Showing papers on "Canonical transformation published in 2000"
TL;DR: In this article, a high density theorem of invariant tori is derived by the classical canonical transformation method extending previous results, and the existence of long heteroclinic chains and of Arnol'd diffusion is proved for systems interacting through a trigonometric polynomial in the angle variables.
Abstract: Interacting systems consisting of two rotators and a point mass near a hyperbolic fixed point are considered, in a case in which the uncoupled systems have three very different characteristic time scales. The abundance of quasi periodic motions in phase space is studied via the Hamilton-Jacobi equation. The main result, a high density theorem of invariant tori, is derived by the classical canonical transformation method extending previous results. As an application the existence of long heteroclinic chains (and of Arnol'd diffusion) is proved for systems interacting through a trigonometric polynomial in the angle variables.
25 citations
TL;DR: In this paper, the internal vibrational states of the large bipolaron are studied by using the strong coupling Bogolyubov-Tyablikov canonical transformation of coordinates, that takes into account the conservation of the total momentum of the system.
Abstract: The internal vibrational states of the large bipolaron are studied by using the strong coupling Bogolyubov-Tyablikov canonical transformation of coordinates, that takes into account the conservation of the total momentum of the system. The complete spectrum is explicitly calculated and the electronic and vibrational properties of the states are discussed. Moreover the comparison of the bipolaron ground state and the binding energies with the results of the path integral and the Lee-Low and Pines methods shows that the proposed approach provides a correct description of the large bipolaron even in the intermediate regime of the electron-phonon interaction.
23 citations
TL;DR: In this article, the manifest form of the symmetry and the corresponding Noether identities are obtained in the first order formalism as well as in the Hamiltonian one, and it is proven that the symmetry can always be presented in the form of canonical transformation for the phase space variables.
Abstract: We study in the Hamiltonian framework the local transformations which leave invariant the Lagrangian action: δeS=div. Manifest form of the symmetry and the corresponding Noether identities is obtained in the first order formalism as well as in the Hamiltonian one. The identities have very simple form and interpretation in the Hamiltonian framework. Part of them allows one to express the symmetry generators which correspond to the primarily expressible velocities through the remaining one. The other part of the identities allows one to select subsystem of constraints with a special structure from the complete constraint system. It means, in particular, that the above written symmetry implies an appearance of the Hamiltonian constraints up to at least ([k]+1) stage. It is proven also that the Hamiltonian symmetries can always be presented in the form of canonical transformation for the phase space variables. The manifest form of the resulting generating function is obtained.
22 citations
TL;DR: In this paper, a conformal sigma model is described by two fields related to the Arnowitt-Deser-Misner mass and to the radius of the (N − 2)-dimensional (hyper)sphere, respectively.
Abstract: Vacuum spherically symmetric Einstein gravity in N ≥ 4 dimensions can be cast in a two-dimensional conformal nonlinear sigma model form by first integrating on the (N − 2)-dimensional (hyper)sphere and then performing a canonical transformation. The conformal sigma model is described by two fields which are related to the ArnowittDeser-Misner mass and to the radius of the (N −2)-dimensional (hyper)sphere, respectively. By quantizing perturbatively the theory we estimate the quantum corrections to the ADM mass of a black hole.
18 citations
TL;DR: In this article, a set of on shell duality equations is proposed that leads to a map between strings moving on symmetric spaces with opposite curvatures, which preserves the energy momentum tensor though it is not a canonical transformation.
18 citations
TL;DR: In this article, the authors derived an effective intra-ladder charge spin Hamiltonian for the quarter-filled ladder compound α'-NaV2O5 by using the standard canonical transformation method.
Abstract: An effective intra- and inter-ladder charge-spin Hamiltonian for the quarter-filled ladder compound α'-NaV2O5 has been derived by using the standard canonical transformation method In the derivation, it is clear that a finite inter-site Coulomb repulsion is needed to get a meaningful result otherwise the perturbation becomes ill-defined Various limiting cases depending on the values of the model parameters have been analyzed in detail and the effective exchange couplings are estimated We find that the effective intra-ladder exchange may become ferromagnetic for the case of zig-zag charge ordering in a purely electronic model We estimate the magnitude of the effective inter-rung Coulomb repulsion in a ladder and find it to be about one-order of magnitude too small in order to stabilize charge-ordering
16 citations
TL;DR: In this paper, a canonical transformation to a new set of non-singular variables is performed, in order to avoid small divisors in the system of equations, and numerical estimations of the effect of the dissipation are given in form of tables and graphics.
