Volume 40, Issue 2, February 1999
Index of content:
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Functional versus canonical quantization of a nonlocal massive vectorgauge theory
View Description Hide DescriptionIt has been shown in literature that a possible mechanism of mass generation for gauge fields is through a topological coupling of vector and tensor fields. After integrating over the tensor degrees of freedom, one arrives at an effective massive theory that, although gauge invariant, is nonlocal. Here we quantize this nonlocal resulting theory both by path integral and canonical procedures. This system can be considered as equivalent to one with an infinite number of time derivatives and consequently an infinite number of momenta. This means that the use of the canonical formalism deserves some care. We show the consistency of the formalism we use in the canonical procedure by showing that the obtained propagators are the same as those of the (Lagrangian) path integral approach. The problem of nonlocality appears in the obtainment of the spectrum of the theory. This fact becomes very transparent when we list the infinite number of commutators involving the fields and their velocities.

Modular groups of quantum fields in thermal states
View Description Hide DescriptionFor a quantum field in a thermal equilibrium state we discuss the group generated by time translations and the modular action associated with an algebra invariant under halfsided translations. The modular flows corresponding to the algebras of the forward light cone and a spacelike wedge admit a simple geometric description in twodimensional models that factorize in lightcone coordinates. At large distances from the domain boundary compared to the inverse temperature, the flow pattern is essentially the same as time translations, whereas the zero temperature results are approximately reproduced close to the edge of the wedge and the apex of the cone. For each domain there is also a oneparameter group with a positive generator, for which the thermal state is a ground state. Formally, this may be regarded as a certain converse of the Unruh effect.

Is there a stable hydrogen atom in higher dimensions?
View Description Hide DescriptionThe Schrödinger equation in higher dimensions is considered. It consists of the kinetic energy part given by the corresponding Laplace operator, and a term describing the interaction with the electrostatic field of a point charge. From Rutherfordtype scattering experiments one can conclude that the potential of a point charge is irrespective of the dimension of the space where the experiment is carried through. Also the structure of the kinetic energy is unchanged in higher dimensions so that one is lead to the result that there exist stable atoms in higher spatial dimensions The solutions and energy eigenvalues to this Schrödinger equation in higher dimensions are presented. As a consequence, the dimensionality of space can be read off from the spectral scheme of atoms: The threedimensionality of space is a consequence of the existence of the Lyman series. Another consequence is that the Maxwell equations in higher dimensions must be modified in order to have the potential as solution for a point charge.

Implementation of an iterative map in the construction of (quasi)periodic instantons: Chaotic aspects and discontinuous rotation numbers
View Description Hide DescriptionAn iterative map of the unit disk in the complex plane is used to explore certain aspects of selfdual, fourdimensional gauge fields (quasi)periodic in the Euclidean time. These fields are characterized by two topological numbers and contain standard instantons and monopoles as different limits. The iterations do not correspond directly to a discretized time evolution of the gauge fields. They are implemented in an indirect fashion. First, being the standard coordinates, the (r,t) halfplane is mapped on the unit disk in an appropriate way. This provides an (r,t) parametrization of the starting point of the iterations and makes the iterates increasingly complex functions of r and t. These are then incorporated as building blocks in the generating function of the fields. We explain in what sense and to what extent some remarkable features of our map (indicated in the title) are thus carried over into the continuous time development of the fields. Special features for quasiperiodicity are studied. Spinor solutions and propagators are discussed from the point of view of the mapping. Several possible generalizations are indicated. Some broader topics are also discussed.

The osp(1,2)covariant Lagrangian quantization of irreducible massive gauge theories
View Description Hide DescriptionThe osp(1,2)covariant Lagrangian quantization of general gauge theories is formulated which applies also to massive fields. The formalism generalizes the Sp(2)covariant Batalin–Lavrov–Tyǔtin (BLT) approach and guarantees symplectic invariance of the quantized action. The dependence of the generating functional of Green’s functions on the choice of gauge in the massive case disappears in the limit Ward identities related to osp(1,2) symmetry are derived. Massive gauge theories with closed algebra are studied as an example.

