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Showing papers on "Canonical transformation published in 1995"


Journal ArticleDOI
01 Apr 1995
TL;DR: In this paper, a general introduction to T-duality is given, in the abelian case the approaches of Buscher and Roucek and Verlinde are reviewed, and some implications of the existence of duality on the cosmological constant and the definition of distance in String Theory are also suggested.
Abstract: In these lectures a general introduction to T-duality is given. In the abelian case the approaches of Buscher, and Roucek 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.

302 citations


Journal ArticleDOI
TL;DR: A new method of obtaining local projective and affine invariants is developed and implemented for real images and is much less sensitive to occlusion than global invariants.
Abstract: Geometric invariants are shape descriptors that remain unchanged under geometric transformations such as projection or changing the viewpoint. A new method of obtaining local projective and affine invariants is developed and implemented for real images. Being local, the Invariants are much less sensitive to occlusion than global invariants. The invariants' computation is based on a canonical method. This consists of defining a canonical coordinate system by the intrinsic properties of the shape, independently of the given coordinate system. Since this canonical system is independent of the original one, it is invariant and all quantities defined in it are invariant. The method was applied without the use of a curve parameter. This was achieved by fitting an implicit polynomial to an arbitrary curve in a vicinity of each curve point. Several configurations are treated: a general curve without any correspondence and curves with known correspondences of one or two feature points or lines. Experimental results for different 2D objects in 3D space are presented. >

94 citations


Journal ArticleDOI
TL;DR: In this article, a connection between the Hirota-Satsuma coupled KdV system and (a generalisation of) the 1:6:1 integrable case quartic potential is made.

63 citations


Journal ArticleDOI
TL;DR: In this paper, path integral expressions for three canonical formalisms (Ostrogradski's one, constrained one and generalized one) of higher-derivative theories are given for both nonsingular and singular cases.
Abstract: Path integral expressions for three canonical formalisms -- Ostrogradski's one, constrained one and generalized one -- of higher-derivative theories are given. For each fomalism we consider both nonsingular and singular cases. It is shown that three formalisms share the same path integral expressions. In paticular it is pointed out that the generalized canonical formalism is connected with the constrained one by a canonical transformation.

40 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that S-duality in four dimensional non-supersymmetric abelian gauge theories can be formulated as a canonical transformation in the phase space of the theory.

39 citations


Journal ArticleDOI
TL;DR: In this paper, a simple compact Lie group $G$ with a bi-invariant metric and a generating function was used to generate a string target-space dual pair at the classical level under the Hamiltonian formalism.
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 group $G$ with a bi-invariant metric and a generating function $\Gamma$ suggested in the physics literature, we follow the above line of thought and work out the canonical transformation $\Phi$ generated by $\Gamma$ together with an $\Ad$-invariant metric and a B-field on the associated Lie algebra $\frak g$ of $G$ so that $G$ and $\frak g$ 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 $\Phi$ including a careful analysis of its domain and image. The geometry of the T-dual structure on $\frak g$ is lightly touched.

31 citations


ReportDOI
01 Feb 1995
TL;DR: In this paper, the authors construct singular eigenfunctions corresponding to the continuous spectrum of eigenvalues for shear flow in a channel and then solve the initial-value problem to establish that these modes, together with any discrete, growing/decaying pairs of modes, comprise a complete basis.
Abstract: The authors construct singular eigenfunctions corresponding to the continuous spectrum of eigenvalues for shear flow in a channel. These modes are irregular as a result of a singularity in the eigenvalue problem at the critical layer of each mode. They consider flows with monotonic shear, so there is only a single critical layer for each mode. They then solve the initial-value problem to establish that these continuum modes, together with any discrete, growing/decaying pairs of modes, comprise a complete basis. They also view the problem within the framework of Hamiltonian theory. In that context, the singular solutions can be viewed as the kernel of an integral, canonical transformation that allows us to write the fluid system, an infinite-dimensional Hamiltonian system, in action-angle form. This yields an expression for the energy in terms of the continuum modes and provides a means for attaching a characteristic signature (sign) to the energy associate with each eigenfunction. They follow on to consider shear-flow stability within the Hamiltonian framework. Next, the authors show the equivalence of integral superpositions of the singular eigenfunctions with the solution derived with Laplace transform techniques. In the long-time limit, such superpositions have decaying integral averages across the channel, revealing phase mixing or continuum damping. Under some conditions, this decay is exponential and is then the fluid analogue of Landau damping. Finally, the authors discuss the energetics of continuum damping.

29 citations


Journal ArticleDOI
TL;DR: In this article, a geometric interpretation of Kuchar's canonical transformation in terms of the theory of quasilocal energy-momentum in general relativity given by Brown and York is discussed.
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. 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 $M_{S}$, expressed in terms of what are essentially Arnowitt-Deser-Misner variables, as a canonical coordinate. 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 Kucha\v{r}'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.