Abstract: The study of the rotation of a non-rigid, non-symmetrical Earth with a heterogeneous and stratified liquid core was recently accomplished by Gonzalez and Getino (1997) through the Hamiltonian formalism. In this work that model is extended by including the effect of the dissipation arising from the mantle–core interaction due to the viscous and electromagnetic coupling. A canonical transformation to a new set of non-singular variables is performed, in order to avoid small divisors in the system of equations. Numerical estimations of the effect of the dissipation are given in form of tables and graphics, and the significance of this effect is discussed.
14 citations
TL;DR: In this article, the reparametrization-invariant generating functional for the unitary and causal perturbation theory in general relativity in a finite space-time is obtained.
Abstract: The reparametrization-invariant generating functional for the unitary and causal perturbation theory in general relativity in a finite space-time is obtained. The region of validity of the Faddeev-Popov-DeWitt functional is studied. It is shown that the invariant content of general relativity as a constrained system can be covered by two "equivalent" unconstrained systems: the "dynamic" (with "dynamic" evolution parameter as the metric scale factor) and "geometric" (given by the Levi-Civita type canonical transformation to the action-angle variables where the energy constraint converts into a new momentum). "Big Bang", the Hubble evolution, and creation of matter fields by the "geometric" vacuum are described by the inverted Levi-Civita (LC) transformation of the geomeric system into the dynamic one. The particular case of the LC transformations are the Bogoliubov ones of the particle variables (diagonalizing the dynamic Hamiltonian) to the quasiparticles (diagonalizing the equations of motion). The choice of initial conditions for the "Big Bang" in the form of the Bogoliubov (squeezed) vacuum reproduces all stages of the evolution of the Friedmann-Robertson-Walker Universe in their conformal (Hoyle-Narlikar) versions.
13 citations
TL;DR: In this article, a set of on shell duality equations is proposed that leads to a map between strings moving on symmetric spaces with opposite curvatures, which preserves the energy momentum tensor though it is not a canonical transformation.
Abstract: A set of on shell duality equations is proposed that leads to a map between strings moving on symmetric spaces with opposite curvatures. The transformation maps "waves" on a riemannian symmetric space to "waves" on its dual riemannian symmetric space. This transformation preserves the energy momentum tensor though it is not a canonical transformation. The preservation of the energy momentum tensor has a natural geometrical interpretation. The transformation maps "particle-like solutions" into static "soliton-like solutions". The results presented here generalize earlier results of E. Ivanov.
12 citations
TL;DR: In this article, a collective-variable theory for nonlinear coherent excitations based on Dirac's formalism of constrained Hamiltonian systems is proposed. No approximations are involved in this theory in the sense that they are able to formulate equations of motion for a new set of variables, including the collective variables themselves, which are equivalent to the original set of Hamilton's equation of motion.
11 citations
TL;DR: In this paper, it is shown how W-algebras emerge from very peculiar canonical transformations with respect to the canonical symplectic structure on a compact Riemann surface.
Abstract: It is shown how W-algebras emerge from very peculiar canonical transformations with respect to the canonical symplectic structure on a compact Riemann surface. The action of smooth diffeomorphisms of the cotangent bundle on suitable generating functions is written in the BRS framework while a W-symmetry is exhibited. Subsequently, the complex structure of the symmetry spaces is studied and the related BRS properties are discussed. The specific example of the so-called W3-algebra is treated in relation to some other different approaches.
TL;DR: In this paper, the procedure for constructing the related inverse counterpart for a given bi-Hamiltonian separable chain is presented, where the inverse counterpart is constructed by constructing the corresponding inverse chain.
Abstract: The procedure for constructing the related inverse counterpart for a given bi-Hamiltonian separable chain is presented.
