Showing papers on "Canonical transformation published in 2011"

Collective dipole model of AdS/CFT and higher spin gravity

Abstract: We formulate a first quantized construction of the AdSd + 1/CFTd correspondence using the bi-local representation of the free d-dimensional large-N vector model. The earlier reconstruction of AdS4 higher spin gravity provides a scheme where the AdS spacetime (and higher spin fields) is given by the composite bi-local fields. The underlying first quantized, world-sheet picture is extracted in the present work and generalized to any dimension. A higher spin AdS particle model is shown to emerge from the collective bi-particle system of Minkowski particles through a canonical transformation. As such, this construction provides a simple explicit mechanism of the AdS/CFT correspondence.

On the Munn-Silbey approach to polaron transport with off-diagonal coupling and temperature-dependent canonical transformations.

Abstract: Improved results using a method similar to the Munn-Silbey approach have been obtained on the temperature dependence of transport properties of an extended Holstein model incorporating simultaneous diagonal and off-diagonal exciton-phonon coupling. The Hamiltonian is partially diagonalized by a canonical transformation, and optimal transformation coefficients are determined in a self-consistent manner. Calculated transport properties exhibit substantial corrections on those obtained previously by Munn and Silbey for a wide range of temperatures thanks to a numerically exact evaluation and an added momentum-dependence of the transformation matrix. Results on the diffusion coefficient in the moderate and weak coupling regime show distinct band-like and hopping-like transport features as a function of temperature. PACS numbers: I. INTRODUCTION Investigations on organic molecular crystals pioneered by Pope and cowokers 1 more than half a century ago made an immense impact on various fields of relevance in physics, chemistry, and materials science. The interplay between geometric and electronic structures has opened up novel materials with unlimited possibilities. The discovery of conducting properties of doped polyacetylene 2

On the Munn-Silbey approach to polaron transport with off-diagonal coupling

Abstract: Improved results using a method similar to the Munn-Silbey approach have been obtained on the temperature dependence of transport properties of an extended Holstein model incorporating simultaneous diagonal and off-diagonal exciton-phonon coupling The Hamiltonian is partially diagonalized by a canonical transformation, and optimal transformation coefficients are determined in a self-consistent manner Calculated transport properties exhibit substantial corrections on those obtained previously by Munn and Silbey for a wide range of temperatures thanks to a numerically exact evaluation and an added momentum-dependence of the transformation matrix Results on the diffusion coefficient in the moderate and weak coupling regime show distinct band-like and hopping-like transport features as a function of temperature

Uniqueness of the Fock quantization of fields with unitary dynamics in nonstationary spacetimes

Abstract: is introduced as part of a linear, time dependent canonical transformation in phase space. In this context, we prove in full detail a uniqueness result about the Fock quantization requiring that the dynamics be unitary and the spatial symmetries of the field equations have a natural unitary implementation. The main conclusion is that, with those requirements, only one particular canonical transformation is allowed, and thus only one choice of the field-momentum pair (up to irrelevant constant scalings). This complements another previous uniqueness result for scalar fields with a time varying mass on S 3 , which selects a specific equivalence class of Fock representations of the canonical commutation relations under the conditions of a unitary evolution and the invariance of the vacuum under the background symmetries. In total, the combination of these two different statements of uniqueness picks up a unique Fock quantization for the system. We also extend our proof of uniqueness to other compact topologies and spacetime dimensions.

Canonical pure spinor (fermionic) T-duality

Abstract: We establish that the recently discovered fermionic T-duality can be viewed as a canonical transformation in phase space. This requires a careful treatment of constrained Hamiltonian systems. Additionally, we show how the canonical transformation approach for bosonic T-duality can be extended to include Ramond–Ramond backgrounds in the pure spinor formalism.

Conformal mechanics inspired by extremal black holes in d = 4

Abstract: A canonical transformation which relates the model of a massive relativistic particle moving near the horizon of an extremal black hole in four dimensions and the conventional conformal mechanics is constructed in two different ways. The first approach makes use of the action-angle variables in the angular sector. The second scheme relies upon integrability of the system in the sense of Liouville.

Collective Dipole Model of AdS/CFT and Higher Spin Gravity

Abstract: We formulate a first quantized construction of the AdS_{d+1}/CFT_d correspondence using the bi-local representation of the free d-dimensional large N vector model. The earlier reconstruction of AdS_4 higher-spin gravity provides a scheme where the AdS spacetime (and higher-spin fields) are given by the composite bi-local fields. The underlying first quantized, world-sheet picture is extracted in the present work and generalized to any dimension. A higher-spin AdS particle model is shown to emerge from the collective bi-particle system of Minkowski particles through a canonical transformation. As such this construction provides a simple explicit mechanism of the AdS/CFT correspondence.

Resonances of the SD oscillator due to the discontinuous phase

Abstract: Resonance phenomena of a harmonically excited system with mul-tiple potential well play an important role in nonlinear dynamics research. In this paper, we investigate the resonant behaviours of a discontinuous dynamical system with double well potential derived from the SD oscillator to gain better understanding of the transition of resonance mechanism. Firstly, the time dependent Hamiltonian is obtained for a Duffing type discontinuous system modelling snap-through buckling. This system comprises two subsystems connected at x = 0, for which the system is discontinuous. We construct a series of generating functions and canonical transformations to obtain the canonical form of the system to investigate the complex resonant behaviours of the system. Furthermore, we introduce a composed winding number to explore complex resonant phenomena. The formulation for resonant phenomena given in this paper generalizes the formulation of n Omega0 = m Omega used in the regular perturbation theory, where n and m are relative prime integers, Omega 0 and Omega are the natural frequency and external frequencies respectively. Understanding the resonant behaviour of the SD oscillator at the discontinuous phase enables us to further reveal the vibrational energy transfer mechanism between smooth and discontinuous nonlinear dynamical systems

