Showing papers on "Canonical transformation published in 1998"
TL;DR: In this paper, the basic properties of the Backlund transformations (BTs) from the Hamiltonian point of view were revised using the n-particle periodic Toda lattice and the relativistic generalization due to Ruijsenaars of the elliptic Calogero-Moser system.
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 (BTs) from the Hamiltonian point of view. The analogy between the BT and Baxter's quantum Q-operator pointed out by Pasquier and Gaudin is exploited to produce a conjugated variable for the parameter of the BT , such that belongs to the spectrum of the Lax operator . As a consequence, the generating function of the composition of n BTs gives rise to another canonical transformation separating variables for the model. For the Toda lattice the dual BT parametrized by is introduced.
118 citations
TL;DR: In this paper, the authors consider a Hamiltonian quantum theory of spherically symmetric, asymptotically flat electrovacuum spacetimes and perform a canonical transformation such that the resulting configuration variables describe the dynamical properties of Reissner-Nordstr\"om black holes in a natural manner.
Abstract: We consider a Hamiltonian quantum theory of spherically symmetric, asymptotically flat electrovacuum spacetimes. The physical phase space of such spacetimes is spanned by the mass and the charge parameters $M$ and $Q$ of the Reissner-Nordstr\"om black hole, together with the corresponding canonical momenta. In this four-dimensional phase space, we perform a canonical transformation such that the resulting configuration variables describe the dynamical properties of Reissner-Nordstr\"om black holes in a natural manner. The classical Hamiltonian written in terms of these variables and their conjugate momenta is replaced by the corresponding self-adjoint Hamiltonian operator and an eigenvalue equation for the Arnowitt-Deser-Misner (ADM) mass of the hole, from the point of view of a distant observer at rest, is obtained. Our eigenvalue equation implies that the ADM mass and the electric charge spectra of the hole are discrete and the mass spectrum is bounded from below. Moreover, the spectrum of the quantity ${M}^{2}\ensuremath{-}{Q}^{2}$ is strictly positive when an appropriate self-adjoint extension is chosen. The WKB analysis yields the result that the large eigenvalues of the quantity $\sqrt{{M}^{2}\ensuremath{-}{Q}^{2}}$ are of the form $\sqrt{2n},$ where $n$ is an integer. It turns out that this result is closely related to Bekenstein's proposal on the discrete horizon area spectrum of black holes.
85 citations
TL;DR: In this article, the canonical transformation for the generalized Hamiltonian system is introduced, which preserves the original Hamiltonian structure of the original system and is expected to provide new insights and useful tools for analysis and synthesis of such systems.
68 citations
TL;DR: In this paper, a detailed examination of numerical results and analytical calculations devoted to a study of two holes doped into a two-dimensional, square lattice described by the t-J model is presented.
Abstract: We report on a detailed examination of numerical results and analytical calculations devoted to a study of two holes doped into a two-dimensional, square lattice described by the t-J model. Our exact diagonalization numerical results represent the first solution of the exact ground state of 2 holes in a 32-site lattice. Using this wave function, we have calculated several important correlation functions, notably the electron momentum distribution function and the hole-hole spatial correlation function. Further, by studying similar quantities on smaller lattices, we have managed to perform a finite-size scaling analysis. We have augmented this work by endeavouring to compare these results to the predictions of analytical work for two holes moving in an infinite lattice. This analysis relies on the canonical transformation approach formulated recently for the t-J model. From this comparison we find excellent correspondence between our numerical data and our analytical calculations. We believe that this agreement is an important step helping to justify the quasiparticle Hamiltonian, and in particular, the quasiparticle interactions, that result from the canonical transformation approach. Also, the analytical work allows us to critique the finite-size scaling ansatzes used in our analysis of the numerical data. One important feature that we can infer from this successful comparison involves the role of higher harmonics in the two-particle, d-wave symmetry bound state -- the conventional (\cos(k_x) - \cos(k_y)) term is only one of many important contributions to the d-wave symmetry pair wave function.
