Path integral approach for superintegrable potentials on spaces of nonconstant curvature: I. Darboux spaces <Emphasis Type="Italic">D</Emphasis><Subscript>I</Subscript> and <Emphasis Type="Italic">D</Emphasis><Subscript>II</Subscript>

In this paper, the Feynman path integral technique is applied for superintegrable potentials on two-dimensional spaces of nonconstant curvature: these spaces are Darboux spaces D _I and D _II. On D _I, there are three, and on D _II four such potentials. We are able to evaluate the path integral in most of the separating coordinate systems, leading to expressions for the Green functions, the discrete and continuous wave-functions, and the discrete energy-spectra. In some cases, however, the discrete spectrum cannot be stated explicitly, because it is either determined by a transcendental equation involving parabolic cylinder functions (Darboux space I), or by a higher order polynomial equation. The solutions on D _I in particular show that superintegrable systems are not necessarily degenerate. We can also show how the limiting cases of flat space (constant curvature zero) and the two-dimensional hyperboloid (constant negative curvature) emerge. The text was submitted by the authors in English.     

Nondegenerate superintegrable systems in <Emphasis Type="Italic">n</Emphasis>-dimensional Euclidean spaces

We analyze the concept of a nondegenerate superintegrable system in n-dimensional Euclidean space. Attached to this idea is the notion that every such system affords a separation of variables in one of the various types of generic elliptical coordinates that are possible in complex Euclidean space. An analysis of how these coordinates are arrived at in terms of their expression in terms of Cartesian coordinates is presented in detail. The use of well-defined limiting processes illustrates just how all these systems can be obtained from the most general nondegenerate superintegrable system in n-dimensional Euclidean space. Two examples help with the understanding of how the general results are obtained. The text was submitted by the authors in English.     

Path integral approach for superintegrable potentials on spaces of non-constant curvature: II. Darboux spaces <Emphasis Type="Italic">D</Emphasis><Subscript>III</Subscript> and <Emphasis Type="Italic">D</Emphasis><Subscript>IV</Subscript>

This is the second paper on the path integral approach of superintegrable systems on Darboux spaces, spaces of non-constant curvature. We analyze five and four superintegrable potentials in the spaces D _III and D _IV, respectively; these potentials were first given by Kalnins et al. We are able to evaluate the path integral in most of the separating coordinate systems, leading to expressions for the Green’s functions, the discrete and continuous wavefunctions, and the discrete energy spectra. In some cases, however, the discrete spectrum cannot be stated explicitly because it is determined by a higher-order polynomial equation. We also show that the free motion in a Darboux space of type III can contain bound states, provided the boundary conditions are appropriate. We can state the corresponding energy spectrum and the wavefunctions. The text was submitted by the authors in English.     

Computing the Maslov index of solitary waves, Part 1: Hamiltonian systems on a four-dimensional phase space

When solitary waves are characterized as homoclinic orbits of a finite-dimensional Hamiltonian system, they have an integer-valued topological invariant, the Maslov index. We develop a new robust numerical algorithm to compute the Maslov index, to understand its properties, and to study the implications for the stability of solitary waves. The algorithm reported here is developed in the exterior algebra representation, which leads to a fast algorithm with some novel properties. New results on the Maslov index for solitary wave solutions of reaction-diffusion equations, the fifth-order Korteweg–de Vries equation, and the long-wave–short-wave resonance equations are presented. Part 1 considers the case of a four-dimensional phase space, and Part 2 considers the case of a 2n-dimensional phase space with n>2.     

A WEIL REPRESENTATION OF sp(4) REALIZED BY DIFFERENTIAL OPERATORS IN THE SPACE OF SMOOTH FUNCTIONS ON S2 × S1

In the space of complex-valued smooth functions on S² × S¹, we explicitly realize a Weil representation of the real Lie algebra sp(4) by means of differential generators. This representation is a rare example of highest weight irreducible representation of sp(4) all whose weight spaces are 1-dimensional. We also show how this space splits into the direct sum of irreducible sl(2)-submodules. Selected applications: complete classification of yrast-band energies in even-even nuclei, the dynamical symmetry in some collective models of nuclear structure, the mapping methods for simplifying initial problem Hamiltonians.     

Classical thermodynamics formulated as a field theory in the representative space

Summary The classical equations of thermodynamics are here rewritten in the form of a field equation in then-dimensional space of the state variables. This result is obtained by proving the validity of a theorem for Pfaffian forms inn-dimensional spaces. By using the field expression thus obtained, it becomes possible to reformulate classical thermodynamics in a very compact way. Due to the relevance of its scientific content, this paper has been given priority by the Journal Direction.     

