Showing papers on "Canonical transformation published in 1996"

On the relaxation of a two-level system: Beyond the weak-coupling approximation

Abstract: The model of two nondegenerate quantum levels coupled linearly and off‐diagonally to a bath of quantum mechanical harmonic oscillators studied previously by Laird, Budimir, and Skinner is re‐examined. Interpretations are made for both the fourth order population relaxation and dephasing processes. Some of the methods used are applied to the standard spin‐boson problem. The question of experimental detection of predicted phenomena is discussed. An approximate method, based on a canonical transformation of the original Hamiltonian is proposed to study the problem.

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. (Contribution to the proceedings of the 30th International Symposium Ahrenshoop on the Theory of Elementary Particles, Buckow, Germany, 26-31 August 1996)

Target Space Duality between Simple Compact Lie Groups and Lie Algebras under the Hamiltonian Formalism: I. Remnants of Duality at the Classical Level

Abstract: It has been suggested that a possible classical remnant of the phenomenon of target-space duality (T-duality) would be the equivalence of the classical string Hamiltonian systems. Given a simple compact Lie groupG with a bi-invariant metric and a generating function Γ suggested in the physics literature, we follow the above line of thought and work out the canonical transformation Φ generated by Γ together with an Ad-invariant metric and a B-field on the associated Lie algebra\(\mathfrak{g}\) ofG so thatG and\(\mathfrak{g}\) form a string target-space dual pair at the classical level under the Hamiltonian formalism. In this article, some general features of this Hamiltonian setting are discussed. We study properties of the canonical transformation Φ including a careful analysis of its domain and image. The geometry of the T-dual structure on\(\mathfrak{g}\) is lightly touched. We leave the task of tracing back the Hamiltonian formalism at the quantum level to the sequel of this paper.

On the canonical reduction of spherically symmetric gravity

Abstract: In a thorough paper Kuchar has examined the canonical reduction of the most general action functional describing the geometrodynamics of the maximally extended Schwarzschild geometry. This reduction yields the true degrees of freedom for (vacuum) spherically symmetric general relativity (SSGR). The essential technical ingredient in Kuchar's analysis is a canonical transformation to a certain chart on the gravitational phase space which features the Schwarzschild mass parameter , expressed in terms of what are essentially Arnowitt - Deser - Misner variables, as a canonical coordinate. (Kuchar's paper complements earlier work by Kastrup and Thiemann, based mostly on Ashtekar variables, which has also explicitly isolated the true degrees of freedom for vacuum SSGR.) In this paper we discuss the geometric interpretation of Kuchar's canonical transformation in terms of the theory of quasilocal energy - momentum in general relativity given by Brown and York. We find Kuchar's transformation to be a `sphere-dependent boost to the rest frame', where the `rest frame' is defined by vanishing quasilocal momentum. Furthermore, our formalism is general enough to cover the case of (vacuum) two-dimensional dilaton gravity. Therefore, besides reviewing Kuchar's original work for Schwarzschild black holes from the framework of hyperbolic geometry, we present new results concerning the canonical reduction of Witten black-hole geometrodynamics. Finally, addressing a recent work of Louko and Whiting, we discuss some delicate points concerning the canonical reduction of the `thermodynamical action', which is of central importance in the path-integral formulation of gravitational thermodynamics.

The Herglotz algorithm for constructing canonical transformations

Abstract: In his 1932 lectures on “Berfihrungstransformationen” [Lecture Notes of the Juliusz Center for Nonlinear Studies, Guenther, Gottsch, and Guenther, Juliusz Center for Nonlinear Studies, 1996], Gustav Herglotz gave an algorithm for constructing canonical transformations. We present Herglotz’s technique, which does not seem to be widely known today, as well as an application of his method. All calculations are carried out using Mathematics.

Contradirectional propagation and canonical transformations

Abstract: We examine the preservation of Poisson brackets and commutation relations in contradirectional propagation. It is shown that the relation between the input and the output amplitudes is a canonical transformation. This result is translated to the quantum case where preservation of the commutation relations is demonstrated in the case of linear propagation equations for the amplitudes.