33 citations
TL;DR: In this paper, the polaronic corrections to the first excited state energies of an electron in a parabolic quantum dot are obtained variationally for the entire range of the electron-phonon coupling constant and for arbitrary confinement length using a canonical transformation method based on the Lee-Low-Pines-Gross formalism.
Abstract: The polaronic corrections to the first excited-state energies of an electron in a parabolic quantum dot are obtained variationally for the entire range of the electron-phonon coupling constant and for arbitrary confinement length using a canonical transformation method based on the Lee-Low-Pines-Gross formalism. Simple analytical results are obtained in some interesting limiting cases and for arbitrary values of the parameters the nature of the excited state is studied numerically. The theory is applied to two- and three-dimensional GaAs quantum dots to obtain information about the existence of both the effective mass and the relaxed excited states of a polaron in these systems.
29 citations
TL;DR: In this paper, the SU(2) gauge invariant Dirac-Yang-Mills mechanics of spatially homogeneous isospinor and gauge fields are considered in the framework of the generalized Hamiltonian approach.
Abstract: The SU(2) gauge invariant Dirac-Yang-Mills mechanics of spatially homogeneous isospinor and gauge fields is considered in the framework of the generalized Hamiltonian approach. The unconstrained Hamiltonian system equivalent to the model is obtained using the gaugeless method of Hamiltonian reduction. The latter includes the Abelianization of the first class constraints, putting the second class constraints into the canonical form and performing a canonical transformation to a set of adapted coordinates such that a subset of the new canonical pairs coincides with the second class constraints and part of the new momenta is equal to the Abelian constraints. In the adapted basis the pure gauge degrees of freedom automatically drop out from the consideration after projection of the model onto the constraint shell. Apart from the elimination of these ignorable degrees of freedom a further Hamiltonian reduction is achieved due to the three dimensional group of rigid symmetry possessed by the system.
29 citations
TL;DR: In this article, the use of Lax pair tensors as a unifying framework for Killing tensors of arbitrary rank is discussed, and some properties of the tensorial Lax Pair formulation are stated.
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.
27 citations
TL;DR: The canonical transformation and its unitary counterpart which relate the rational Calogero-Moser system to the free one were constructed in this article, and the canonical transformation was shown to be a unitary transformation.
Abstract: The canonical transformation and its unitary counterpart which relate the rational Calogero-Moser system to the free one are constructed.
24 citations
01 Jan 1998
TL;DR: In this paper, the authors introduce the canonical transformation for generalized Hamiltonian systems, which preserves the Hamiltonian structure of the original system and is expected to provide new insights and useful tools for analysis and synthesis of such systems.
Abstract: The generalized Hamiltonian systems are generalization of well-known Hamiltonian systems, which include various passive electric circuit systems as well as mechanical ones. This paper introduces the canonical transformation for the generalized Hamiltonian systems, which preserves the Hamiltonian structure of the original system and is expected to provide new insights and useful tools for analysis and synthesis of such systems. First, the class of such transformations and some of their properties are clarified. Second, we show how to stabilize the generalized Hamiltonian systems by using the transformation. This method works even when we can not stabilize them by conventional unity feedback without canonical transformation. Furthermore, it is shown that the proposed stabilization method includes the well-known one which exploits the virtual potential energy.
20 citations
TL;DR: In this paper, a canonical formalism for higher-curvature gravity was proposed, in which the Lagrangian density was given in terms of a function of the salar curvature.
Abstract: Following the method of Buchbinder and Lyahovich, we carry out a canonical formalism for a higher-curvature gravity in which the Lagrangian density ${\cal L}$ is given in terms of a function of the salar curvature $R$ as ${\cal L}=\sqrt{-\det g_{\mu
u}}f(R)$. The local Hamiltonian is obtained by a canonical transformation which interchanges a pair of the generalized coordinate and its canonical momentum coming from the higher derivative of the metric.
