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Showing papers on "Canonical transformation published in 1991"


Journal ArticleDOI
TL;DR: The Lissajous transformation as discussed by the authors is a canonical transformation for elliptic oscillators, which is defined explicitly in terms of Cartesian variables, and implicitly by resolution of a partial differential equation separable in polar variables.
Abstract: A new canonical transformation is proposed to handle elliptic oscillators, that is, Hamiltonian systems made of two harmonic oscillators in a 1-1 resonance. Lissajous elements pertain to the ellipse drawn with a light pen whose coordinates oscillate at the same frequency, hence their name. They consist of two pairs of angle-action variables of which the actions and one angle refer to basic integrals admitted by an elliptic oscillator, namely, its energy, its angular momentum and its Runge-Lenz vector. The Lissajous transformation is defined in two ways: explicitly in terms of Cartesian variables, and implicitly by resolution of a partial differential equation separable in polar variables. Relations between the Lissajous variables, the common harmonic variables, and other sets of variables are discussed in detail.

65 citations


Journal ArticleDOI
TL;DR: The theory of Hamiltonian systems with first-class constraints has been studied in this article, where it is possible to separate the physical part from the gauge part by transforming to canonical coordinates in which the constraints are a subset of the new momenta; this construction is accomplished by algebraic methods and the use of a set of Hamilton-Jacobi-like equations.
Abstract: Some new results are presented on the theory of Hamiltonian systems with first‐class constraints. In these systems it is possible to separate the physical part from the gauge part by transforming to canonical coordinates in which the constraints are a subset of the new momenta; this construction is accomplished by algebraic methods and the use of a set of Hamilton–Jacobi‐like equations. Finally, the problem and meaning of evolution in systems with weakly vanishing Hamiltonian is commented on.

31 citations


Journal ArticleDOI
TL;DR: The systematic method of generating all ocal canonical transformations enables us to discover a ``nonlinear'' local U(1) gauge symmetry of the Heisenberg-Hubbard model that remains a local symmetry away from half filling.
Abstract: By permitting canonical transformations that are nonlinear in fermion creation and annihilation operators, we show that the space of canonical transformations of ordinary spin-1/2 operators local to a point in space is SU(2)\ensuremath{\bigotimes}SU(2)\ensuremath{\bigotimes}U(1)\ensuremath{\bigotimes}${\mathit{Z}}_{2}$. We identify those subgroups that form local and global gauge symmetries of the Hubbard-Heisenberg model on and off half filling. Our systematic method of generating all ocal canonical transformations enables us to discover a ``nonlinear'' local U(1) gauge symmetry of the Heisenberg-Hubbard model that remains a local symmetry away from half filling. The paper presents this group together with all other known canonical transformations in a unified framework.

25 citations


Journal ArticleDOI
Stephen Hwang1
TL;DR: In this article, an explicit ansatz for the abelian constraints is proposed, and it is shown that under comparatively weak conditions, this ansatz will solve the problem of ABELIANization.

14 citations


Book ChapterDOI
01 Jan 1991
TL;DR: For generic rational coadjoint orbits in the dual of the positive half of the loop algebra, the natural divisor coordinates associated to the eigenvector line bundles over the spectral curves project to Darboux coordinates on the Gl(r)-reduced space as discussed by the authors.
Abstract: For generic rational coadjoint orbits in the dual \(\tilde gl(r)^{ + *}\) of the positive half of the loop algebra \(\tilde gl(r)^{ + *}\), the natural divisor coordinates associated to the eigenvector line bundles over the spectral curves project to Darboux coordinates on the Gl(r)-reduced space. The geometry of the embedding of these curves in an ambient ruled surface suggests an intrinsic definition of symplectic structure on the space of pairs (spectral curves, duals of eigenvector line bundles) based on Serre duality. It is shown that this coincides with the reduced Kostant-Kirillov structure. For all Hamiltonians generating isospectral flows, these Darboux coordinates allow one to deduce a completely separated Liouville generating function, with the corresponding canonical transformation to linearizing variables identified as the Abel map.

