Showing papers on "Canonical transformation published in 1997"

Few remarks on Backlund transformations for many-body systems

Abstract: Using the n-particle periodic Toda lattice and the relativistic generalization due to Ruijsenaars of the elliptic Calogero-Moser system as examples, we revise the basic properties of the Backlund transformations (BT's) from the Hamiltonian point of view. The analogy between BT and Baxter's quantum Q-operator pointed out by Pasquier and Gaudin is exploited to produce a conjugated variablefor the parameter λ of the BT Bsuch thatbelongs to the spectrum of the Lax operator L(λ). As a consequence, the generating function of the composition B�1 ◦ . . . ◦ Bn of n BT's gives rise also to another canonical transformation separating variables for the model. For the Toda lattice the dual BT parametrized byis introduced.

Dirac constraint quantization of a dilatonic model of gravitational collapse

Abstract: We present an anomaly-free Dirac constraint quantization of the string-inspired dilatonic gravity (the CGHS model) in an open 2-dimensional spacetime. We show that the quantum theory has the same degrees of freedom as the classical theory; namely, all the modes of the scalar field on an auxiliary flat background, supplemented by a single additional variable corresponding to the primordial component of the black hole mass. The functional Heisenberg equations of motion for these dynamical variables and their canonical conjugates are linear, and they have exactly the same form as the corresponding classical equations. A canonical transformation brings us back to the physical geometry and induces its quantization.

Poisson-Lie T-duality and supersymmetry

Abstract: We review aspects of Poisson-Lie T-duality which we explicitly formulate as a canonical transformation on the world-sheet. Extensions of previous work on T-duality in relation to supersymmetry are also discussed.

Two-hole problem in the t-J model: A canonical transformation approach

Abstract: The t-J model in the spinless-fermion representation is studied. An effective Hamiltonian for the quasiparticles is derived using a canonical transformation approach. It is shown that the rather simple form of the transformation generator allows one to take into account the effect of hole interactions with the short-range spin waves and to describe the single-hole ground state. Obtained results are very close to ones of the self-consistent Born approximation. Further accounting of the long-range spin-wave interaction is possible on a perturbative basis. Spin-wave exchange and an effective interaction due to minimization of the number of broken antiferromagnetic bonds are included in the effective quasiparticle Hamiltonian. The two-hole bound state problem is solved using a Bethe-Salpeter equation. The only bound state found to exist in the region of 1{lt}(t/J){lt}5 is the d wave. Both types of the hole-hole interaction are important for its formation. A discussion of the possible relation of the obtained results to the problem of superconductivity in real systems is presented. {copyright} {ital 1997} {ital The American Physical Society}

Hamiltonian thermodynamics of a Lovelock black hole

Abstract: We consider the Hamiltonian dynamics and thermodynamics of spherically symmetric spacetimes within a one-parameter family of five-dimensional Lovelock theories. We adopt boundary conditions that make every classical solution part of a black hole exterior, with the spacelike hypersurfaces extending from the horizon bifurcation three-sphere to a timelike boundary with fixed intrinsic metric. The constraints are simplified by a Kucha\ifmmode \check{r}\else \v{r}\fi{}-type canonical transformation, and the theory is reduced to its true dynamical degrees of freedom. After quantization, the trace of the analytically continued Lorentzian time evolution operator is interpreted as the partition function of a thermodynamical canonical ensemble. Whenever the partition function is dominated by a Euclidean black hole solution, the entropy is given by the Lovelock analogue of the Bekenstein-Hawking entropy; in particular, in the low temperature limit the system exhibits a dominant classical solution that has no counterpart in Einstein's theory. The asymptotically flat space limit of the partition function does not exist. The results indicate qualitative robustness of the thermodynamics of five-dimensional Einstein theory upon the addition of a nontrivial Lovelock term.

Gauge invariant Hamiltonian formalism for spherically symmetric gravitating shells

Abstract: The dynamics of a spherically symmetric thin shell with arbitrary rest mass and surface tension interacting with a central black hole is studied. A careful investigation of all classical solutions reveals that the value of the radius of the shell and of the radial velocity as an initial datum does not determine the motion of the shell; another configuration space must, therefore, be found. A different problem is that the shell Hamiltonians used in the literature are complicated functions of momenta (nonlocal) and they are gauge dependent. To solve these problems, the existence is proved of a gauge-invariant super-Hamiltonian that is quadratic in momenta and that generates the shell equations of motion. The true Hamiltonians are shown to follow from the super-Hamiltonian by a reduction procedure including a choice of gauge and solution of constraint; one important step in the proof is a lemma stating that the true Hamiltonians are uniquely determined (up to a canonical transformation) by the equations of motion of the shell, the value of the total energy of the system, and the choice of time coordinate along the shell. As an example, the Kraus-Wilczek Hamiltonian is rederived from the super-Hamiltonian. The super-Hamiltonian coincides with that of a fictitious particle moving in a fixed two-dimensional Kruskal spacetime under the influence of two effective potentials. The pair consisting of a point of this spacetime and a unit timelike vector at the point, considered as an initial datum, determines a unique motion of the shell.