Gravity Waves on the Surface of the Sphere

Abstract: We propose a Hamiltonian model for gravity waves on the surface of a fluid layer surrounding a gravitating sphere. The general equations of motion are nonlocal and can be used as a starting point for simpler models, which can be derived systematically by expanding the Hamiltonian in dimensionless parameters. In this paper, we focus on the small wave amplitude regime. The first-order nonlinear terms can be eliminated by a formal canonical transformation. Similarly, many of the second order terms can be eliminated. The resulting model has the feature that it leaves invariant several finite-dimensional subspaces on which the motion is integrable.

Admissible gauges for constrained systems.

Abstract: Gauge-fixing and gaugeless methods for reducing the phase space in generalized Hamiltonian dynamics are compared with the aim to define the class of admissible gauges. In the gaugeless approach, the reduced phase space of a Hamiltonian system with first class constraints is constructed locally, without any gauge fixing, using the following procedure: Abelianization of constraints with a subsequent canonical transformation so that some of the new momenta are equal to the new Abelian constraints. As a result, the corresponding conjugate coordinates are ignorable (nonphysical) while the remaining canonical pairs correspond to the true dynamical variables. This representation of the phase space prompts the definition of the subclass of admissible gauges, canonical gauges, as functions depending only on the ignorable coordinates. A practical method to determine the canonical gauge is proposed. \textcopyright{} 1996 The American Physical Society.

Geometry and Duality in Supersymmetric sigma-Models

Abstract: The Supersymmetric Dual Sigma Model (SDSM) is a local field theory introduced to be nonlocally equivalent to the Supersymmetric Chiral nonlinear sigma-Model (SCM), this dual equivalence being proven by explicit canonical transformation in tangent space. This model is here reconstructed in superspace and identified as a chiral-entwined supersymmetrization of the Dual Sigma Model (DSM). This analysis sheds light on the Boson-Fermion Symphysis of the dual transition, and on the new geometry of the DSM.

Ground-state description of quasi-one-dimensional polarons with arbitrary electron-phonon coupling strength

Abstract: We consider the interaction of a confined electron with bulk polar-optical phonons in a cylindrical quantum well wire with infinite boundary potential. Expressions for the polaron self-energy and mass are derived within a variational scheme over reasonably broad ranges of the wire radius and the phonon-coupling strength. The formulation is based on the standard canonical transformation of the strong-coupling ansatz and consists of a variationally determined perturbative extension serving for the theory to interpolate in the overall range of the coupling constant. Contrary to the general trend that the electron-phonon interaction is inherently stronger in systems of lower dimensionality, our results indicate that, at weak coupling, the binding energy of the polaron can be smaller and its mass less inertial compared with the bulk case when the wire is made narrow.

A uniform asymptotic expansion for the jacobi polynomials with explicit remainder

Abstract: A new asymptotic expansion is derived for the Jacobi polynonmial which holds uniformly for .An explicit expression is also given for the error term associated with the expansion. Our approach begins with a contour integral representation, followed by a suitable canonical transformation.

Particle trapping by oscillating fields: influence of dissipation upon the squeezing effect

Abstract: Previous papers in recent literature for particle trapping by oscillating fields employ a model by Glauber and a time-dependent canonical transformation to investigate the occurrence of squeezing and instabilities in this system. Here we extend this study to a more realistic case including the presence of dissipation. This influence upon the squeezing effect is investigated, the previous results in the absence of dissipation becoming a particular case of the present work.

Equivalence of Faddeev-Jackiw and Dirac approaches for gauge theories

Abstract: The equivalence between the Dirac method and Faddeev-Jackiw analysis for gauge theories is proved. In particular we trace out, in a stage by stage procedure, the standard classification of first and second class constraints of Dirac's method in the F-J approach. We also find that the Darboux transformation implied in the F-J reduction process can be viewed as a canonical transformation in Dirac approach. Unlike Dirac's method the F-J analysis is a classical reduction procedure, then the quantization can be achieved only in the framework of reduce and then quantize approach with all the know problems that this type of procedures presents. Finally we illustrate the equivalence by means of a particular example.