Einstein metrics onS 3,R 3 andR 4 bundles

Starting from a 4n-dimensional quaternionic Kähler base space, we construct metrics of cohomogeneity one in (4n+3) dimensions whose level surfaces are theS ² bundle space of almost complex structures on the base manifold. We derive the conditions on the metric functions that follow from imposing the Einstein equation, and obtain solutions both for compact and non-compact (4n+3)-dimensional spaces. Included in the non-compact solutions are two Ricci-flat 7-dimensional metrics withG ₂ holonomy. We also discuss two other Ricci-flat solutions, one on theR ⁴ bundle overS ³ and the other on anR ⁴ bundle overS ⁴. These haveG ₂ and Spin(7) holonomy respectively.     

Multidimensional mixmaster models

The problem of chaotic behaviour in multidimensional mixmaster models is discussed. We classify n-dimensional homogeneous spaces possessing the structure of the product M³×B, where M³ is a three-dimensional homogenous space. We show that compactness of the microspace B is for dimension 3<n⩽10 the sufficient condition for the chaotic regime to disappear. We characterize the class of spaces of dimension n>10 for which the chaotic regime does not exist.     

Ambient space formulations and statistical mechanics of holonomically constrained Langevin systems

The most classic approach to the dynamics of an n-dimensional mechanical system constrained by d independent holonomic constraints is to pick explicitly a new set of (n − d) curvilinear coordinatesparametrizingthe manifold of configurations satisfying the constraints, and to compute the Lagrangian generating the unconstrained dynamics in these (n − d) configuration coordinates. Starting from this Lagrangian an unconstrained Hamiltonian H(q,p) on 2(n−d) dimensional phase space can then typically be defined in the standard way via a Legendre transform. Furthermore, if the system is in contact with a heat bath, the associated Langevin and Fokker-Planck equations can be introduced. Provided that an appropriate fluctuation-dissipation condition is satisfied, there will be a canonical equilibrium distribution of the Gibbs form exp(−βH) with respect to the flat measure dqdp in these 2(n − d) dimensional curvilinear phase space coordinates. The existence of (n − d) coordinates satisfying the constraints is often guaranteed locally by an implicit function theorem. Nevertheless in many examples these coordinates cannot be constructed in any tractable form, even locally, so that other approaches are of interest. In ambient space formulations the dynamics are defined in the full original n-dimensional configuration space, and associated 2n-dimensional phase space, with some version of Lagrange multipliers introduced so that the 2(n − d) dimensional sub-manifold of phase space implied by the holonomic constraints and their time derivative, is invariant under the dynamics. In this article we review ambient space formulations, and explain that for constrained dynamics there is in fact considerable freedom in how a Hamiltonian form of the dynamics can be constructed. We then discuss and contrast the Langevin and Fokker-Planck equations and their equilibrium distributions for the different forms of ambient space dynamics.     

Super p-branes

It is shown that the extension of the spacetime supersymmetric Green-Schwarz covariant action p-dimensional extended objects (p-branes) is possible if and only if the on-shell p-dimensional Bose and Fermi degrees of freedom are equal. This is further evidence for world-tube supersymmetry in these models. All the p-branes models are related to superstring actions in d=3, 4, 6 or 10 dimensions by double dimensional reduction (which we generalise to reduction on arbitrary compact spaces), and we also show how they may be considered as topological defects of supergravity theories.     

On classification of conformally flat spaces

Classification of conformally flat n-dimensional pseudo-Riemannian spaces via Plebanski's method is discussed. It is based on embedding these spaces into a flat (n + 2)-dimensional space and on finding their minimal isometry groups which are subgroups of the conformal group. In particular, the case n = 4 is given in full detail and compared with incomplete results known in the literature. The found conformally flat spacetimes are identified with the associated solutions of the Einstein equations and with the spacetimes used in various cosmological considerations.     

Discrete monopoles and instantons over projective spaces

We generalize Manton's construction of discrete monopoles in Minkowski space to their analog in CP(n). Topological charge, analogous to the first Chern number in the smooth bundle, is obtained for the corresponding discrete bundle and is shown to be Q=±1. We also discuss the discretization of the smooth sphere bundles over the real projective space RP(n) and the quaternionic projective space HP(n). Finally, we make a conjecture of the discretization of the smooth sphere bundles over the discrete projective spaces R ^2k P(n) for all positive integers k and n.     

