalar products of the Bethe wave function.

A new Hamilton operator for a massive relativistic particle with spin one in a generalized Heisenberg/Schrödinger picture
View Description Hide DescriptionWe consider a particular fourdimensional generalization of the transition from the Heisenberg to the Schrödinger picture. The space–time independent expansion with respect to the unitary irreducible representations of the Lorentz group is applied, as Fourier transformation in the Heisenberg picture, to the states of a massive relativistic particle. A new Hamilton operator has been found for such a particle with spin one.

Effective action of composite fields for general gauge theories in Batalin, Lavrov, and Tyutin covariant formalism
View Description Hide DescriptionThe gauge dependence of the effective action of composite fields for general gauge theories in the framework of the quantization method by Batalin, Lavrov and Tyutin is studied. The corresponding Ward identities are obtained. The variation of composite fields effective action is found in terms of new set of generators depending on composite field. The theorem of the onshell gauge fixing independence for the effective action of composite fields in such formalism is proven. A brief discussion of gravitationalvector induced interaction for Maxwelltheory with composite fields is given.

SU(1,1) coherent states and associated Wick symbol calculus
View Description Hide DescriptionWe display the relationship of Perelomov and Barut–Girardello coherent states associated with the discrete series of SU(1,1) by establishing the interwining operators. Both Perelomov and Barut–Girardello coherent states are deformations of Glauber coherent states of Heisenberg–Weyl group, but they approximate the latter from different sides and exhibit a certain duality. We also develop a symbol calculus of SU(1,1) algebra in terms of Barut–Girardello coherent states (Perelomov coherent states are not suitable for this purpose) and rederive the disentanglement (Baker–Campbell–Hausdorff) formula for SU(1,1) generators by the symbol calculus and parameter differentiation technique.

Quantum canonical transformations and exact solution of the Schrödinger equation
View Description Hide DescriptionTimedependent unitary transformations are used to study the Schrödinger equation for explicitly timedependent Hamiltonians of the form where R is an arbitrary real vectorvalued function of time and J is the angular momentum operator. The solution of the Schrödinger equation for the most general Hamiltonian of this form is shown to be equivalent to the special case This corresponds to the problem of a driven twolevel atom for the spin half representation of J. It is also shown that by requiring the magnitude of R to depend on its direction in a particular way, one can solve the Schrödinger equation exactly. In particular, it is shown that for every Hamiltonian of the form there is another Hamiltonian with the same eigenstates for which the Schrödinger equation is exactly solved. The application of the results to the exact solution of the parallel transport equation and exact holomony calculation for SU(2) principal bundles (Yang–Mills gauge theory) is also pointed out.

Recovery of singularities from amplitude information
View Description Hide DescriptionLet denote a potential in the onedimensional Schrödinger equation, without bound states, for which (1) for and (2) is piecewise continuous with adequate decay as goes to infinity. We are interested in the problem of determining given the reflectivity where is the usual lefthand reflection coefficient. For very special classes of potentials it is known that determines uniquely. Here we show that under much more general (although still restrictive) assumptions, the location and magnitude of discontinuities of can be determined from The nature of the restrictions is related to the behavior of for in the upper half of the complex plane.

Nonlocal symmetries and nonlocal conservation laws of Maxwell’s equations
View Description Hide DescriptionNonlocal symmetries are obtained for Maxwell’sequations in three space–time dimensions through the use of two potential systems involving scalar and vector potentials for the electromagnetic field. Corresponding nonlocal conservation laws are derived from these symmetries. The conservation laws yield nine functionally independent constants of motion which cannot be expressed in terms of the constants of motion arising from local conservation laws for space–time symmetries. These nine constants of motion represent additional conserved quantities for the electromagnetic field in three space–time dimensions.

Phase flow of an axially symmetrical gyrostat with one constant rotor
View Description Hide DescriptionWe analyze the attitude dynamics of an axially symmetric gyrostat under no external forces and one constant internal spin. We introduce coordinates to represent the orbits of constant angular momentum as a flow on a sphere. With these coordinates, we realize that the problem belongs to a general class of Hamiltonian systems, namely the problem here considered is the one parameter Hamiltonian that is a polynomial of at most degree two in a base of the Lie algebra so (3). The parametric bifurcations are found for both cases, when the rotor is spinning about the axis of symmetry of the gyrostat, and when it is spinning about another axis of inertia. The general solution for the global general flow is expressed in terms of the Jacobian elliptic functions.