Group theoretical approach in using canonical transformations and symplectic geometry in the control of approximately modelled mechanical systems interacting with an unmodelled environment

Abstract: In spite of its simpler structure than that of the Euler-Lagrange equations-based model, the Hamiltonian formulation of Classical Mechanics (CM) gained only limited application in the Computed Torque Control (CTC) of the rather conventional robots. A possible reason for this situation may be, that while the independent variables of the Lagrangian model are directly measurable by common industrial sensors and encoders, the Hamiltonian canonical coordinates are not measurable and can also not be computed in the lack of detailed information on the dynamics of the system under control. As a consequence, transparent and lucid mathematical methods bound to the Hamiltonian model utilizing the special properties of such concepts as Canonical Transformations, Symplectic Geometry, Symplectic Group, Symplectizing Algorithm, etc. remain out of the reach of Dynamic Control approaches based on the Lagrangian model. In this paper the preliminary results of certain recent investigations aiming at the introduction of these methods in dynamic control are summarized and illustrated by simulation results. The proposed application of the Hamiltonian model makes it possible to achieve a rigorous deductive analytical treatment up to a well defined point exactly valid for a quite wide range of many different mechanical systems. From this point on it reveals such an ample assortment of possible non-deductive, intuitive developments and approaches even within the investigations aiming at a particular paradigm that publication of these very preliminary and early results seems to have definite reason, too.

D-brane Dualities as Canonical Transformations

Abstract: We show that the SL(2,R) duality symmetry of type IIB superstring theory can be formulated as the canonical transformation interchanging momenta and magnetic degrees of freedom associated to the abelian world-volume gauge field of the D-3-brane. D-strings are shown to be connected under the corresponding transformation in the world-sheet to the (m,n) family of string solutions of type IIB supergravity constructed by Schwarz. For the type IIA superstring the D-2-brane is mapped under the three dimensional world-volume electric-magnetic duality to the dimensional reduction of the membrane of M-theory.

Canonical transform for tracking with kinematic models

Abstract: A canonical transform is presented that converts a coupled or uncoupled kinematic model for target tracking into a decoupled dimensionless canonical form. The coupling is due to non-zero off-diagonal terms in the covariance matrices of the process noise and/or the measurement noise, which can be used to model the coupling of motion and/or measurement between coordinates. The decoupled dimensionless canonical form is obtained by simultaneously diagonalizing the noise covariance matrices, followed by a spatial-temporal normalization procedure. This canonical form is independent of the physical specifications of an actual system. Each subsystem corresponding to a canonical coordinate is characterized by its process noise standard deviation, called the maneuver index as a generalization of the tracking index for target tracking, which characterizes completely the performance of a steady-state Kalman filter. A number of applications of this canonical form are discussed. The usefulness of the canonical transform is illustrated via an example of performance analysis of maneuvering target tracking in an air traffic control (ATC) system.

Analysis and stabilization of nonlinear chaotic systems

Abstract: The qualitative analysis problem for chaotic dynamics of nonlinear control systems and stabilization of unstable periodic orbits (UPO stabilization) and stationary states were considered in the paper. Nonlinear control systems are analysed by the canonical transformation methods of affine systems and by the localization of periodic orbits method based on properties of Lie derivatives along smooth vector fields. It is proved that Lorenz and Rossler systems are not equivalent in the affine classification. For chaotic systems, a general approach to UPO stabilization and stabilization of stationary states based on canonical transformations was proposed. Estimates for chaotic regions in the phase space for Lorenz and Rossler systems were obtained. Relations between canonical transformations, localization, and UPO stabilization problems were studied.