TL;DR: In this article, the authors studied T-duality for open strings with non-zero B-field on them from the point of view of canonical transformations, and showed that the Sl (2, Z ) N transformations can be understood as a sub-class of canonical transformation on the open strings in the B field background.
TL;DR: In this paper, a general stabilization strategy for port-controlled Hamiltonian systems based on the generalized canonical transformation is presented, which is a natural generalization of well-known passivity based control.
TL;DR: In this paper, the authors studied the Cu5O4 cluster by exact diagonalization of a three-band Hubbard model and showed that bound electron or hole pairs are obtained at appropriate fillings, and produce superconducting flux quantization.
Abstract: We study the Cu5O4 cluster by exact diagonalization of a three-band Hubbard model and show that bound electron or hole pairs are obtained at appropriate fillings, and produce superconducting flux quantization. The results extend earlier cluster studies and illustrate a canonical transformation approach to pairing that we have developed recently for the full plane. The quasiparticles that in the many-body problem behave like Cooper pairs are W
=0 pairs, that is, two-hole eigenstates of the Hubbard Hamiltonian with vanishing on-site repulsion. The cluster allows W
=0 pairs of d symmetry, due to a spin fluctuation, and ssymmetry, due to a charge fluctuation. Flux quantization is shown to be a manifestation of symmetry properties that hold for clusters of arbitrary size.
TL;DR: The Herglotz algorithm for constructing autonomous canonical transformations is extended to include nonautonomous canonical transformations and a remainder function that is related to a generating function by ${\cal R}^* = \partial F/\partial t$.
Abstract: The Herglotz algorithm for constructing autonomous canonical transformations that was presented by Guenther, Gottsch, and Kramer in [SIAM Rev., 38 (1996), pp. 287--293] is extended to include nonautonomous canonical transformations and a remainder function ${\cal R}^*$ that is related to a generating function by ${\cal R}^* = \partial F/\partial t$.
TL;DR: In this article, an effective spin Hamiltonian for the two-band Hubbard model of the high-Tc materials is derived by a canonical transformation that was calculated and summed up to infinite order.
Abstract: An effective spin Hamiltonian for the two-band Hubbard model of the high-T
c materials is derived by a canonical transformation that was calculated and summed up to infinite order. The spin interaction is obtained to be of Kondo type rather than t–J. We also discuss briefly the impact of the obtained result on understanding the spin gap.
TL;DR: In this paper, the authors derive a kernel in phase space that gives us the evolution of the probability density associated with the canonical transformation, and illustrate their analysis through several simple examples.
12 Dec 2000
TL;DR: In this article, the output feedback stabilization of a class of nonholonomic systems in port-controlled Hamiltonian formulae via generalized canonical transformations is discussed. But the authors do not consider the use of an integrator.
Abstract: The paper is concerned with the output feedback stabilization of a class of nonholonomic systems in port-controlled Hamiltonian formulae via generalized canonical transformations. In order to obtain a dynamic feedback, an integrator is added to the system firstly. Then the generalized canonical transformation is utilized to let the integrator play the role of an estimator of the unmeasurable state based on passivity. This technique can derive a time-varying output feedback stabilizing controller under a certain assumption. Furthermore the effectiveness of the proposed technique is demonstrated via a well known knife edge example.
TL;DR: In this paper, an iterative algorithm for the transformation of a three-way array to multiple orthogonality is presented, and a proof of monotonical convergence when only two modes are orthonormalized.
Abstract: This paper is concerned with the question to what extent the concept of rowwise or columnwise orthonormality can he generalized to three-way arrays. Whereas transforming a three-way array to multiple orthogonality is immediate, transforming it to multiple orthonormality is far from straightforward. The present paper offers an iterative algorithm for such transformations, and gives a proof of monotonical convergence when only two modes are orthonormalized. Also, it is shown that a variety of three-way arrays do not permit double orthonormalization. This is due to the order of the arrays, and holds regardless of the particular elements of the array. Studying three-way orthonormality has proven useful in exploring the possibilities for simplifying the core, to guide the search for equivalent direct transformations to simplicity; see Murakami et al. (Psychometrika 1998; 63: 255-261) as an example. Also, it appears in various contexts of the mathematical study of three-way analysis. Copyright (C) 2000 John Wiley & Sons, Ltd.