Symmetry and Topology in Quantum Logic

A test space is a collection of non-empty sets, usually construed as the catalogue of (discrete) outcome sets associated with a family of experiments. Subject to a simple combinatorial condition called algebraicity, a test space gives rise to a “quantum logic”—that is, an orthoalgebra. Conversely, all orthoalgebras arise naturally from algebraic test spaces. In non-relativistic quantum mechanics, the relevant test space is the set ℱ F(H) of frames (unordered orthonormal bases) of a Hilbert space H. The corresponding logic is the usual one, i.e., the projection lattice L(H) of H. The test space ℱ F(H) has a strong symmetry property with respect to the unitary group of H, namely, that any bijection between two frames lifts to a unitary operator. In this paper, we consider test spaces enjoying the same symmetry property relative to an action by a compact topological group. We show that such a test space, if algebraic, gives rise to a compact, atomistic topological orthoalgebra. We also present a construction that generates such a test space from purely group-theoretic data, and obtain a simple criterion for this test space to be algebraic. PACS: 02.10.Ab; 02.20.Bb; 03.65.Ta.     

Phase structure of two-dimensional spin models and percolation

For a class of classical spin models in 2D satisfying a certain continuity constraint it is proven that some of their correlations do not decay exponentially. The class contains discrete and continuous spin systems with Abelian and non-Abelian symmetry groups. For the discrete models our results imply that they show either long-range order or are in a soft phase characterized by powerlike decay of correlations; for the continuous models only the second possibility exists. The continuous models include a version of the plane rotator [O(2)] model; for this model we rederive, modulo two conjectures, the Fröhlich-Spencer result on the existence of the Kosterlitz-Thouless phase in a very simple way. The proof is based on percolation-theoretic and topological arguments.     

The topological meaning of non-abelian anomalies

We show how the non-abelian anomaly for gauge fields coupled to Weyl fermions in 2n dimensions is related to the non-trivial topology of gauge orbit space. The form of the anomaly and its normalization are shown to follow from a familiar index theorem for a certain (2n + 2)-dimensional Dirac operator. We are thus able to recover and give topological meaning to a variety of results concerning anomalies in 4- and higher-dimensional theories.     

Exact solutions in gravity with a sigma model source

We consider a D-dimensional model of gravity with non-linear “scalar fields” as a matter source. The model is defined on the product manifold M, which contains n Einstein factor spaces. General cosmological type solutions to the field equations are obtained when n − 1 factor spaces are Ricci-flat, e.g. when one space M ₁ of dimension d ₁ > 1 has nonzero scalar curvature. The solutions are defined up to solutions to geodesic equations corresponding to a sigma model target space. Several examples of sigma models are presented. A subclass of spherically symmetric solutions is studied and a restricted version of “no-hair theorem” for black holes is proved. For the case d ₁ = 2 a subclass of latent soliton solutions is singled out.     

Self-Similar Asymptotics for the Boltzmann Equation with Inelastic and Elastic Interactions

We consider some questions related to the self-similar asymptotics in the kinetic theory of both elastic and inelastic particles. In the second case we have in mind granular materials, when the model of hard spheres with inelastic collisions is replaced by a Maxwell model, characterized by a collision frequency independent of the relative speed of the colliding particles. We first discuss how to define the n-dimensional (n = 1,2,...) inelastic Maxwell model and its connection with the more basic Boltzmann equation for inelastic hard spheres. Then we consider both elastic and inelastic Maxwell models from a unified viewpoint. We prove the existence of (positive in the inelastic case) self-similar solutions with finite energy and investigate their role in large time asymptotics. It is proved that a recent conjecture by Ernst and Brito devoted to high energy tails for inelastic Maxwell particles is true for a certain class of initial data which includes Maxwellians. We also prove that the self-similar asymptotics for high energies is typical for some classes of solutions of the classical (elastic) Boltzmann equation for Maxwell molecules. New classes of (not necessarily positive) finite-energy eternal solutions of this equation are also studied.     

Space-time as a micromorphic continuum

We review generally-covariant Lagrangians for the field of linear coframes in ann-dimensional manifold. Discussed are Lagrangians invariant under the internal groupGL(n, ) and under its pseudo-Euclidean subgroups. It is shown that group spaces of semisimple Lie groups and certain of their modifications are natural vacuumlike solutions for allGL(n, )-invariant models. In some sense the signature of space-time may be interpreted as a consequence of differential equations; the velocity of light is an integration constant.     

Non-Abelian isometries of multidimensional universes

A field theory on a(d + n)-dimensional manifold in the presence of ann-dimensional isometry group spanningn-dimensional orbit spaces may be reduced to a field theory on ad-dimensional manifold. The field content of such reduced theories is completely worked out when the isometries may be non-Abelian and the resultant space may have torsion. The equations of motion of the dimensionally reduced theory are obtained directly from the higher-dimensional theory. The reduced theory is given in terms of the metric tensor, a set of scalar fields, and a set of antisymmetric tensor fields.Supported in part by the Department of Energy under Contract DE-AS-2-76ER02978 and in part by the National Science Foundation under Grant NSF Phy 83 134 10.