29 citations


Journal ArticleDOI
TL;DR: The quantum-mechanical version of the four kinds of classical canonical transformations is investigated by using non-Hermitian operator techniques, showing that when one uses the classical action as a generating function of the quantum canonical transformation of time evolutions of state vectors, the corresponding propagator can easily be obtained.
Abstract: The quantum-mechanical version of the four kinds of classical canonical transformations is investigated by using non-Hermitian operator techniques. To help understand the usefulness of this approach, the eigenvalue problem of a harmonic oscillator is solved in two different types of canonical transformations. The quantum form of the classical Hamilton-Jacobi theory is also employed to solve time-dependent Schr\"odinger wave equations, showing that when one uses the classical action as a generating function of the quantum canonical transformation of time evolutions of state vectors, the corresponding propagator can easily be obtained.

22 citations


Posted Content
TL;DR: In this article, a canonical formalism for spherical symmetry, originally developed by Kucha\v{r} to describe vacuum Schwarzschild black holes, is extended to include a spherically symmetric, massless, scalar field source.
Abstract: A canonical formalism for spherical symmetry, originally developed by Kucha\v{r} to describe vacuum Schwarzschild black holes, is extended to include a spherically symmetric, massless, scalar field source. By introducing the ADM mass as a canonical coordinate on phase space, one finds that the super-Hamiltonian and supermomentum constraints for the coupled system simplify considerably. Yet, despite this simplification, it is difficult to find a functional time formalism for the theory. First, the configuration variable that played the role of time for the vacuum theory is no longer a spacetime scalar once spherically symmetric matter is coupled to gravity. Second, although it is possible to perform a canonical transformation to a new set of variables in terms of which the super-Hamiltonian and supermomentum constraints can be solved, the new time variable also fails to be a spacetime scalar. As such, our solutions suffer from the so-called {\it spacetime problem of time}. A candidate for a time variable that {\it is} a spacetime scalar is presented. Problems with turning this variable into a canonical coordinate on phase space are discussed.

13 citations


Journal ArticleDOI
TL;DR: In this paper, the authors discuss the quantum potential approach of Bohm in the context of quantum cosmological model, which makes it possible to convert the wave function of the universe to a set of equations describing the time evolution.
Abstract: In this paper we discuss the quantum potential approach of Bohm in the context of quantum cosmological model. This approach makes it possible to convert the wavefunction of the universe to a set of equations describing the time evolution of the universe. Following Ashtekar et.\ al., we make use of quantum canonical transformation to cast a class of quantum cosmological models to a simple form in which they can be solved explicitly, and then we use the solutions do recover the time evolution.

Journal ArticleDOI
TL;DR: In this article, the authors use Boulware's Hamiltonian formalism of quadratic gravity theories in order to analyse the classical behaviour of Bianchi cosmological models for a Lagrangian density in four spacetime dimensions.
Abstract: We use Boulware's Hamiltonian formalism of quadratic gravity theories in order to analyse the classical behaviour of Bianchi cosmological models for a Lagrangian density in four spacetime dimensions. For this purpose we define a canonical transformation which leads to a clear distinction between two main variants of the general quadratic theory, i.e. for or conformal Lagrangian densities. In this paper we restrict the study to the first variant. For the Bianchi-type I and IX models, we give the explicit forms of the super-Hamiltonian constraint, of the ADM Hamiltonian density and of the corresponding canonical equations. In the case of a pure quadratic theory , we solve them analytically for the Bianchi I model. For the Bianchi-type IX model, we reduce the first-order equations of the Hamiltonian system to three coupled second-order equations for the true physical degrees of freedom. This discussion is extended to isotropic FLRW models.

Journal ArticleDOI
TL;DR: This paper uses the Batalin-Fradkin-Vilkovisky (BFV) formalism to study a recently proposed nonlocal symmetry of QED and shows that this symmetry stems from a canonical transformation in the ghost sector.
Abstract: In this paper we use the Batalin-Fradkin-Vilkovisky (BFV) formalism to study a recently proposed nonlocal symmetry of QED. In the BFV extended phase space we show that this symmetry stems from a canonical transformation in the ghost sector. {copyright} 1995 The American Physical Society.

Journal ArticleDOI
01 Sep 1995
TL;DR: In this article, it was shown that both the minimal coupling and the multipolar Hamiltonians are two forms of the same Hamiltonian corresponding to two choices of gauge: div A = 0 and r · A( r ) = 0 respectively.
Abstract: The multipolar Hamiltonian has many advantages for describing the electrodynamics of nonrelativistic material systems. Usually it is derived by performing a canonical transformation on the minimal coupling Hamiltonian. We show that both the minimal coupling and the multipolar Hamiltonians are two forms of the same Hamiltonian corresponding to two choices of gauge: div A = 0 and r · A( r ) = 0 respectively. We further discuss the use of the multipolar Hamiltonian in electronically extended systems.