Constructive inversion of energy trajectories in quantum mechanics
View Description Hide DescriptionWe suppose that the groundstateeigenvalue of the Schrödinger Hamiltonian in one dimension is known for all values of the coupling The potential shape is assumed to be symmetric, bounded below, and monotone increasing for A fast algorithm is devised which allows the potential shape to be reconstructed from the energy trajectory Three examples are discussed in detail: a shifted powerpotential, the exponential potential, and the sechsquared potential are each reconstructed from their known exact energy trajectories.

Superintegrability in threedimensional Euclidean space
View Description Hide DescriptionPotentials for which the corresponding Schrödinger equation is maximally superintegrable in threedimensional Euclidean space are studied. The quadratic algebra which is associated with each of these potentials is constructed and the bound statewave functions are computed in the separable coordinates.

Bethe ansatz solution of a closed spin 1 XXZ Heisenberg chain with quantum algebra symmetry
View Description Hide DescriptionA quantum algebra invariant integrable closed spin 1 chain is introduced and analyzed in detail. The Bethe ansatz equations as well as the energy eigenvalues of the model are obtained. The highest weight property of the Bethe vectors with respect to is proved.

Deconstructing supersymmetry
View Description Hide DescriptionTwo supersymmetric classical mechanical systems are discussed. Concrete realizations are obtained by supposing that the dynamical variables take values in the Grassmann algebra with two generators. The equations of motion are explicitly solved, and the action of the supergroup on the space of solutions is elucidated. The Lie algebra of the supergroup is the even part of the tensor product where G is the super Lie algebra of supersymmetries and time translations. For each system, the solutions with zero energy need to be constructed separately. For these Bogomolnytype solutions, the orbit of the supergroup is smaller than in the generic case.

Stochastic quantum geometry within a (1+1) manifold: A basic construction
View Description Hide DescriptionA bounded, globally hyperbolic, space–time manifold with topology is presented as a simple classical, deterministic, geometry, with ADM (1+1) slicing via a lapse function and shift vectors. The Cauchy development is then equivalent to the evolution of a spatial Riemann 1metric for a bounded spatial interval such that for all for all The 1metric is considered a parametrized, stochastically fluctuating variable for some critical (microscopic) correlation scale l, where is the space of all 1metrics on I. If the metric fluctuations are constrained by a scaledependent probability kernel or density distribution on [0,1], then a Fokker–Planck equation can be developed for the Cauchy evolution of the kernel. The stationary (Cauchy invariant) equilibrium limit solution is obtained. The equilibrium limit correlations at second order derived from the stochastic model, can be directly identified with the general, wellknown form of the metric twopoint (equal time) correlations obtained from linearized general relativity treated as a quantum field theory. The metric diffusion coefficients of the stochastic model are then correctly identified. The uncertainty relation for nonzero metric fluctuations δβ, emerges from the solution and is a necessary condition for the kernel to be constrained on [0,1]. The 1metric fluctuations are exponentially damped or amplified as the spatial interval is expanded or contracted with respect to the Planck length

On rotational coherent states in molecular quantum dynamics
View Description Hide DescriptionCoherent states suitable for the description of molecular rotations are developed and their connection to similar coherent states in the literature are explored. In particular their quasiclassical properties are developed. The use of such coherent states in timedependent electron nuclear dynamics studies of molecular collision processes is discussed.

Gauge theories with graded differential Lie algebras
View Description Hide DescriptionWe present a mathematical framework of gauge theories that is based upon a skewadjoint Lie algebra and a generalized Dirac operator, both acting on a Hilbert space.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Reduction of constrained systems with symmetries
View Description Hide DescriptionA general model is proposed for constrained dynamical systems on a symplectic manifold which covers, among others, the description of Lagrangian and Hamiltonian systems with nonholonomic constraints and the canonical description of mechanical systems with a singular Lagrangian. The reduction properties of these systems in the presence of symmetry are investigated within this general framework.