Canonical transformations to warped surfaces: Correction of aberrated optical images

Abstract: The projection of optical images on warped screens is a canonical transformation of phase space between flat and warped evolution-parameter surfaces. In mechanics, the evolution parameter is time; in geometric optics it is the optical axis of coordinate space. We consider the specific problem of bringing to focus an axis-symmetric aberrating optical system by warping the output screen. The solution for the surface curvature coefficients is given in terms of the Lie aberration coefficients of the system; a linear optimization strategy applies.

q-Schrödinger Equations for V = u 2 + 1/u 2 and Morse Potentials in Terms of the q-Canonical Transformation

Abstract: The realizations of the Lie algebra corresponding to the dynamical symmetry group SO(2,1) of the Schrodinger equations for the Morse and the V = u2 + 1/u2 potentials were known to be related by a canonical transformation. q-deformed analog of this transformation connecting two different realizations of the slq(2) algebra is presented. By the virtue of the q-canonical transformation, a q-deformed Schrodinger equation for the Morse potential is obtained from the q-deformed V = u2 + 1/u2 Schrodinger equation. Wave functions and eigenvalues of the q-Schrodinger equations yielding a new definition of the q-Laguerre polynomials are studied.

Canonical gauges in the path integral for parametrized systems

Abstract: It is well known that—differing from ordinary gauge systems—canonical gauges are not admissible in the path integral for parametrized systems. This is the case for the relativistic particle and gravitation. However, a time dependent canonical transformation can turn a parametrized system into an ordinary gauge system. It is shown how to build a canonical transformation such that the fixation of the new coordinates is equivalent to the fixation of the original ones; this aim can be achieved only if the Hamiltonian constraint allows for an intrinsic global time. Thus the resulting action, describing an ordinary gauge system and allowing for canonical gauges, can be used in the path integral for the quantum propagator associated with the original variables.

Finding the Hamiltonian for Cosmological Models in Fourth-Order Gravity Theories Without Resorting to the Ostrogradski or Dirac Formalism

Abstract: The Hamilton formalism of cosmological models in fourth-order theories of gravity is considered. An approach to constructing the Hamilton function is presented which starts by replacing the second order derivatives of configuration space coordinates by functions depending on these coordinates, its first order derivatives, and additional variables playing the role of configuration space coordinates. This formalism, which does not resort to the Ostrogradski or Dirac formalism, is elucidated and applied to examples. For a special class of Lagrange functions, it is demonstrated that the canonical coordinates of the considered formalism and of the Ostrogradski formalism are related via a canonical transformation. The canonical transformation is a transformation of the configuration space coordinates and a transformation of momentum components induced by the transformation of the configuration space coordinates for a special element of the class of Lagrange functions mentioned. The Wheeler-DeWitt equations belonging to this Lagrange function are related via minisuperspace coordinate transformations.

Non Abelian T-duality for open strings

Abstract: In the first part of the talk we discuss T-duality for a free boson on a world sheet with boundary in a setting suitable for the generalization to non-trivial backgrounds. The gauging method as well as the canonical transformation are considered. In both cases Dirichlet strings as T-duals of Neumann strings arise in a generic way. In the second part the gauging method is employed to construct the T-dual of a model with non-Abelian isometries.

Gauge transformations are canonical transformations

Abstract: It is shown that a gauge transformation is also a canonical transformation which preserves Hamilton's equation of motion, where and L are two different choices of Lagrangian of a dynamical system, and f is an arbitrary function of coordinates q and time t. Resumen. Sa presenta una derivacion sencilla y directa de los propiadades de transformaciones canonicas empezando de transformaciones `gauge'. Dicha derivacion no necesita la aplicacion de principios de Hamilton ni el calculo de variaciones.

Canonical equivalence of a generic 2-d dilaton gravity model and a free field theory

Abstract: We show that a canonical transformation converts, up to a boundary term, a generic 2-D dilaton gravity model into a free field theory with a Minkowskian target space.

Fast symplectic mapping and long-term stability near broad resonances

Abstract: Fast symplectic mapping, based on a canonical generator of the full-turn map in polar coordinates (I, {Phi}), is a powerful tool to study long-term stability in large hadron storage rings. Accurate maps for realistic lattices can be constructed in a few hours on a workstation computer, and can be iterated to follow orbits for 10{sup 7} turns in less than 4 hours. Orbits of the map can also be used in a non-perturbative construction of quasi-invariant actions. By bounding the small changes in quasi-invariants along many short orbits, one can derive conservative estimates of survival time for long orbits, for any initial condition in a region of phase space. A first quasi-invariant vector, J, arises from a canonical transformation (1, {Phi}) {r_arrow} (J, {Psi}), based on interpolation of invariant tori surrounding the origin. The variation of J is relatively large near a broad resonance. In such a region a second canonical transformation, associated with pendulum-like motion in appropriate variables, is required to produce a good quasi-invariant. This quasi-invariant is used to set a long-term bound on motion near a broad, large-amplitude resonance in a realistic model of the Large Hadron Collider (LHC). Interesting general properties of the pseudo-pendulum motion are studied.

