Showing papers on "Canonical transformation published in 1994"
TL;DR: In this paper, the authors present a reduced integrodifferential Hamiltonian reduced equation for weakly nonlinear surface waves, called five-wave, where the non-resonant quadratic, cubic and fourth-order nonlinear terms are eliminated by suitable canonical transformation.
Abstract: Many studies of weakly nonlinear surface waves are based on so-called reduced integrodifferential equations. One of these is the widely used Zakharov four-wave equation for purely gravity waves. But the reduced equations now in use are not Hamiltonian despite the Hamiltonian structure of exact water wave equations. This is entirely due to shortcomings of their derivation. The classical method of canonical transformations, generalized to the continuous case, leads automatically to reduced equations with Hamiltonian structure. In this paper, attention is primarily paid to the Hamiltonian reduced equation describing the combined effects of four- and five-wave weakly nonlinear interactions of purely gravity waves. In this equation, for brevity called five-wave, the non-resonant quadratic, cubic and fourth-order nonlinear terms are eliminated by suitable canonical transformation. The kernels of this equation and the coefficients of the transformation are expressed in explicit form in terms of expansion coefficients of the gravity-wave Hamiltonian in integral-power series in normal variables. For capillary–gravity waves on a fluid of finite depth, expansion of the Hamiltonian in integral-power series in a normal variable with accuracy up to the fifth-order terms is also given.
283 citations
CERN1
TL;DR: In this paper, it was shown that Buscher's Abelian duality transformation rules can be recovered in a very simple way by performing a canonical transformation first suggested by Giveon, Rabinovici and Veneziano.
188 citations
TL;DR: The conventional algorithm is refined to more efficiently produce the nonlocal symmetries of the pseudodual chiral model, and the complete local current algebra for the Pseudodual theory is discussed.
Abstract: We discuss the pseudodual chiral model to illustrate a class of two-dimensional theories which have an infinite number of conservation laws but allow particle production, at variance with naive expectations. We describe the symmetries of the pseudodual model, both local and nonlocal, as transmutations of the symmetries of the usual chiral model. We refine the conventional algorithm to more efficiently produce the nonlocal symmetries of the model, and we discuss the complete local current algebra for the pseudodual theory. We also exhibit the canonical transformation which connects the usual chiral model to its fully equivalent dual, further distinguishing the pseudodual theory.
160 citations
TL;DR: In this paper, the importance of non-unitary canonical transformations for constructing solutions of the Schrodinger equation is discussed, and three elementary canonical transformations are shown both to have quantum implementations as finite transformations and to generate, classically and infinitesimally, the full canonical algebra.
119 citations
TL;DR: A new derivation from first principle is given of the energy-time uncertainty relation in quantum mechanics, and a canonical transformation is made in clusical mechanic that creates a new canonical coordinate T, which is called tempu, co~ugate to the energy.
Abstract: A derivation from first principles is given of the energy-time uncertainty relation in quantum mechanics. A canonical transformation is made in classical mechanics to a new canonical momentum, which is energy E, and a new canonical coordinate T, which is called tempus, conjugate to the energy. Tempus T, the canonical coordinate conjugate to the energy, is conceptually different from the time t in which the system evolves. The Poisson bracket is a canonical invariant, so that energy and tempus satisfy the same Poisson bracket as do p and q. When the system is quantized, we find the energy-time uncertainty relation \ensuremath{\Delta}E\ensuremath{\Delta}T\ensuremath{\ge}\ensuremath{\Elzxh}/2. For a conservative system the average of the tempus operator T^ is the time t plus a constant. For a free particle and a particle acted on by a constant force, the tempus operators are constructed explicitly, and the energy-time uncertainty relation is explicitly verified.
67 citations
TL;DR: In this article, the authors show how to remove the divergences in an arbitrary gauge-field theory (possibly non-renormalizable, i.e. involving infinitely many parameters) in the contex of the Batalin-Vilkovisky formalism.
Abstract: We show how to remove the divergences in an arbitrary gauge-field theory (possibly nonrenormalizable, i.e. involving infinitely many parameters) in the contex of the Batalin-Vilkovisky formalism. We show that this can be achieved by performing, order by order in the loop expansion, a redefinition of the parameters of the classical Lagrangian (possibly infinitely many) and a canonical transformation (in the sense of Batalin and Vilkovisky) of fields and BRS sources. Gauge-invariance is turned into a suitable quantum generalization of BRS invariance. We define quantum observables in this formal context and study their properties. We show the independence of the on-shell physical amplitudes from gauge fixing. We apply the result to renormalizable gauge-field theories that are gauge-fixed with a non-renormalizable gauge fixing and prove that their predictivity is retained. A corollary is that topological field theories are predictive.