Gaugeinvariant variationally trivial problems on T ^{*}M
View Description Hide DescriptionA classification of variationally trivial Lagrangians on which are invariant under the Lie algebra of infinitesimal gauge transformations of the principal bundle is given. A characterization of Lagrangian densities on which are invariant under the Lie algebra of all infinitesimal automorphisms of is also obtained.

 STATISTICAL PHYSICS AND STOCHASTIC PROCESSES


On the geometric structure of thermodynamics
View Description Hide DescriptionIn this paper we consider symmetry transformations of the generalized entropy function that preserve the Gibbs oneform (Gibbs relation). We show that this symmetry consideration naturally leads to the geometric structure of thermodynamics in terms of contact geometry. We also construct an example based on the van der Waals’ fluid to illustrate the method discussed in the paper.

Green’s function of the Fokker–Planck equation: General formula of frequency expansion
View Description Hide DescriptionThe onevariable Fokker–Planck equation is studied in its general form by means of an algebraic method. An expression of the Green’s function is derived as an expansion in powers of the square root of frequency. The expansion coefficient of arbitrary order is expressed as a functional of the potential in terms of integrals.

 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


The Camassa–Holm equation as a geodesic flow on the diffeomorphism group
View Description Hide DescriptionMisiolek [J. Geom. Phys. 24, 203–208 (1998)] has shown that the Camassa–Holm equation is a geodesic flow on the Bott–Virasoro group. In this paper it is shown that the Camassa–Holm equation for the case κ=0 is the geodesic spray of the weak Riemannian metric on the diffeomorphism group of the line or the circle obtained by right translating the inner product over the entire group. This paper uses the righttrivialization technique to rigorously verify that the Euler–Poincaré theory for Lie groups can be applied to diffeomorphism groups. The observation made in this paper has led to physically meaningful generalizations of the CHequation to higher dimensional manifolds.

Large smoke rings with concentrated vorticity
View Description Hide DescriptionIn this paper we study an incompressible inviscid fluid when the initial vorticity is sharply concentrated in N disjoint regions. This problem has been well studied when a planar symmetry is present, i.e., the fluid moves in In this case we know that, when the diameter σ of each region supporting the vorticity is very small, the time evolution of the fluid is quite well described by a dynamical system with finite degrees of freedom called the “point vortex model.” In particular the connection between this model and the Euler equation has been proved rigorously as σ→0. In the present paper we discuss the “stability” of the point vortex model with respect to a particular small perturbation of the planar symmetry. More precisely we consider a fluid moving in with a cylindrical symmetry without swirl in which each vortex is no longer a straight tube, but a vorticity ring. We prove that large annuli of radii for any β>0 remain “localized” and hence we obtain the point vortex model as σ→0.

Exact periodic solutions of the complex Ginzburg–Landau equation
View Description Hide DescriptionThree new exact periodic solutions of the complex Ginzburg–Landau equation are obtained in terms of the Weierstrass elliptic function ℘. Furthermore, the new periodic solutions and other shock solutions appear as their bounded limits (along the real axis) for particular relationships between the coefficients in the equation. In the general case, bounded limits are nothing but the already known pulse, hole, and shock solutions. It is also shown that the shapes of the solutions are quite different from the shape of the usual envelope wave solution. In particular, the spatial structure of the new bounded periodic solutions varies in time, while the pulse solution may exhibit breatherlike behavior.

Similarity reduction for a class of algebraically special perfect fluids
View Description Hide DescriptionFor a class of perfect fluids first considered by Wainwright [Int. J. Theor. Phys. 10, 39 (1974)], a complete symmetry analysis of the field equations is performed. The results are used for a symmetry reduction of the field equations and the construction of (new) similarity solutions.