Nonconducting electromagnetic media with rotational invariance: Transition operators and Green’s functions
View Description Hide DescriptionFor an accurate description of multiple scattering of electromagnetic waves, such as in random dielectrics or dielectric lattices, the full offshell single scatterer transition (T) operator is required. Here we study this quantity for the case of a nonconducting medium with rotational invariant permeabilities. We start with the scattering problem where rotational invariance is not yet assumed and find a class of mutually different Toperators which all have the same onshell restriction. Next we extend the usual method of expressing scattered fields for rotational invariant systems in terms of the solutions of two scalar wave equations to the resolvent (Green’s function) associated with the vector wave equation. We find that it can be expressed in terms of the resolvents of two scalar operators. Finally we turn to the Mie case (a dielectric sphere in vacuum) for which we obtain explicit expressions for the corresponding Green’s functions and the general offshell Tmatrix elements.

Euler and Navier–Stokes limits of the Uehling–Uhlenbeck quantum kinetic equations
View Description Hide DescriptionThe Uehling–Uhlenbeck evolution equations for gases of identical quantum particles either fermions or bosons, in the case in which the collision kernel does not depend on the distribution function, are considered. The existence of solutions and their asymptotic relations with solutions of the hydrodynamic equations both at the level of the Euler system and at the level of the Navier–Stokes system are proved.

On the moment methods and irreversible thermodynamics
View Description Hide DescriptionIn this paper we investigate the kinetic foundation of irreversible thermodynamics by means of the moment methods proposed by Grad and by Eu, respectively. First we show that the moment methods yield a weak solution of the Boltzmann equation. On the other hand, the entropy balance equation can be satisfied only by the strong solution of the Boltzmann equation. Second, we reformulate the energy balanceequation in an alternative form where dissipative energy as well as a generalized work 1form are included in this new equation. Assume that the dissipative energy is semipositive definite. The local form of the second law of thermodynamics is then formulated in terms of the inaccessibility condition of Caratheodory. We then show that our new formulation of the second law is equivalent to Kelvin’s principle and Clausius’ principle. Finally we obtain a calotropy balance equation where the calotropy density function is a state function in the thermodynamic space.

The eight tetrahedron equations
View Description Hide DescriptionIn this paper we derive from arguments of string scattering a set of eight tetrahedron equations, with different index orderings. It is argued that this system of equations is the proper system that represents integrable structures in three dimensions generalizing the Yang–Baxter equation. Under additional restrictions this system reduces to the usual tetrahedron equation in the vertex form. Most known solutions fall under this class, but it is by no means necessary. Comparison is made with the work on braided monoidal 2categories also leading to eight tetrahedron equations.

Painlevé classification of coupled Korteweg–de Vries systems
View Description Hide DescriptionIn this work, we give a classification of coupled Korteweg–de Vries equations. We found new systems of equations that are completely integrable in the sense of Painlevé.

The structure of spherically symmetric Yang–Mills fields
View Description Hide DescriptionWe summarize the algebraic structure of spherically symmetric Yang–Mills potentials for a general compact gauge group, and investigate the particular case of gauge groups with Lie algebra in detail. We develop techniques that lead to a complete classification of the possible spherical symmetry ansätze, including descriptions of the reduced gauge group Z, the space of magnetic potentials H, and for those ansätze that admit extensions across the symmetry axis, a description of the space of vacuum potentials and its little group These results are illustrated by listing all irreducible models for

Classification of spherically symmetric static space–times by their curvature collineations
View Description Hide DescriptionA complete classification of all spherically symmetric static space–times according to their curvature collineations is presented and compared with Ricci collineations of corresponding space–times.

Selfgravitating nonlinear scalar fields
View Description Hide DescriptionWe investigate the Cauchy problem for the Einstein  scalar fieldequations in asymptotically flat spherically symmetric spacetimes, in the standard formulation. We prove the local existence and uniqueness of solutions for initial data given on a spacelike hypersurface in the Sobolev space. Solutions exist globally if a central (integral) singularity does not form and/or outside an outgoing null hypersurface. An explicit example demonstrates that there exists a local evolution with a naked initial curvature singularity at the symmetry centre.