43 citations
01 Jan 1994
TL;DR: In this paper, the authors used Geogdzhaev's method to derive the solution to the initial value problem of any dispersionless Lax equation, including the Boussinesq equation.
Abstract: Geogdzhaev’s method is used to derive the solution to the initial value problem of any dispersionless Lax equation. The particular case of the dispersionless Boussinesq equation is worked out in detail and possible generalisations are considered.
42 citations
TL;DR: The extended Wick's theorem for fermion operators, which is used to compute matrix elements of an arbitrary operator between two different quasiparticle vacuums, is reformulated to deal with quasips expanded in a finite single particle basis not closed under the canonical transformation relating them.
Abstract: The extended Wick's theorem for fermion operators, which is used to compute matrix elements of an arbitrary operator between two different quasiparticle vacuums, is reformulated to deal with quasiparticle vacuums expanded in a finite single particle basis not closed under the canonical transformation relating them. A new expression for the overlap of those quasiparticle vacuums is also given.
39 citations
TL;DR: In this article, the authors apply the Tokuyama-Mori theory together with canonical transformation to the exciton-phonon system with site-diagonal linear coupling and full exciton or electron density matrix; finite concentrations of the carriers are admitted.
Abstract: Tokuyama-Mori theory together with canonical transformation but with quantities of interest remaining untransformed is applied to the exciton- or electron-phonon system with site-diagonal linear coupling and full exciton or electron density matrix; finite concentrations of the carriers are admitted. The result is applied to a particle on a symmetric dimer. In contrast to recent calculations showing that time-convulution (Mori or generalized master equation) second-order (in coupling to phonons) approaches fail in providing genuine relaxation, independent (up to the initial time interval) of the degree of initial polaron formation, the Tokuyama-Mori theory is illustrated to be fully acceptable in this respect. Correspondence with a recent generalization of the Haken-Strobl-Reineker model is found.
33 citations
TL;DR: A canonical transformation between two known integrable cases of the H\'enon-Heiles systems is given and the separation of variables for the corresponding Hamilton-Jacobi equations is discussed.
Abstract: A canonical transformation between two known integrable cases of the H\'enon-Heiles systems is given and the separation of variables for the corresponding Hamilton-Jacobi equations is discussed.
22 citations
TL;DR: The hamiltonian formulation of the string with a dynamical geometry and two-dimensional gravity with torsion is given in this paper, which is the semidirect sum of the Virasoro algebra and the abelian subalgebra corresponding to local Lorentz rotation.
TL;DR: In this paper, the authors examined the possibilities of improving the Wigner method by propagating the Weyl-Wigner function in an interaction picture, and showed that the classical interaction picture in a natural way comes into play.
Abstract: published in Advance ACS Abstracts, March 15, 1994. 0022-3654/94/2098-3212~04.50~0 successive propagations of the state vector within the Gaussian wavepacket method using the full Hamiltonian or a zeroth-order channel Hamiltonian, respectively. In the second approach introduced by Skodje'4 a canonical transformation of the classical coordinates into a set of classical interaction coordinates is performed, and the interaction Hamiltonian to be used in the Gaussian wavepacket method is found by applying the quanti- zation procedure for generalized canonical coordinates proposed by Heller.lz In this paper we examine the possibilities of improving the Wigner method by propagating the Wigner function in an interaction picture. It is organized in the following way: We begin by recalling the main features of the Weyl-Wigner representation of quantum mechanics. Then we give a short description of the quantum and the classical interaction pictures. We discuss subsequently the Wigner method in the interaction picture and show that the classical interaction picture, in a natural way, comes into play. Finally we present a numerical application and discuss the results. For simplicity all derivations are kept in one dimension, the extension to several dimensions being straightforward. To avoid confusion, quantum mechanical operators are written in capital letters (with the exception of the position and momentum operators) supplied with a hat, the corresponding phase space functions denoted by the same letter without the hat and classical functions are written in script.
TL;DR: In this paper, a new canonical transformation of freedom two was found and three new sets of canonical variables for the orbital motion and two for the rotational motion were derived, which remain well-defined in the case when the classical sets become ill-defined, for example, when the eccentricity and/or the inclination is small for the elliptic orbital motion.