A Canonical Diffraction Problem With Two Media

Abstract: A Wiener-Hopf equation in L 2 being equivalent [5] to a boundary value problem (of the first kind) for a wave-scattering Sommerfeld half-plane Σ = R + x {0} which faces two different media Ω- : x 2 0, as a special configuration in [3], is solved by canonical Weiner-Hopffactorization of its L 2 -regular scalar symbol γ 0 = γ 0- γ 0+ . The factors are calculated by solving a Riemann-Hilbert boundary value problem on the semi-infinite branch cuts of tj(ξ) := (ξ 2 - k j 2 ) 1/2 , k j ∈ C ++ for j = 1,2 : taken parallel to the imaginary axis. The procedure following this idea is known as the Wiener-Hopf-Hilbert(-Hurd) method [2] and requires the evaluation of elliptic-type integrals. Formula (3.7) seems not to be contained in tables of integrals.

Study of the Dynamics of a Particle in a Constant Homogeneous Magnetic Field and a Transverse Homogeneous Rotating Electric Field in the Development of an X-Ray Source

Abstract: The relativistic motion of a charged particle in a homogeneous constant magnetic field and a transverse circularly polarized electric field is reduced to an integrable form. Using canonical transformations, it is shown that the equations of motion can be derived from a one degree of freedom time-dependent Hamiltonian which has a first integral. As a consequence the system can be shown to be integrable. An equation governing the energy of the particle is obtained. A simple approximate expression for its maximum value is derived for the case when the particle is initially resonant and at rest. This gives the upper limit in frequency of the X-rays emitted when, for instance, an electron hits a high-Z material. The relativistic motion of an electron in a constant homogeneous magnetic field and a transverse electric field is studied. This problem has already been explored by other authors[1,2]. One of the aims of this paper is to bring some enlightenment to their discussion by using the Hamiltonian formalism. Another aim is to derive a simple approximate expression for the maximum energy the particle can reach in the interesting physical situation, when the electron is initially resonant and at rest.This gives the upper limit in frequency of the X-rays emitted when the particle hits a high-Z material. Let us reduce the motion of an electron in an homogeneous constant magnetic field and a transverse circularly polarized electric field to a problem with a single degree of freedom. The constant magnetic field is assumed to be along the zaxis, and the electric field has the following components Ex = E0 cosω0t, Ey = E0 sinω0t, Ez =0, (1) where E0 and ω 0 are constants. The following gauge is chosen for the electromagnetic field A = B0 2 y + E0 ω0 sin ω0t ex + B0 2 x + E0 ω0 cos ω0t ey. (2) Assuming that the motion of an electron is in the x-y plane, its relativistic Hamiltonian is H = Px eE0 ω0 sin ω0t eB0 2 y 2 c2 + Py+ eE0 ω0 cos ω0t + eB0 2 x 2 c2 + m2c4 1 2 . (3) The Hamilton equations allow us to readily find two constants of motion C1 = Px + eB0 2 y, C2 = Py eB0 2 x. (4) Another constant of motion can be obtained by using Noether's theorem[3,4]. It is simple to show that the Lagrangian of the system is invariant under the following transformation t → t e ω0, x → x + ey, y → y ex, (5) where e is an infinitesimal quantity. Therefore, a third first integral is C3 = yPx xPy + H ω0. (6) It can be noted that the two first constants are canonically conjugated C1 , C2 eB0 = 1. (7) This property can be used to reduce the dimension of the problem by choosing the two constants as new conjugated momentum and coordinate. The following dimensionless variables and parameters are introduced x = x ω0 c , y = y ω0 c , Px,y = Px,y mc , t = ω0t, H = γ = H mc2 , a = eE0 mcω0 , Ω0 = eB0 mω0 , where m is the mass of the charged particle. A first canonical transformation is then introduced: x, y, Px, Py → x, y, Px, Py , given by the following type 2 generating function[3,5] F2 = Px Ω0 2 y x + Pyy. (8) A second canonical transformation is introduced: x, y, Px, Py → Q1, Q2, P1, P2 , generated by F2 = P2+ Ω0x y + P1 x + P2 Ω0 . (9) The product of the two transformations yield

Canonical and D-transformations in theories with constraints

Abstract: We describe a class of transformations in a super phase space (we call them D-transformations) which play the role of ordinary canonical transformations in theories with second-class constraints. Namely, in such theories they preserve the form invariance of equations of motion, their quantum analogs are unitary transformations, and the measure of integration in the corresponding Hamiltonian path integral is invariant under these transformations.