15 citations
TL;DR: In this article, a canonical tranformation approach to the effective interaction between two holes, based on the three-band Hubbard model but ready to include extra interactions as well, was proposed.
TL;DR: In this article, self-duality of super D3-brane theory was established as an exact symmetry of the action both in the Lagrangian and Hamiltonian formalisms.
TL;DR: In this paper, a canonical transformation is performed such that in terms of the new variables the model has a linear and homogeneous constraint, and therefore canonical gauges are admissible in the path integral.
Abstract: Simple cosmological models are used to show that gravitation can be quantized as an ordinary gauge system if the Hamilton–Jacobi equation for the model under consideration is separable. In this situation, a canonical transformation can be performed such that in terms of the new variables the model has a linear and homogeneous constraint, and therefore canonical gauges are admissible in the path integral. This has the additional practical advantage that gauge conditions that do not generate Gribov copies are then easy to choose.
TL;DR: In this paper, a canonical transformation into a set of free fields with a Minkowskian target space was proposed, and the quantization in terms of these free-field variables was discussed.
TL;DR: In this paper, the authors developed an extended time dependent Hartree-Bogoliubov (TDHB) theory in which paired and unpaired modes are treated in an equal manner.
Abstract: An induced representation of an SO(2N + 1) group has been obtained from a group extension of the SO(2N) Bogoliubov transformation for fermions to a new canonical transformation group. Embedding the SO(2N + 1) group into an SO(2N + 2) group and using the SO(2N + 2)/U(N + 1) coset variables, we develop an extended time dependent Hartree–Bogoliubov (TDHB) theory in which paired and unpaired modes are treated in an equal manner. The extended TDHB theory applicable to both even and odd fermion systems is a time dependent self-consistent field (TDSCF) theory with the same level of the mean field approximation as the usual TDHB theory for even fermion systems. We start from the Hamiltonian of the fermion system which includes, however, the Lagrange multiplier terms to select the physical spinor subspace. The extended TDHB equation is derived from the classical Euler–Lagrange equation of motion for the SO(2N + 2)/U(N + 1) coset variables in the TDSCF. The final form of the extended TDHB equation can be expressed through the variables as the representatives of the paired mode and the unpaired mode. We introduce the quasi anti-commutation relation approximation for the fermion. The parameters included in the Lagrangian multiplier terms are determined under the quasi anti-commutation relation approximation.
TL;DR: In this article, it was shown that the super D-string action is canonically equivalent to the type IIB superstring action with a world-sheet gauge field, and a transformation to the Type IIB theory with dynamical tension was constructed to establish the covariance beyond the semi-classical approximations.
Abstract: We show that the super D-string action is canonically equivalent to the type IIB superstring action with a world-sheet gauge field. Canonical transformation to the type IIB theory with dynamical tension is also constructed to establish the SL(2,Z) covariance beyond the semi-classical approximations.
TL;DR: In this article, the duality transformation described by Arnol-d, which relates the orbits for different central potentials through a change of variables, is shown to be an example of a generalized canonical transformation, in an extended phase space that includes time as a dynamical variable.
Abstract: The duality transformation described by Arnol’d, which relates the orbits for different central potentials through a change of variables, is shown to be an example of a generalized canonical transformation, in an extended phase space that includes time as a dynamical variable.
TL;DR: In this paper, a quasi-Shanmugadhasan canonical transformation related to the existence of a Poincare group action is used to construct the Dirac observables in the stratum for relativistic free particles.
Abstract: The construction of the Dirac observables in the stratum for a system of N relativistic free particles is carried out on the basis of a quasi-Shanmugadhasan canonical transformation related to the existence of a Poincare group action. The explicit form of the Dirac observables is derived by exploiting an internal Euclidean group having the Poincare canonical spin as generator of rotations. This procedure provides the symplectic version of the conventional angular momentum composition.