TL;DR: In this article, a relationship between the action-angle variables and the canonical transformation relating the rational Calogero-Moser system to the free one is discussed, and the authors also discuss the relationship between action angle variables and action angle transformations.
TL;DR: In this paper, the Coulomb interaction effects on W = 0 pairs are dynamically small, and repulsive or attractive, depending on the filling, and the analytical approach allows us to identify the Cooper pairs.
Abstract: We have recently developed a canonical transformation of the Hubbard and related models, valid for systems of arbitrary size and for the full plane; this is particularly suited to study hole pairing. In this work we show that exact diagonalization results of the one band Hubbard model for small clusters with periodic boundary conditions agree well with the analytical ones obtained by means of our canonical transformation. In the presence of a pairing instability, the analytic approach allows us to identify the Cooper pairs. They are W = 0 pairs, that is, singlet two-hole eigenstates of the Hubbard Hamiltonian with vanishing on-site repulsion. Indeed, we find that the Coulomb interaction effects on W = 0 pairs are dynamically small, and repulsive or attractive, depending on the filling.
TL;DR: In this article, the nonlinear supersymmetry of one-dimensional systems is investigated in the context of the quantum anomaly problem, and the most general one-parametric Calogero-like solution with the second order supercharges is found.
Abstract: The nonlinear supersymmetry of one-dimensional systems is investigated in the context of the quantum anomaly problem. Any classical supersymmetric system characterized by the nonlinear in the Hamiltonian superalgebra is symplectomorphic to a supersymmetric canonical system with the holomorphic form of the supercharges. Depending on the behaviour of the superpotential, the canonical supersymmetric systems are separated into the three classes. In one of them the parameter specifying the supersymmetry order is subject to some sort of classical quantization, whereas the supersymmetry of another extreme class has a rather fictive nature since its fermion degrees of freedom are decoupled completely by a canonical transformation. The nonlinear supersymmetry with polynomial in momentum supercharges is analysed, and the most general one-parametric Calogero-like solution with the second order supercharges is found. Quantization of the systems of the canonical form reveals the two anomaly-free classes, one of which gives rise naturally to the quasi-exactly solvable systems. The quantum anomaly problem for the Calogero-like models is ``cured'' by the specific superpotential-dependent term of order $\hbar^2$. The nonlinear supersymmetry admits the generalization to the case of two-dimensional systems.
TL;DR: Based on these steps a solution algorithm is described for any equation with this symmetry type which resembles a similar scheme for second order equations with projective symmetry group.
Abstract: The largest group of Lie symmetries that a third-order ordinary differential equation (ode) may allow has seven parameters. Equations sharing this property belong to a single equivalence class with a canonical representative v′′′(u)=0. Due to this simple canonical form, any equation belonging to this equivalence class may be identified in terms of certain constraints for its coefficients. Furthermore a set of equations for the transformation functions to canonical form may be set up for which large classes of solutions may be determined algorithmically. Based on these steps a solution algorithm is described for any equation with this symmetry type which resembles a similar scheme for second order equations with projective symmetry group.
TL;DR: In this paper, a finite-dimensional canonical Hamiltonian system depending on a finite number of parameters is given for the description of each such solution, which provides a solution of the initial-value problem for the Riemann-theta functions.
Abstract: The KP equation has a large family of quasiperiodic multiphase solutions. These solutions can be expressed in terms of Riemann-theta functions. In this paper, a finite-dimensional canonical Hamiltonian system depending on a finite number of parameters is given for the description of each such solution. The Hamiltonian systems are completely integrable in the sense of Liouville. In effect, this provides a solution of the initial-value problem for the theta-function solutions. Some consequences of this approach are discussed.
01 Jun 2000
TL;DR: In this article, the Shanmugadhasan canonical tetrad gravity on globally hyperbolic, asymptotically flat at spatial infinity, spacetimes with simultaneity spacelike hypersurfaces diffeomorphic to R3 is studied.