12 citations


Journal ArticleDOI
TL;DR: In this article, a quantum mechanical canonical transformation was applied to eliminate a Coriolis term from the rotational-vibrational Hamiltonian of a polyatomic molecule to permit the computation of rotational energy levels for highly excited vibrational states.
Abstract: A quantum mechanical canonical transformation, applied previously to eliminate a Coriolis term from the rotational–vibrational Hamiltonian of a polyatomic molecule [J. Chem. Phys. 94, 461 (1991)], is simplified to permit the computation of rotational energy levels for highly excited vibrational states. An approximate matrix representation of the transformed Hamiltonian is presented which is shown to be very accurate for a two‐mode model of formaldehyde. The quantum dynamics of the two nearly degenerate vibrational modes, strongly coupled by a Coriolis term, is studied using the approximate representation. It is shown that the second order Coriolis term and quartic anharmonic terms in the potential can be treated effectively as perturbations using the transformed Hamiltonians as the zeroth‐order model.

12 citations


Journal ArticleDOI
TL;DR: In this paper, a quantum mechanical rotational-vibrational Hamiltonian with one Coriolis term which couples vibration and rotation is transformed by employing a Bogoliubov-Tyablikov transformation.
Abstract: A quantum mechanical rotational–vibrational Hamiltonian with one Coriolis term which couples vibration and rotation is transformed to eliminate the Coriolis term. This is achieved by employing a Bogoliubov–Tyablikov transformation. A closed‐form energy level expression is obtained for the vibrational–rotational energy levels of a rigid symmetric top coupled to harmonic oscillators. For an asymmetric rotor the transformation introduces small off‐diagonal matrix elements which couple vibrational states. Nearly degenerate vibrational states would be strongly coupled by the original Coriolis term, but the small off‐diagonal matrix elements of the transformed Hamiltonian may be treated perturbatively to obtain an effective rotational Hamiltonian for each vibrational state. The new theoretical method is compared with variational calculations.

11 citations


Journal ArticleDOI
TL;DR: An extension of the Hurwitz transformation to a canonical transformation between phase spaces allows conversion of the five-dimensional Kepler problem into that of a constrained harmonic oscillator problem in eight dimensions.
Abstract: An extension of the Hurwitz transformation to a canonical transformation between phase spaces allows conversion of the five-dimensional Kepler problem into that of a constrained harmonic oscillator problem in eight dimensions. Thus a new regularization of the Kepler problem is established. Then, following Dirac, we quantize the extended phase space, imposing constraint conditions as super-selection rules. In that way the interchangeability of the reduction and the quantization procedures is proved

10 citations



01 Jan 1991
TL;DR: In this paper, the energy eigenfunctions for the simple linear potential were discussed using time-independent canonical transformation methods, pedagogically setting the stage for some field theory calculations to follow.
Abstract: We use time-independent canonical transformation methods to discuss the energy eigenfunctions for the simple linear potential, pedagogically setting the stage for some field theory calculations to follow. We then discuss the Schr\"odinger wave-functional method of calculating correlation functions for Liouville field theory. We compare this approach to earlier treatments, in particular we check against known weak-coupling results for the Liouville field defined on a cylinder. Finally, we further set the stage for future Liouville calculations on curved two-manifolds and briefly discuss simple quantum mechanical systems with time-dependent Hamiltonians.

8 citations


Journal ArticleDOI
TL;DR: In this article, a method for decoupling the collisional and vibrational variables in the collision process is developed based on a canonical transformation in phase space, where the basic principle employed is the "maximal decoupled condition".
Abstract: A method for decoupling the collisional and vibrational variables in the collision process is developed The present procedure of decoupling is based on a canonical transformation in phase space The basic principle employed is the ‘‘maximal decoupling condition’’ [∂H/∂u]u=v=0=[∂H/∂v]u=v=0=0, where H is the Hamiltonian and u and v represent new vibrational coordinates and momenta ‘‘Decoupling surface’’ is defined as the surface determined by u=v=0 The partial differential equation to be satisfied by this decoupling surface is derived This partial differential equation can be solved easily by utilizing classical trajectories Local vibrational frequencies along the decoupling surface are defined and their stability analysis is shown to provide the criterion for the separability ‘‘Adiabatic approximation’’ which assumes the conservation of locally defined actions of the new vibrational variables leads to an ‘‘effective Hamiltonian’’ that describes the collision process under a given set of the initial v

Journal ArticleDOI
TL;DR: A fresh analysis of Berry's-phase calculations for some coherent states of general theoretical interest is presented, though semiclassical, which lends itself easily for comparison with the exact results.
Abstract: Motivated by some recent remarks on the "removability" of Berry's phase [Ph. De Sousa Gerbert, Ann. Phys. (N.Y.) 189, 155 (1989)] or its classical counterpart, the Hannay angle, we present here a fresh analysis of Berry's-phase calculations for some coherent states of general theoretical interest. Our reasoning, though semiclassical, lends itself easily for comparison with the exact results. Unlike spin-coherent states, both the (ordinary) harmonic-oscillator and squeezed coherent states are contained in Hamiltonians for which Berry's phase can be removed globally by a canonical transformation.