Abstract: A new canonical transformation of freedom two was found. By using this, we derived three new sets of canonical variables for the orbital motion and two for the rotational motion. New canonical variables have clear physical meanings and remain well-defined in the case when the classical sets become ill-defined, for example, when the eccentricity and/or the inclination is small for the elliptic orbital motion.
TL;DR: In this paper, the action of (2+1)-dimensional gravity with negative cosmological constant Λ is written uniquely in terms of the time-independent traces of holonomies around two intersecting noncontractible paths on T2.
TL;DR: In this paper, the authors reviewed classical Floquet theory with careful attention to the case of repeated eigenvalues common in Hamiltonian systems and proposed a canonical transformation to modal variables if the periodic matrix can be made symplectic at the initial time.
Abstract: Classical Floquet theory is reviewed with careful attention to the case of repeated eigenvalues common in Hamiltonian systems Floquet theory generates a canonical transformation to modal variables if the periodic matrix can be made symplectic at the initial time It is shown that this symplectic normalization can always be carried out, again with careful attention to the degenerate case The periodic modal vectors and canonical modal variables can always be chosen to be purely real It is possible to introduce real valued action-angle variables for all modes Physical interpretation of the canonical degenerate normal modal variables are offered Finally, it is shown that this transformation enables canonical perturbation theory to be carried out using Floquet modal variables
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TL;DR: In this article, a connection between two different action principles for perfect fluids in the context of general relativity was established by expressing the fluid four-velocity as a sum of products of scalar fields and their gradients.
Abstract: The connection is established between two different action principles for perfect fluids in the context of general relativity. For one of these actions, $S$, the fluid four--velocity is expressed as a sum of products of scalar fields and their gradients (the velocity--potential representation). For the other action, ${\bar S}$, the fluid four--velocity is proportional to the totally antisymmetric product of gradients of the fluid Lagrangian coordinates. The relationship between $S$ and ${\bar S}$ is established by expressing $S$ in Hamiltonian form and identifying certain canonical coordinates as ignorable. Elimination of these coordinates and their conjugates yields the action ${\bar S}$. The key step in the analysis is a point canonical transformation in which all tensor fields on space are expressed in terms of the Lagrangian coordinate system supplied by the fluid. The canonical transformation is of interest in its own right. It can be applied to any physical system that includes a material medium described by Lagrangian coordinates. The result is a Hamiltonian description of the system in which the momentum constraint is trivial.
TL;DR: Relevant aspects of the Hamiltonian path integral and its measure are discussed, and used to show that the quantum-mechanical version of the classical transformation does not leave the measure of the path-integral invariant, instead inducing an anomaly.
Abstract: This paper is a generalization of previous work on the use of classical canonical transformations to evaluate Hamiltonian path integrals for quantum-mechanical systems. Relevant aspects of the Hamiltonian path integral and its measure are discussed, and used to show that the quantum-mechanical version of the classical transformation does not leave the measure of the path-integral invariant, instead inducing an anomaly. The relation to operator techniques and ordering problems is discussed, and special attention is paid to incorporation of the initial and final states of the transition element into the boundary conditions of the problem. Classical canonical transformations are developed to render an arbitrary power potential cyclic. The resulting Hamiltonian is analyzed as a quantum system to show its relation to known quantum-mechanical results. A perturbative argument is used to suppress ordering-related terms in the transformed Hamiltonian in the event that the classical canonical transformation leads to a nonquadratic cyclic Hamiltonian. The associated anomalies are analyzed to yield general methods to evaluate the path integral's prefactor for such systems. The methods are applied to several systems, including linear and quadratic potentials, the velocity-dependent potential, and the time-dependent harmonic oscillator.
TL;DR: In this article, the BRST quantisation of chiral Virasoro and $W_3$ worldsheet gravities was reformulated using momenta in order to put the ghost action back into first-order form.
Abstract: We reformulate the BRST quantisation of chiral Virasoro and $W_3$ worldsheet gravities. Our approach follows directly the classic BRST formulation of Yang-Mills theory in employing a derivative gauge condition instead of the conventional conformal gauge condition, supplemented by an introduction of momenta in order to put the ghost action back into first-order form. The consequence of these simple changes is a considerable simplification of the BRST formulation, the evaluation of anomalies and the expression of Wess-Zumino consistency conditions. In particular, the transformation rules of all fields now constitute a canonical transformation generated by the BRST operator $Q$, and we obtain in this reformulation a new result that the anomaly in the BRST Ward identity is obtained by application of the anomalous operator $Q^2$, calculated using operator products, to the gauge fermion.