TL;DR: In this paper, the Lagrange equations of motion are formulated in the undefined and general variable set {Q,P} and the general Hamilton equations are derived from the Lagrangian equations by using a variational principle.
Abstract: The classical derivation of the canonical transformation theory [H. Goldstein, Classical Mechanics, 2nd ed. (Addison–Wesley, Reading, 1981)] is based on Hamilton’s principle which is only valid for conservative systems. This paper avoids this principle by using an approach that is basically reversed compared to the classical derivation. The Lagrange equations of motion are formulated in the undefined and general variable set {Q,P}, and the general Hamilton equations of motion are derived from the Lagrange equations by using a variational principle. The undefined general variables {Q,P} are defined through a transformation to a special (defined) variable set {q,p}. The transformation equations connecting the two sets are derived by using the invariants property of the value of the Lagrangian. This approach results in a more general interpretation of the generator function.
TL;DR: In this article, a simple construction of a group of birational canonical transformations of the fifth Painleve equation is given, which isomorphic to the affine Weyl group of the root system of type A3.
Abstract: The aim of this paper is to give a simple construction of a group of birational canonical transformations of the fifth Painleve equation isomorphic to the affine Weyl group of the root system of type A3 (Theorem 2 and Corollary 3). In previous works [6] (especially, §1) and [7], we constructed such groups of the sixth and the fourth equations by a method based on the synunetry of their Hamiltonian structures on their respective defining varieties with respect to Affine Weyl groups. To apply this method to the fifth equation, we reconstruct its defining variety to make clear the syrrunetry of the Hamiltonian structure on it Coming from the cyclic group of order four (Theorem 1). Prem this construction, we easily obtain generators of the desired group of birational canonical transformations without Okamoto's differential equation which is satisfied by a Hamiltonian auxiliary function ([3]). Our method clarifies the geometric meaning of birational canonical transformations of the fifth equation, and releases us from complicated calculation. AMS 1991 Subject Classification: 34A20, 34A26, 34A34. 1. Two polvnomial Hamiltonians According to Okamoto's theory of the isomonodromic deformation [2], each Painleve equation has its Hamiltonian structure, and is written in a Hamiltonian system with a polynomial Hamiltonian. For the fifth Pcdnleve equation, Okamoto [3] introduces a polynomial Hamiltonian K by the equation
TL;DR: In this article, the authors consider the problem of evolving a quantum field between any two (in general, curved) Cauchy surfaces and show that functional evolution of the quantum state can be satisfactorily described using algebraic quantum field theory.
Abstract: We consider the problem of evolving a quantum field between any two (in general, curved) Cauchy surfaces. Classically, this dynamical evolution is represented by a canonical transformation on the phase space for the field theory. We show that this canonical transformation cannot, in general, be unitarily implemented on the Fock space for free quantum fields on flat spacetimes of dimension greater than 2. We do this by considering time evolution of a free Klein-Gordon field on a flat spacetime (with toroidal Cauchy surfaces) starting from a flat initial surface and ending on a generic final surface. The associated Bogolubov transformation is computed; it does not correspond to a unitary transformation on the Fock space. This means that functional evolution of the quantum state as originally envisioned by Tomonaga, Schwinger, and Dirac is not a viable concept. Nevertheless, we demonstrate that functional evolution of the quantum state can be satisfactorily described using the formalism of algebraic quantum field theory. We discuss possible implications of our results for canonical quantum gravity.
25 Sep 1998
TL;DR: A novel approach for feature extraction of radar targets is proposed in this paper, which explores canonical analysis on a matrix formed by the range profiles (RPs) of training targets in different aspect angles.