Book ChapterDOI
01 Jan 1991


Journal ArticleDOI
TL;DR: In this article, it was shown that Berry's phase and Hannay's angle can both be derived from the same quantum unitary transformation; their relationship is easily established in this framework.
Abstract: Using an 'action-angle' coherent states formalism, introduced by the authors in a preceding paper, it is shown that Berry's phase and Hannay's angle can both be derived from the same quantum unitary transformation; their relationship is easily established in this framework.

Journal ArticleDOI
TL;DR: In this article, the authors consider canonical transformations which map the Hubbard model into an effective Hamiltonian that conserves the number of doubly occupied sites, and they show that such transformations may be varied arbitrarily by appropriate choice of canonical transformation.

Journal ArticleDOI
TL;DR: The link between the radiation cross-section and the S -matrix of scattering of a particle in the external field on the basis of the canonical transformation method in the quantum theory of radiation of high energy charged particles in an external field was established in this article.

Journal ArticleDOI
TL;DR: In this paper, a Lagrangian and compressible representation of the flow is used in order to avoid difficulties associated with non-canonical coordinates and constrained systems, and a quadratic Hamiltonian for the linearized motion around a state of no flow is derived.
Abstract: A Hamiltonian representation for the interaction between vortical and internal wave motion in an inviscid, stratified fluid in a rotating frame is formulated. A Lagrangian and compressible representation of the flow is used in order to avoid difficulties associated with non-canonical coordinates and constrained systems. A quadratic Hamiltonian for the linearized motion around a state of no flow is derived. Canonical transformations are then employed in order to isolate normal mode coordinates and write the Hamiltonian as a sum of independent harmonic oscillators. The three possible modes of linearized motion are the acoustic, internal wave and potential vorticity carrying motions. The acoustic modes are of little interest in the problem at hand and are retained in order to avoid difficulties associated with the incompressibility constraint. We discuss ways in which these modes may be eliminated or rendered harmless. The coordinate corresponding to the potential vorticity carrying mode is absent f...


Journal ArticleDOI
TL;DR: In this article, the quantum unitary transformation corresponding to the classical canonical transformation is presented and the removability of the topological term in the O(3) nonlinear δ-model is discussed.
Abstract: It has been shown that the topological term in theO(3) nonlinear δ-model can be removed by a suitable canonical transformation using classical theory. In this paper, the quantum unitary transformation corresponding to the classical canonical transformation is presented. The meaning of the unitary transformation and the removability of the topological term are then discussed.

Book ChapterDOI
G. Rosensteel1
01 Jan 1991
TL;DR: The GCM(3) as mentioned in this paper is the dynamical group for the classical Riemann ellipsoids, which is a semidirect product of GI(3,R).
Abstract: A dynamical group is constructed from a Lie algebra of physically relevant observables which characterize a system. A dynamical group may be exact so that the algebra's observables engird all the degrees of freedom of the physical system. Examples of exact dynamical groups include the Heisenberg group for a spinless non-relativistic particle, the Poincar~ group for a relativistic free particle, and gauge groups such as color SU(2). On the other hand, an approximate dynamical group models a complex system by including only the most important degrees of freedom in the algebra and omitting observables which are adiabatically deeoupled. Examples of approximate dynamical groups in nuclear physics include U(6) of the interacting boson model [1] and Sp(3,R) of the symplectie collective model [2]. Dynamical groups are realized in quantum mechanics as unitary irreducible representations. Less familiar, but no less significant, dynamical groups arise in classical mechanics as transitive groups of canonical transformations on phase space [3]. In this article, GCM(3), which is an acronym for general collective motion in 3 spatial dimensions, is shown to be the dynamical group for the classical Riemann ellipsoids. The Riemann ellipsoids model rotating galaxies (period ~ 1015 see), stars (10ss), gaseous plasmas (?s), and rapidly-rotating atomic nuclei (10Z~s) [4-6]. The Lie algebra GCM(3) = [R6]GI(3,R) is a semidirect product of GI(3,R),