On the Canonical Equivalence of Liouville and Free Fields

Abstract: We obtain the parity invariant generating functional for the canonical transformation mapping the Liouville theory into a free scalar field and explain how it is related to the pseudoscalar transformation

Smooth Bosonization as a Quantum Canonical Transformation

Abstract: We consider a 1+1 dimensional field theory which contains both a complexfermion field and a real scalar field. We then construct a unitary operatorthat, by a similarity transformation, gives a continuum of equivalent theorieswhich smoothly interpolate between the massive Thirring model and the sine-Gordon model. This provides an implementation of smooth bosonizationproposed by Damgaard et al. as well as an example of a quantum canonicaltransformation for a quantum field theory.I. INTRODUCTION Duality, or the quantum equivalence of field theories, allows one to relate quantities inone theory, such as the particle spectrum and Green’s functions, to those of another theory.This concept is most useful when duality maps a theory with a strong coupling, in whichperturbation theory is invalid, to one with a weak coupling, in which a perturbative calcula-tion may be performed. Unfortunately, most such duality transformations are hypotheticalsince an explicit operator mapping is absent. Duality is usually demonstrated either by

T duality for boundary-non-critical strings

Abstract: Recent work on the action of T duality on Dirichlet-branes is generalized to the case in which the open string satisfies boundary conditions that are neither Neumann nor Dirichlet. This is achieved by implementing T duality as a canonical transformation of the $\sigma$-model path integral. A class of boundary interactions that violate conformal symmetry is found to be T-dual of a correspondingly non-conformal class of boundary conditions. The analogy with some problems in boundary-non-critical quantum mechanics of interest for condensed matter is pointed out.

Normal forms of quadratic bose operators

Abstract: We construct a proper canonical transformation that reduces the quadratic Bose operator to a direct sum of finite-dimensional quadratic operators each of which can be reduced by a finite-dimensional canonical transformation to one of the standard forms corresponding to the standard forms of real quadratic Hamiltonians.

Lax pair tensors and integrable spacetimes

Abstract: The use of Lax pair tensors as a unifying framework for Killing tensors of arbitrary rank is discussed. Some properties of the tensorial Lax pair formulation are stated. A mechanical system with a well-known Lax representation -- the three-particle open Toda lattice -- is geometrized by a suitable canonical transformation. In this way the Toda lattice is realized as the geodesic system of a certain Riemannian geometry. By using different canonical transformations we obtain two inequivalent geometries which both represent the original system. Adding a timelike dimension gives four-dimensional spacetimes which admit two Killing vector fields and are completely integrable.

The annular twist mapping of the restricted three-body problem

Abstract: This paper presents a new derivation of the twist mapping in the planar restricted problem. It differs from other treatments in the use of a novel canonical transformation which allows for the utilization of symplectic reduction techniques.

1+1 string with quarks at the ends revisited

Abstract: The canonical description is presented for the string with pointlike masses at the ends in 1+1 dimensions in two different gauges: in the proper time gauge and in the light cone one. The classical canonical transformation is written out explicitly, which relates physical variables in both gauges, and equivalence of two classical theories is demonstrated in such a way. Both theories are quantized, and it is shown that quantum theories are not unitary equivalent. It happens due to the fact that the canonical transformation depends on interaction. The quantum Poincare algebra proves to be closed in both cases, so that the requirement of Poincare covariance is not able to distinguish between two versions of the theory.

Path Integral Localization of the Laplacian on Lie Groups and Selberg's Trace Formula

Abstract: We introduce a new localization principle which is a generalized canonical transformation. It unifies BRST localization, the non-abelian localization principle and a special case of the conformal Duistermaat-Heckman integration formula of Paniak, Semenoff and Szabo. The heat kernel on compact Lie groups is localized in two ways. First using a non-abelian generalization of the derivative expansion localization of Palo and Niemi and secondly using the BRST localization principle and a configuration space path integral. In addition we present some new formulas on homogeneous spaces which might be useful in a possible localization of Selberg's trace formula on locally homogeneous spaces.

Analytical Effects of Gravitational Waves on the Motion of an Artificial Satellite

Abstract: The motion of an artificial satellite in the Earth’s gravitational field is discussed in the post-Newtonian framework including the effect of weak gravitation waves using the perturbation technique of the canonical Lie-transformations Two successive canonical transformations are used to derive analytical expressions for the short-period, long-period and secular perturbations of orbital elements The solution is expressed in terms of the Delaunay variables