Abstract: A novel approach for feature extraction of radar targets is proposed in this paper. This approach explores canonical analysis on a matrix formed by the range profiles (RPs) of training targets in different aspect angles. A subspace is obtained in this analysis. Projection of a RP into this subspace forms canonical profile (CP). The CPs of a training target in different aspect angles are averaged into feature vector for this target in data base. Using CP of an unknown target as the feature vector for recognition, experimental results on simulated data are presented.© (1998) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.
TL;DR: It is shown theoretically and numerically that momentum projections are better and more efficient than position projections, and lead to smaller error growth rates and affect the energy error much less.
Abstract: We study the effect of position and momentum projections in the numerical integration of constrained Hamiltonian systems. We show theoretically and numerically that momentum projections are better and more efficient. They lead to smaller error growth rates and affect the energy error much less, as they define a canonical transformation. As a concrete example, the planar pendulum is treated.
KEK1
TL;DR: In this article, the authors derive dispersive lattice functions in a way totally invariant under canonical transformation, which bridges the gap between invariant treatments that use only the coefficients of the coupled Courant-Snyder invariants as lattice function and treatments that introduce dispersive functions that depend on particular parametrizations.
Abstract: We derive dispersivelike lattice functions in a way totally invariant under canonical transformation. This bridges the gap between invariant treatments that use only the coefficients of the coupled Courant-Snyder invariants as lattice functions and treatments that introduce dispersive lattices functions that depend on particular parametrizations.
TL;DR: In this article, the parity invariant generating functional for the canonical transformation mapping the Liouville theory into a free scalar field was obtained, which is related to the pseudoscalar transformation.
TL;DR: In this paper, a canonical transformation is presented in the form of an asymptotic series to remove the time dependence up to some order in the small parameter of a Hamiltonian system with rapidly oscillating explicit dependence on time.
Abstract: Hamiltonian systems with rapidly oscillating explicit dependence on time are considered. The wavelength of this oscillation is treated as a small parameter and it is shown how to remove the time dependence up to some order in the small parameter by means of a canonical transformation presented in the form of an asymptotic series. The result has applications for the study of pulse propagation for high bit-rate transmission in optical fibres. PACS number 4279Sz
TL;DR: In this paper, it is shown that the Bogolyubov?Tyablikov canonical transformation allows collective coordinates to be introduced in a regular way for two particles in a quantum field and that for the case of the bipolaron the resulting electron?electron interaction in a phonon field depends on the electron coordinate difference alone.
Abstract: Differences between translation-invariant and broken-symmetry bipolaron theories are analyzed in detail. It is shown that the Bogolyubov?Tyablikov canonical transformation allows collective coordinates to be introduced in a regular way for two particles in a quantum field and that for the case of the bipolaron the resulting electron?electron interaction in a phonon field depends on the electron coordinate difference alone. Predictions using a revised solution of the nonlinear differential equations for a bipolaron are given. It is shown that solving bipolaron equations numerically reduces the total bipolaron energies compared to known variational results.
TL;DR: In this article, it was shown that the generators of canonical transformations in the triplectic manifold must satisfy constraints that have no parallel in the usual field antifield quantization, and the existence of a wide class of solutions to this relation nicely reflects the large freedom of the gauge fixing process in triplectic quantization.
TL;DR: In this article, the authors give an explicit canonical transformation which transforms a generic chiral 2D dilaton gravity model into a free field theory, based on an explicit transformation from a chiral model to a non-chiral model.
Abstract: We give an explicit canonical transformation which transforms a generic chiral 2D dilaton gravity model into a free field theory.
TL;DR: In this paper, the contribution of Kolmogorov to classical mechanics is illustrated through the famous KAM theorem, which solves a longstanding problem regarding stability in non-linear Hamiltonian dynamics.
Abstract: The contribution of Kolmogorov to classical mechanics is illustrated through the famous Kolmogorov-Arnold-Moser (KAM) theorem. This theorem solves a longstanding problem regarding stability in non-linear Hamiltonian dynamics. Various concepts required to understand the KAM theorem are also developed.