Proceedings ArticleDOI
06 May 1991
TL;DR: In this article, two canonical transformations are implemented to find approximate invariant surfaces for a nonlinear time-periodic Hamiltonian, and the residual angle dependence remaining after performing the transformation is mostly eliminated by a second, perturbative transformation.
Abstract: Two canonical transformations are implemented to find approximate invariant surfaces for a nonlinear time-periodic Hamiltonian. The first transformation is found from the nonperturbative, iterative solution of the Hamilton-Jacobi equation. The residual angle dependence remaining after performing the transformation is mostly eliminated by a second, perturbative transformation. This refinement can improve the accuracy or the speed, of the invariant surface calculation. The motion of a single particle in one transverse dimension is studied in a storage ring example where strong sextupole magnets are the source of the nonlinearity. >

Journal ArticleDOI
TL;DR: In this article, the authors show that the strongly correlated one-dimensional Hubbard model with infinite (U= infinity ) on-site repulsion is related to a corresponding spinless free-fermion lattice model.
Abstract: The authors show with the aid of a canonical transformation that separates charge and spin degrees of freedom that the strongly correlated one-dimensional Hubbard model with infinite (U= infinity ) on-site repulsion is related to a corresponding spinless free-fermion lattice model. Thermodynamic properties of the two models are shown to be closely related whereas the single-particle two-site function for the U= infinity model is shown to be related to a modified many site function for the spinless model which involves many-point correlations. The latter is shown to be expressible in terms of the inverse of a certain matrix which should be amenable to numerical and analytical analysis.

Book ChapterDOI
01 Jan 1991
TL;DR: In this article, a generalized canonical Delaunay-similar (CDS) set wsls derived by means of a canonical transformation whose generating function is inspired by Deprit's radial intermediary and can be considered as a generalization of the canonical set introduced by Scheifele.
Abstract: In a previous paper we had defined a set of eight generalized canonical Delaunay- Similar (CDS) variables incorporating the first-order secular effects present in the Main Problem in the Theory of Earth’s Artificial Satellite. The new CDS set wsls derived by means of a canonical transformation whose generating function is inspired by Deprit’s radial intermediary and can be considered as a generalization of a canonical set introduced by Scheifele. When applied to Deprit’s intermediary, the proposed variables lead to a simple solution, the momenta being constant and the co-ordinates being either a constant or a linear function of the independent variable. As a further step, a set of generalized Poincare-Similar (PS) canonical variables corresponding to the aforesaid DS ones is now constructed; the new GPS set also exhibits the same feature of containing the whole first-order secular contribution of the J2 zonal harmonic of the Earth’s potential and is free from singularities.

Journal ArticleDOI
TL;DR: In this paper, a two-level atom interacting with an electromagnetic field is investigated using a canonical transformation, and a Hamiltonian analysis is carried out using the transformed Hamiltonian manifestly shows various features, in particular, the effects of the interaction Hamiltonian on both the atom and the field are obtained simultaneously.
Abstract: A two-level atom interacting with an electromagnetic field is investigated. Using a canonical transformation, a Hamiltonian analysis is carried out. The transformed Hamiltonian manifestly shows various features, in particular, the effects of the interaction Hamiltonian on both the atom and the field are obtained simultaneously.

Journal ArticleDOI
TL;DR: Using the tunneling Hamiltonian approach, a theoretical model of the system including electron-phonon interaction was presented in this paper, where the relevant coupling constants were determined from realistic wave functions for the expected confinement potentials.

Journal ArticleDOI
TL;DR: In this article, a canonical transformation relating the positive U and negative U sectors of the one band Hubbard model was used to study the global SU(2) pseudospin symmetry in the model, and to investigate the structure of the collective excitation spectrum from a superconducting ground state.

Journal ArticleDOI
TL;DR: In this article, the authors apply Hamilton-Jacobi theory to find appropriate canonical transformations for the calculation of the phase-space path integrals of the relativistic particle equations, and they also introduce canonical transformations into quantum mechanics.
Abstract: Hamilton-Jacobi theory is applied to find appropriate canonical transformations for the calculation of the phase-space path integrals of the relativistic particle equations. Hence, canonical transformations and Hamilton-Jacobi theory are also introduced into relativistic quantum mechanics. Moreover, from the classical physics viewpoint, it is very interesting to find and to solve the Hamilton-Jacobi equations for the relativistic particle equations.