Showing papers on "Symplectic vector space published in 1993"

On some explicit formulas in the theory of Weil representation

Abstract: The object of this paper is to derive some explicit formulae concerning the Weil representation that allow us to define this projective representation in a unique manner for each choice of symplectic basis. Let F be a self-dual locally compact field of char φ 2 and X a symplectic vector space over F. Let V, V* be two transversal Lagrangian subspaces. Then a classical construction due to Shale-SegalWeil gives a projective representation of the symplectic group Sp(JSΓ) in the Schwartz-space of V. The operators ξ(σ) corresponding to each σ e Sp(X) are determined uniquely only up to a scalar multiple. The starting point of this paper is an explicit integral formula for these operators ξ(σ), valid for all σ e Sp(X). In fact (see Lemma 3.2) we have for each σ e Sp(ΛΓ) ξ(σ)φ :x-+ fσ(x, x*)φ(xa + x*y) dμσ JV*/kevγ

Recent Progress in the Theory and Application of Symplectic Integrators

Abstract: In this paper various aspect of symplectic integrators are reviewed. Symplectic integrators are numerical integration methods for Hamiltonian systems which are designed to conserve the symplectic structure exactly as the original flow. There are explicit symplectic schemes for systems of the form H = T (p) + V (q), and implicit schemes for general Hamiltonian systems. As a general property, symplectic integrators conserve the energy quite well and therefore an artificial damping (excitation) caused by the accumulation of the local truncation error cannot occur. Symplectic integrators have been applied to the Kepler problem, the motion of minor bodies in the solar system and the long-term evolution of outer planets.

Relations Among Lie Formal Series and Construction of Symplectic Integrators

Abstract: Symplectic integrators are numerical integration schemes for hamiltonian systems. The integration step is an explicit symplectic map. We find symplectic integrators using universal exponential identities or relations among formal Lie series. We give here general methods to compute such identities in a free Lie algebra. We recover by these methods all the previously known symplectic integrators and some new ones. We list all possible solutions for integrators of low order.

Loop Algebra Moment Maps and Hamiltonian Models for the Painleve Transcendants

Abstract: The isomonodromic deformations underlying the Painleve transcendants are interpreted as nonautonomous Hamiltonian systems in the dual $\gR^*$ of a loop algebra $\tilde\grg$ in the classical $R$-matrix framework. It is shown how canonical coordinates on symplectic vector spaces of dimensions four or six parametrize certain rational coadjoint orbits in $\gR^*$ via a moment map embedding. The Hamiltonians underlying the Painleve transcendants are obtained by pulling back elements of the ring of spectral invariants. These are shown to determine simple Hamiltonian systems within the underlying symplectic vector space.

Construction of symplectic schemes for wave equations via hyperbolic functions sinh(x), cosh(x) and tanh(x)

Abstract: Hamiltonian systems are canonical systems on phase space endowed with symplectic structures. The dynamical evolutions, i.e., the phase flow of the Hamiltonian systems are symplectic transformations which are area-preserving. The importance of the Hamiltonian systems and their special property require the numerical algorithms for them should preserve as much as possible the relevant symplectic properties of the original systems. Feng Kang [1–3] proposed in 1984 a new approach to computing Hamiltonian systems from the view point of symplectic geometry. He systematically described the general method for constructing symplectic schemes with any order accuracy via generating functions. A generalization of the above theory and methods for canonical Hamiltonian equations in infinite dimension can be found in [4]. Using self-adjoint schemes, we can construct schemes of arbitrary even order [5]. These schemes can be applied to wave equation [6,7] and the stability of them can be seen in [7,8]. In this paper, we will use the hyperbolic functions sinh( x ), cosh( x ) and tanh( x ) to construct symplectic schemes of arbitrary order for wave equations and stabilities of these constructed schemes are also discussed.

Isospectral flow in loop algebras and quasiperiodic solutions of the sine‐Gordon equation

Abstract: The sine‐Gordon equation is considered in the Hamiltonian framework provided by the Adler–Kostant–Symes theorem. The phase space, a finite dimensional coadjoint orbit in the dual space g* of a loop algebra g , is parameterized by a finite dimensional symplectic vector space W embedded into g* by a moment map. Real quasiperiodic solutions are computed in terms of theta functions using a Liouville generating function which generates a canonical transformation to linear coordinates on the Jacobi variety of a suitable hyperelliptic curve.

On special submanifolds in symplectic geometry

Abstract: We study the symplectic geometry of submanifolds of a manifold P equipped with a Kahler metric of non-positive curvature. We show that totally geodesic complex or isotropic submanifolds of P are standard.

Variables of the action-angle type on symplectic manifolds stratified by coisotropic tori

Abstract: A symplectic manifold is considered under the assumption that a smooth symplectic action of a commutative Lie group with compact coisotropic orbits is defined on it. The problem of existence of variables of the action-angle type is investigated with a view to giving a detailed description of flows in Hamiltonian systems with invariant Hamiltonians. We introduce the notion of a nonresonance symplectic structure for which the problem of recognition of resonance and nonresonance tori is solved.

Self-gravitating symplectic systems

Abstract: The Riemann ellipsoidal model of self-gravitating systems assumes that the velocity field is linear. The symplectic model eliminates this restrictive ansatz while still retaining much of the simplicity of the original Riemann model. The kinetic energy of a Riemann ellipsoid is its linear velocity field value; the kinetic energy in the symplectic model is the exact expression. The Riemann hydrodynamic equations form a Hamiltonian dynamical system on co-adjoint orbits of a 15-dimensional Lie subgroup GCM(3) of the noncompact symplectic group Sp(3,R). A symplectic model phase space is a co-adjoint orbit of the more general 21-dimensional group Sp(3,R)

A symplectic structure for the space of quantum field theories

Abstract: We use the formal Lie algebraic structure in the ``space'' of hamiltonians provided by equal time commutators to define a Kirillov-Konstant symplectic structure in the coadjoint orbits of the associated formal group. The dual is defined via the natural pairing between operators and states in a Hilbert space.