Posted Content
TL;DR: In this article, the authors examined the electric-magnetic duality for a U(1) gauge theory on a general four manifold which generates the SL(2, Z) group.
Abstract: We examine the electric-magnetic duality for a U(1) gauge theory on a general four manifold which generates the SL(2, Z) group. The partition functions for such a theory transforms as a modular form of specific weight. However, in the canonical approach, we show that S-duality for the abelian theory, like T-duality, is generated by a canonical transformation leading to a modular invariant partition function.

Journal ArticleDOI
TL;DR: GITA as discussed by the authors is a program for computing the normal Birkhoff-Gustavson form in Cartesian as well as angle-action coordinates and the formal integral of the motion for a polynomial Hamiltonian near an equilibrium point.

Journal ArticleDOI
TL;DR: In this paper, 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.

Journal ArticleDOI
TL;DR: In this paper, a transition from Lagrangian to Hamiltonian description of constrained dynamics was proposed, which allows one to clarify the canonical transformation sense in phase space for these systems and, in particular, to describe the Lagrangians which give the same dynamics.
Abstract: A standard procedure for transition from Lagrangian to Hamiltonian description of constrained dynamics was proposed by the author in [1]. This work shows that the procedure allows one to clarify the canonical transformation sense in phase space for these systems and, in particular, to describe the Lagrangian class which gives the same dynamics.

Journal ArticleDOI
TL;DR: In this article, it was shown that one can construct an exactly soluble two-dimensional higher-derivative fermionic quantum field theory with only derivatives of even order whose classical Lagrangian exhibits chiralgauge invariance.
Abstract: It has been suggested that the chiral symmetry can be implemented only in classical Lagrangians containing higher covariant derivatives of odd order. Contrary to this belief, it is shown that one can construct an exactly soluble two-dimensional higher-derivative fermionic quantum field theory containing only derivatives of even order whose classical Lagrangian exhibits chiralgauge invariance. The original field solution is expressed in terms of usual Dirac spinors through a canonical transformation, whose generating function allows the determination of the new Hamiltonian. It is emphasized that the original and transformed Hamiltonians are different because the mapping from the old to the new canonical variables depends explicitly on time. The violation of cluster decomposition is discussed and the general Wightman functions satisfying the positive-definiteness condition are obtained.

Posted Content
TL;DR: In this paper, the Shanmugadhasan canonical transformation was used to find a canonical basis of Dirac's observables for Yang-Mills theory with fermions.
Abstract: Relevant physical models are described by singular Lagrangians, so that their Hamiltonian description is based on the Dirac theory of constraints. The qualitative aspects of this theory are now understood, in particular the role of the Shanmugadhasan canonical transformation in the determination of a canonical basis of Dirac's observables allowing the elimination of gauge degrees of freedom from the classical description of physical systems. This programme was initiated by Dirac for the electromagnetic field with charged fermions. Now Dirac's observables for Yang-Mills theory with fermions (whose typical application is QCD) have been found in suitable function spaces where the Gribov ambiguity is absent. Also the ones for the Abelian Higgs model are known and those for the $SU(2) \times U(1)$ electroweak theory with fermions are going to be found with the same method working for the Abelian case. The main task along these lines will now be the search of Dirac's observables for tetrad gravity in the case of asymptotically flat 3-manifolds. The philosophy behind this approach is ``first reduce, then quantize": this requires a global symplectic separation of the physical variables from the gauge ones so that the role of differential geometry applied to smooth field configurations is dominating, in contrast with the standard approach of ``first quantizing, then reducing", where, in the case of gauge field theory, the reduction process takes place on distributional field configurations, which dominate in quantum measures. This global separation has been accomplished till now, at least at a heuristic level, and one is going to have a classical (pseudoclassical for the fermion) variables basis for the physical description of the $SU(3)\times SU(2)\times U(1)$ standard model; instead, with tetrad gravity one expects to

Journal ArticleDOI
TL;DR: In this paper, the authors used reduced density matrix methods of general applicability in manybody physics to study one-particle properties of a simple, soluble quantum system, the fermionic anharmonic oscillator.
Abstract: We use reduced density matrix methods of general applicability in many‐body physics to study one‐particle properties of a simple, soluble quantum system, the fermionic anharmonic oscillator. The time evolution of a general initial state of this system is worked out and analyzed in detail from this point of view. We use a canonical transformation of the Bogoliubov type to enhance the power of the one‐particle reduced description and show that the reduced density matrix corresponds in general to a statistical mixture even if the state of the entire many‐body system is described in terms of a state vector (pure state).