TL;DR: Chern-Simons field theory coupled to a nonrelativistic matter field on a sphere is analyzed using canonical transformation on the fields with special attention to the role of the rotation symmetry.
Abstract: We analyze Chern-Simons field theory coupled to a nonrelativistic matter field on a sphere using canonical transformation on the fields with special attention to the role of the rotation symmetry: SO(3) invariance restricts the Hilbert space to the one with a definite number of charges and dictates the Dirac quantization condition to the Chern-Simons coefficient, whereas SO(2) invariance does not. The corresponding Schroedinger equation for many anyons (and for multispecies) on the sphere are presented with appropriate boundary conditions. In the presence of an external magnetic monopole source, the ground state solutions of anyons are compared with monopole harmonics. The role of the translation and modular symmetry on a torus is also expounded.
TL;DR: In this paper, a classical canonical transformation can convert the Poincare group generators from the usual 'instant form' into expressions associated with a light-cone, but there is no consistent Hamiltonian dynamics for quantum wave functions on a lightcone.
Abstract: A classical canonical transformation can convert the Poincare group generators from the usual 'instant form' into expressions associated with a light-cone. However, there is no consistent Hamiltonian dynamics for quantum wave functions on a light-cone, because the radial momentum operator is not self-adjoint.
TL;DR: In this paper, a general introduction to T-duality is given, in the abelian case the approaches of Buscher, and Ro\u{c}ek and Verlinde are reviewed.
Abstract: In these lectures a general introduction to T-duality is given. In the abelian case the approaches of Buscher, and Ro\u{c}ek and Verlinde are reviewed. Buscher's prescription for the dilaton transformation is recovered from a careful definition of the gauge integration measure. It is also shown how duality can be understood as a quite simple canonical transformation. Some aspects of non-abelian duality are also discussed, in particular what is known on relation to canonical transformations. Some implications of the existence of duality on the cosmological constant and the definition of distance in String Theory are also suggested.
TL;DR: In this paper, the authors extend the results of a previous paper to fluids of finite depth, and derive the canonical transformation that eliminates the leading order of nonlinearity for finite depth.
Abstract: We extend the results of a previous paper to fluids of finite depth. We consider the Hamiltonian theory of waves on the free surface of an incompressible fluid, and derive the canonical transformation that eliminates the leading order of nonlinearity for finite depth. As in the previous paper we propose using the Lie transformation method since it seems to include a nearly correct implementation of short waves interacting with long waves. We show how to use the Eikonal method for slowly varying currents and/or depths in combination with the nonlinear transformation. We note that nonlinear effects are more important in water of finite depth. We note that a nonlinear action conservation law can be derived. 10 refs.
CERN1
TL;DR: In this article, it was shown that Buscher's abelian duality transformation rules can be recovered in a very simple way by performing a canonical transformation first suggested by Giveon, Rabinovici and Veneziano.
Abstract: We show that Buscher's abelian duality transformation rules can be recovered in a very simple way by performing a canonical transformation first suggested by Giveon, Rabinovici and Veneziano. We explore the properties of this transformation, and also discuss some aspects of non-abelian duality.
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TL;DR: In this paper, the authors investigated nonperturbative canonical quantization of two dimen-sional dilaton gravity theories with an emphasis on the CGHS model, where a canonical transformation is constructed such that the constraints take a quadratic form.
Abstract: Theoretical Physics Group, Blackett Laboratory, Imperial College,Prince Consort Road, London SW7 2BZ, U.K.ABSTRACTWe investigate nonperturbative canonical quantization of two dimen-sional dilaton gravity theories with an emphasis on the CGHS model. Weuse an approach where a canonical transformation is constructed such thatthe constraints take a quadratic form. The required canonical transforma-tion is obtained by using a method based on the B¨acklund transformationfrom the Liouville theory. We quantize dilaton gravity in terms of the newvariables, where it takes a form of a bosonic string theory with backgroundcharges. Unitarity is then established by going into a light-cone gauge. Asa direct consequence, black holes in this theory do not violate unitarity,and there is no information loss. We argue that the information escapesduring the evaporation process. We also discuss the implications of thisquantization scheme for the quantum fate of real black holes. The mainconclusion is that black holes do not have to violate quantum mechanics.