Abstract: After a study of the Hamiltonian group of gauge transformations, whose infinitesimal generators are the 14 first class constraints of a new formulation of canonical tetrad gravity on globally hyperbolic, asymptotically flat at spatial infinity, spacetimes with simultaneity spacelike hypersurfaces Στ diffeomorphic to R3, the multitemporal equations associated with the constraints generating space rotations and space diffeomorphisms on the cotriads are given. Their solutions give the dependence of the cotriads on Στ and of their momenta on the six parameters associated with such transformations. The choice of 3-coordinates on Στ , namely the gauge fixing to the space diffeomorphisms constraints, is equivalent to the choice of how to parametrize the dependence of the cotriad on the last three degrees of freedom: namely to the choice of a parametrization of the superspace of 3-geometries. The Shanmugadhasan canonical transformation, corresponding to the choice of 3-orthogonal coordinates on Στ and adapted to 13 of the 14 first class constraints, is found, the superhamiltonian constraint is rewritten in this canonical basis and the interpretation of the gauge transformations generated by it is given. Some interpretational problems connected with Dirac’s observables are discussed. In particular the gauge interpretation of tetrad gravity based on constraint theory implies that a “Hamiltonian kinematical gravitational field” is an equivalence class of pseudo-Riemannian spacetimes modulo the Hamiltonian group of gauge transformations: it includes a conformal 3-geometry and all the different 4-geometries (standard definition of a kinematical gravitational field, Riem M4/Diff M4) connected to it by the gauge transfor-
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TL;DR: In this paper, a theory of nonunitary-invertible and unitary canonical transformations is formulated in the context of Weyl's phase space representations, and exact solutions of the transformation kernels and the phase space propagators are given for the three fundamental canonical maps as fractional-linear, gauge and contact (point) transformations.
Abstract: A theory of nonunitary-invertible as well as unitary canonical transformations is formulated in the context of Weyl's phase space representations. Exact solutions of the transformation kernels and the phase space propagators are given for the three fundamental canonical maps as fractional-linear, gauge and contact (point) transformations. Under the nonlinear maps a phase space representation is mapped to another phase space representation thereby extending the standard concept of covariance. This extended covariance allows Dirac-Jordan transformation theory to naturally emerge from the Hilbert space representations in the Weyl quantization.
TL;DR: In this article, an explicit canonical transformation is constructed to relate the physical subspace of Yang-Mills theory to the phase space of the ADM variables of general relativity, which maps (3+1)-dimensional YMM theory to local evolution of metrics on 3-manifolds.
TL;DR: In this paper, a canonical transformation of the time variable and the Hamiltonian of a given completely integrable system is constructed for the Toda chains, Holt systems, and Stackel-type systems.
Abstract: We investigate the explicit construction of a canonical transformation of the time variable and the Hamiltonian whereby a given completely integrable system is mapped into another integrable system. The change of time induces a transformation of the equations of motion and of their solutions, the integrals of motion, the methods of separation of variables, the Lax matrices, and the correspondingr-matrices. For several specific families of integrable systems (Toda chains, Holt systems, and Stackel-type systems), we construct canonical transformations of time in the extended phase space that preserve the integrability property.
TL;DR: In this paper, a new approach to the study of quantum dynamics in condensed phases is presented, which allows the description of the liquid as a collection of coupled (through kinetic and potential coupling) harmonic modes.
Abstract: This paper presents a new approach to the study of quantum dynamics in condensed phases. The methodology is comprised of two main components. First, a formally exact method is described which allows the description of the liquid as a collection of coupled (through kinetic and potential coupling) harmonic modes. The modes are related to the Fourier modes of the component particle densities. Once the modes have been defined, a canonical transformation from the standard classical interparticle Hamilton function describes a new Hamilton function, which is exactly equivalent and defined on these harmonic coordinates. The final step in this section is the transformation of this Hamilton function into a quantum Hamiltonian operator. The second step in the process is the derivation of a new quantum mechanical evolution operator which is exact and allows the correction from a reference evolution operator, which is formed by adiabatic evolution on an approximate potential. A particular approximate potential which w...