Batalin-Vilkovisky field-antifield quantisation of fluctuations around classical field configurations

Abstract: The Lagrangian field-antifield formalism of Batalin and Vilkovisky (BV) is used to investigate the application of the collective coordinate method to soliton quantisation. It is shown how Noether identities and local symmetries of the Lagrangian arise when collective coordinates are introduced in order to avoid divergences related to zero modes. This transformation to collective and fluctuation degrees of freedom is interpreted as a canonical transformation in the symplectic field-antifield space which induces a time-local gauge symmetry. Separating the corresponding Lagrangian path integral of the BV scheme in lowest order into harmonic quantum fluctuations and a free motion of the collective coordinate with the classical mass of the soliton, we show how the BV approach clarifies the relation between zero modes, collective coordinates, gauge invariance and the center-of-mass motion of classical solutions in quantum fields. Finally, we apply the procedure to the reduced nonlinear O(3) sigma-model.

Self-consistent green's function approach to dissipative two-state system

Abstract: We study the properties of a dissipative two-state fermionic system embedded in a phonon bath at zero temperature. In contrast to the traditional path-integral and canonical transformation approaches, we propose a self-consistent Green's function approach. Our results show that the mean field theory overestimates the polarization of the system. To justify our approach, we consider a simplest case in which there is only one phonon mode. This simple model can be ‘exactly’ diagonalized. We find that the results from the self-consistent Green's function approach agree well with the quantum mechanical diagonalization results.

Dynamics and Dynamical Systems — Quod Satis

Abstract: In some cases it is easier to describe what something is not, rather than what it is or what it wants to be We shall therefore adhere to this principle when introducing this chapter

Conservation Laws Associated to Same Dynamical Systems

Abstract: It is well-known that a significant role in the study of the dynamical systems is occupied by the finding of conservation laws (first integrals) expressed by functions with null-derivatives along the trajectories. There are known a few methods to determine conservation laws. We quote in the Lagrangian formalism, the method given by Noether’s theorem [6].

Batalin‐Vilkovisky field‐antifield quantisation of fluctuations around classical field configurations

Abstract: The Lagrangian field-antifield formalism of Batalin and Vilkovisky (BV) is used to investigate the application of the collec- tive coordinate method to soliton quantisation. In field theories with soliton solutions, the Gaussian fluctuation operator has zero modes due to the breakdown of global symmetries of the Lagrangian in the soliton solutions. It is shown how Noether identities and local symmetries of the Lagrangian arise when collective coordinates are introduced in order to avoid divergences related to these zero modes. This transformation to collective and fluctuation degrees of freedom is interpreted as a canonical transformation in the symplectic field-antifield space which induces a time-local gauge symmetry. Separating the corresponding Lagrangian path integral of the BV scheme in lowest order into harmonic quantum fluctuations and a free motion of the collective coordinate with the classical mass of the soliton, we show how the BV approach clarifies the relation between zero modes, collective coordinates, gauge invariance and the center- of-mass motion of classical solutions in quantum fields. Finally, we apply the procedure to the reduced nonlinear O(3) oe-model.^L

Hamiltonian center of mass variables in an arbitrary geometric form of dynamics

Abstract: In the context of the relativistic theory of direct interactions we study the transition from the Lagrangian to the Hamiltonian description of a relativistic system of N particles in an arbitrary geometric form of dynamics in terms of center of mass variables. Such a Hamiltonian description is constructed under the assumption of predicative external motion. We find the canonical transformations that reduce it to the known Bakamjien-Thomas model of the instantaneous form of dynamics.