TL;DR: For a model pairing potential, involving a temperature-dependent energy cutoff [ital c]([ital T]) as a variational parameter, the thermodynamical properties of the conventional intermediate- and strong-coupling superconductors are determined.
Abstract: To treat the phonon-induced electron pairing in a way which allows for vortex corrections and retardation effects, we develop a method using a canonical transformation involving a variational function. This method is expected to be useful for developing a unified theory for phonon-induced electron pairing ranging between adiabatic and nonadiabatic behavior, or, respectively, between Copper-pair condensation in momentum space and bipolaron condensation in real space. To demonstrate the power of our variational treatment we determine for a model pairing potential, involving a temperature-dependent energy cutoff [ital c]([ital T]) as a variational parameter, the thermodynamical properties of the conventional intermediate- and strong-coupling superconductors. Thus, we obtain results which compare well with those obtained by solving the Eliashberg equations.
TL;DR: The SU(m) symmetry underlying the degeneracies in the energy levels of the m-dimensional anisotropic oscillator with commensurate frequencies discussed by Rosensteel and Draayer as mentioned in this paper, in the context of models for super-deformed nuclei, is related to the non-bijective canonical transformation found by Moshinsky and his group.
Abstract: The SU(m) symmetry underlying the degeneracies in the energy levels of the m-dimensional anisotropic oscillator with commensurate frequencies discussed by Rosensteel and Draayer(1989), in the context of models for super-deformed nuclei, is related to the non-bijective canonical transformation found by Moshinsky and his group(1981).
TL;DR: In this article, a general method to extend a point transformation to a canonical transformation increasing the number of variables, in the sense of Scheifele, has been proposed, which has the property of becoming completely automatic after a choice of some adequate constraints.
Abstract: Given a point transformation from a certain domain in ℝ
m
onto a space of lower dimension, we offer a general method to extend it to a canonical transformation increasing the number of variables, in the sense of Scheifele. It has the property of becoming completely automatic after a choice of some adequate constraints, and can even be performed by symbolic manipulation. Questions of degeneration associated to the introduction of redundant variables are also considered, as well as the relations with the corresponding generalized Lagrangean formalism. The paper is completed by examples showing the most popular among those transformations, so that the convenience of the new approach can be easily appreciated.
25 Sep 1994
TL;DR: This paper presents a categorical data-specification mechanism, and it shows that, for an important subset of these specifications, equivalence is indeed decidable: it is shown that every specification can be transformed to a canonical form such that two specifications are equivalent iff they have isomorphic canonical forms.
Abstract: Semantic data-specifications, like Chen's Entity-Relationship specification mechanism ([Ch 76]) have been used for many years in the early stages of database design. More recently, they have become key ingredients of objectoriented software development methodologies ([Co 90, VB 91]). Since most of the data-specification mechanisms used in practice have only weak expressive power, we may hope to find an algorithm to decide wether two specifications are equivalent, in the sense that they have essentially the same models. In this paper, we present a categorical data-specification mechanism, and we show that, for an important subset of these specifications, equivalence is indeed decidable: we show that every specification (in the abovementioned subset) can be transformed to a canonical form such that two specifications are equivalent iff they have isomorphic canonical forms. This property is of utmost importance if r e u s e of data-specifications among software engineers is ever to be put in practice: it ensures us that two specifications of different parts of a given reality can always be combined by identifying the overlapping parts of their canonical forms.
01 Jan 1994
TL;DR: In this article, it was shown that it is possible to transform the equations of motion, expressed in terms of a given set of coordinates, to another set of generalized coordinates for which all the generalized coordinates are ignorable, and that the required transformation may be obtained from a single function governed by a differential equation referred to as the Hamilton-Jacobi equation.
Abstract: We have seen that the equations of motion for dynamical systems can take a variety of forms depending upon the generalized coordinates used. It will also be recalled that when ignorable coordinates arise the associated equations may be partially integrated. Recognizing the equivalence of different forms and attributing the presence of ignorable coordinates to the particular set of generalized coordinates used, it is natural to enquire whether it is possible to transform the equations of motion, expressed in terms of a given set of coordinates, to another set of coordinates for which all the generalized coordinates are ignorable. In this way the problem of solving the equations of motion reduces to a transformation. The researches of Hamilton and Jacobi revealed that the required transformation may be obtained from a single function which is governed by a differential equation referred to as the Hamilton-Jacobi equation.