Equivariant localization: BV geometry and supersymmetric dynamics

Abstract: It is shown, that the geometrical objects of Batalin-Vilkovisky formalism-- odd symplectic structure and nilpotent operator $\Delta$ can be naturally uncorporated in Duistermaat--Heckman localization procedure. The presence of the supersymmetric bi-Hamiltonian dynamics with even and odd symplectic structure in this procedure is established. These constructions can be straightly generalized for the path-integral case.

Dual Isomonodromic Deformations and Moment Maps to Loop Algebras

Abstract: The Hamiltonian structure of the monodromy preserving deformation equations of Jimbo {\it et al } is explained in terms of parameter dependent pairs of moment maps from a symplectic vector space to the dual spaces of two different loop algebras. The nonautonomous Hamiltonian systems generating the deformations are obtained by pulling back spectral invariants on Poisson subspaces consisting of elements that are rational in the loop parameter and identifying the deformation parameters with those determining the moment maps. This construction is shown to lead to ``dual'' pairs of matrix differential operators whose monodromy is preserved under the same family of deformations. As illustrative examples, involving discrete and continuous reductions, a higher rank generalization of the Hamiltonian equations governing the correlation functions for an impenetrable Bose gas is obtained, as well as dual pairs of isomonodromy representations for the equations of the Painleve transcendents $P_{V}$ and $P_{VI}$.

Geometrization of classical mechanics

Abstract: In this paper, it is shown that the symplectic description of the Hamiltonian mechanics contains a gauge structure, where the symplectic form acts as connection form.

Algebraic approach to quantum field theory on non-globally-hyperbolic spacetimes

Abstract: The mathematical formalism for linear quantum field theory on curved spacetime depends in an essential way on the assumption of global hyperbolicity. Physically, what lie at the foundation of any formalism for quantization in curved spacetime are the canonical commutation relations, imposed on the field operators evaluated at a global Cauchy surface. In the algebraic formulation of linear quantum field theory, the canonical commutation relations are restated in terms of a well-defined symplectic structure on the space of smooth solutions, and the local field algebra is constructed as the Weyl algebra associated to this symplectic vector space. When spacetime is not globally hyperbolic, e.g. when it contains naked singularities or closed timelike curves, a global Cauchy surface does not exist, and there is no obvious way to formulate the canonical commutation relations, hence no obvious way to construct the field algebra. In a paper submitted elsewhere, we report on a generalization of the algebraic framework for quantum field theory to arbitrary topological spaces which do not necessarily have a spacetime metric defined on them at the outset. Taking this generalization as a starting point, in this paper we give a prescription for constructing the field algebra of a (massless or massive) Klein-Gordon field on an arbitrary background spacetime. When spacetime is globally hyperbolic, the theory defined by our construction coincides with the ordinary Klein-Gordon field theory on a

Symmetric integrable-polynomial factorization for symplectic one-turn-map tracking

Abstract: It was found that any homogeneous polynomial can be written as a sum of integrable polynomials of the same degree by which Lie transformations can be evaluated exactly. By utilizing symplectic integrators, an integrable polynomial factorization is developed to convert a symplectic map in the form of Dragt-Finn factorization into a product of Lie transformations associated with integrable polynomials. A small number of factorization bases of integrable polynomials enables one to use high-order symplectic integrators so that the high-order spurious terms can be greatly suppressed. A symplectic map can thus be evaluated with desired accuracy. >

On Hamiltonian Matrices, Symplectic Transformations, and Invariant Subspaces

Abstract: In this paper, the reduction of a Hamiltonian matrix to a condensed form using a combination of orthogonal and non-orthogonal symplectic similarity transformations is considered. Two applications of this condensed form are described. One is concerned with the computation of the eigenvalues of the Hamiltonian matrix, and the other involves the reduction of the Hamiltonian matrix to a block upper triangular (Hamiltonian-Schur) form.

Field models from two-cocycles on infinite-dimensional Lie groups and symplectic structures: 2D-gravity and Chern-Simons theory

Abstract: In this paper the connection between field models and infinite-dimensional Lie groups is widely analysed on the bases of a new group quantization approach. We also relate the Poincare-Cartan form of variational calculus to the symplectic current/structure of the covariant phase-space formulation of (higher-derivative) field theory. The Virasoro and Kac-Moody groups are considered. In the first case the action functional of the 2D-induced gravity in the light-cone formulation is derived. The hidden SL(2, R) simply appears as generated by the kernel of the Lie algebra two-cocycle and plays the role of a gauge-type symmetry. Nevertheless, it is shown that a proper space-like formulation is out of reach of the Virasoro group. The corresponding symplectic structure of the (nonlocal)action functional is determined showing that it is related to the symplectic structure associated with the SL(2, R)-Kac-Moody group. This unravels the proper geometrical meaning of the hidden symmetry and differs from the analysis in related works based on the coadjoint-orbit approach. The relation between the Kac-Moody groups and the Chern-Simons gauge theory on a disc in the presence of a source is considered using the new approach.

An optimized formulation for Deprit-type Lie transformations of Taylor maps for symplectic systems

Abstract: We present an optimized iterative formulation for directly transforming a Taylor map of a symplectic system into a Deprit-type Lie transformation, which is a composition of a linear transfer matrix and a single Lie transformation, to an arbitrary order. >