Journal ArticleDOI
S. F. Hassan1
TL;DR: In this article, the authors studied the non-localization of extended worldsheet supersymmetry under T-duality, when the associated complex structure depends on the coordinate with respect to which duality is performed.
Abstract: We study the non-localization of extended worldsheet supersymmetry under T-duality, when the associated complex structure depends on the coordinate with respect to which duality is performed. First, the canonical transformation which implements T-duality is generalized to the supersymmetric non-linear $\sigma$-models. Then, we obtain the non-local object which replaces the complex structure in the dual theory and write down the condition it should satisfy so that the dual action is invariant under the non-local supersymmetry. For the target space, this implies that the supersymmetry transformation parameter is a non-local spinor. The analogue of the Killing equation for this non-local spinor is obtained. It is argued that in the target space, the supersymmetry is no longer realized in the standard way. The string theoretic origin of this phenomenon is briefly discussed.

Journal ArticleDOI
TL;DR: In this paper, it was shown that one can construct an exactly soluble two-dimensional higher-derivative fermionic quantum field theory with only derivatives of even order whose classical Lagrangian exhibits chiral-gauge invariance.
Abstract: It has been suggested that the chiral symmetry can be implemented only in classical Lagrangians containing higher covariant derivatives of odd order. Contrary to this belief, it is shown that one can construct an exactly soluble two-dimensional higher-derivative fermionic quantum field theory containing only derivatives of even order whose classical Lagrangian exhibits chiral-gauge invariance. The original field solution is expressed in terms of usual Dirac spinors through a canonical transformation, whose generating function allows the determination of the new Hamiltonian. It is emphasized that the original and transformed Hamiltonians are different because the mapping from the old to the new canonical variables depends explicitly on time. The violation of cluster decomposition is discussed and the general Wightman functions satisfying the positive-definiteness condition are obtained.


Journal ArticleDOI
TL;DR: In this paper, a new pairing theory for many-fermion systems is obtained via a framework related to the one recently introduced to describe Dirac particles in external potentials, and it is shown that the standard Bogoliubov-Valatin canonical transformation treatment of the quasi-particle BCS singlet pairing mechanism naturally falls within this framework.


Journal ArticleDOI
TL;DR: In this paper, a model of spinless fermions on the square latticeZ2 with an interaction potential of strengthU>0 at distance one and strengthJ at distance two, in the largeU limit |t|, |J|≪U, wheret is the hopping amplitude.
Abstract: We consider a model of spinless fermions on the square latticeZ2 with an interaction potential of strengthU>0 at distance one and strengthJ at distance two, in the largeU limit |t|, |J|≪U, wheret is the hopping amplitude. As the chemical potential μ is varied, ift=T=0 we find three different phases corresponding to full, half and zero filling fractions. We study the system at low temperatureT≧0 by a method involving a canonical transformation and a functional integral representation. IfT=0 we locate the phase boundaries of the Mott metal-insulator transition for all |J|≪U with upper and lower bounds, show that mean field theory is valid ifJ 0 we have only one sided bounds for the phase boundaries and we can't validate mean field theory in caseJ<0. We introduce a new resummation scheme for low temperature expansions which yields finite and convergent perturbation series and permits us to study issues like the sign problem. Our algorithm gives an optimal canonical transformation for the functional integral such that the expectation of the sign observableS is ≧ exp\(( - c \cdot t \cdot Ve^{ - \frac{\beta }{2}} )\), whereV is the volume and β=T−1.

Journal ArticleDOI
TL;DR: In this article, the authors derived simple semi-analytical expressions for the Stark energy levels of a general symmetric-top molecule placed in a uniform electric field of arbitrary strength.
Abstract: We derive simple semi-analytical expressions for the Stark energy levels of a general symmetric-top molecule placed in a uniform electric field of arbitrary strength These are obtained by means of third-order perturbation theory following a suitable canonical transformation of the system Hamiltonian, which is conveniently written in terms of the generators of either of the Lie algebras SO(3) or SO(2,1)

Posted Content
TL;DR: In this article, 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.

Dissertation
01 Jan 1995
TL;DR: Guillemin et al. as discussed by the authors studied the Radon transform from RP3 to the Lagrangian Grassmanian, and used the representation theory of Sp(4, R) to characterize the kernel and the range of the transform.
Abstract: In this paper, we study the Radon transform from RP3 to the Lagrangian Grassmanian. We use the representation theory of Sp(4, R) to characterize the kernel and the range of the Radon transform. We explicitly construct a Fourier integral operator on RP3 which picks off the kernel for us and we give a number of descriptions of its associated canonical transformation. Thesis Supervisor: Victor Guillemin Title: Professor of Mathematics