Mutation-periodic quivers, integrable maps and associated Poisson algebras

Abstract: We consider a class of map, recently derived in the context of cluster mutation. In this paper, we start with a brief review of the quiver context, but then move onto a discussion of a related Poisson bracket, along with the Poisson algebra of a special family of functions associated with these maps. A bi-Hamiltonian structure is derived and used to construct a sequence of Poisson-commuting functions and hence show complete integrability. Canonical coordinates are derived, with the map now being a canonical transformation with a sequence of commuting invariant functions. Compatibility of a pair of these functions gives rise to Liouville’s equation and the map plays the role of a Backlund transformation.

Relativistic Statistical Mechanics vs. Relativistic Thermodynamics

Abstract: Based on a covariant theory of equilibrium Thermodynamics, a Statistical Relativistic Mechanics is developed for the non-interacting case. Relativistic Thermodynamics and Juttner Relativistic Distribution Function in a moving frame are obtained by using this covariant theory. A proposal for a Relativistic Statistical Mechanics is exposed for the interacting case.

Hamiltonian formulation for the theory of gravity and canonical transformations in extended phase space

Abstract: A starting point for this work was the statement recently discussed in the literature that two Hamiltonian formulations for the theory of gravity, the one proposed by Dirac and the other by Arnowitt?Deser?Misner, may not be related by a canonical transformation. In its turn, it raises a question about the equivalence of these two Hamiltonian formulations and their equivalence to the original formulation of general relativity. We argue that, since the transformation from the components of a metric tensor to the ADM variables touches gauge degrees of freedom, which are non-canonical from the point of view of Dirac, the problem cannot be resolved in the limits of the Dirac approach. The proposed solution requires the extension of phase space by treating gauge degrees of freedom on an equal footing with other variables and introducing missing velocities into the Lagrangian by means of gauge conditions in differential form. We illustrate with a simple cosmological model the features of Hamiltonian dynamics in extended phase space. Then, we give a clear proof for the full gravitational theory that the ADM-like transformation is canonical in extended phase space in a wide enough class of possible parametrizations.

Generating functions for volume-preserving transformations

Abstract: A general implicit solution for determining volume-preserving transformations in the n -dimensional Euclidean space is obtained in terms of a set of 2 n generating functions in mixed coordinates. For n =2, the proposed representation corresponds to the classical definition of a potential stream function in a canonical transformation. For n =3, the given solution defines a more general class of isochoric transformations, when compared to existing methods based on multiple potentials. Illustrative examples are discussed both in rectangular and in cylindrical coordinates for applications in mechanical problems of incompressible continua. Solving exactly the incompressibility constraint, the proposed representation method is suitable for determining three-dimensional isochoric perturbations to be used in bifurcation theory. Applications in non-linear elasticity are envisaged for determining the occurrence of complex instability patterns for soft elastic materials.

Hamiltonian formulation for the theory of gravity and canonical transformations in extended phase space

Abstract: A starting point for the present work was the statement recently discussed in the literature that two Hamiltonian formulations for the theory of gravity, the one proposed by Dirac and the other by Arnowitt - Deser - Misner, may not be related by a canonical transformation. In its turn, it raises a question about the equivalence of these two Hamiltonian formulations and their equivalence to the original formulation of General Relativity. We argue that, since the transformation from components of metric tensor to the ADM variables touches gauge degrees of freedom, which are non-canonical from the point of view of Dirac, the problem cannot be resolved in the limits of the Dirac approach. The proposed solution requires the extension of phase space by treating gauge degrees of freedom on an equal footing with other variables and introducing missing velocities into the Lagrangian by means of gauge conditions in differential form. We illustrate with a simple cosmological model the features of Hamiltonian dynamics in extended phase space. Then, we give a clear proof for the full gravitational theory that the ADM-like transformation is canonical in extended phase space in a wide enough class of possible parametrizations.

Lie algebraic approach and quantum treatment of an anisotropic charged particle via the quadratic invariant

Abstract: We consider the problem of a charged harmonic oscillator under the influence of a constant magnetic field. The system is assumed to be anisotropic and the magnetic field is applied along z-axis. A canonical transformation is invoked to remove the interaction term and the system is reduced to a model contains two uncoupled harmonic oscillators. Two classes of real and complex quadratic invariants (constants of motion) are obtained. We employ the Lie algebraic technique to find the most general solution for the wave-function for both real and complex invariants. The quadratic invariant is also used to derive two classes of creation and annihilation operators from which the wave-functions in the coherent states and number states are obtained. Some discussion related to the advantage of using the quadratic invariants to solve the Cauchy problem instead of the direct use of the Hamiltonian itself is also given.

Path integral treatment of the one-dimensional Klein–Gordon oscillator with minimal length

Abstract: The path integral method is used to present an exact treatment of the one-dimensional Klein–Gordon oscillator in the context of quantum mechanics with a deformed Heisenberg algebra of the form , leading to the existence of a minimal observable length . We tackle the problem in the momentum space and we use the Schwinger proper-time method to represent Green's function. Calculations are run with the help of the point canonical transformation technique. The bound-state energy spectrum and the associated momentum space eigenfunctions are obtained, and a detailed comparison with the results for the undeformed case is made.

An accurate symplectic calculation of the inboard magnetic footprint from statistical topological noise and field errors in the DIII-D

Abstract: Any canonical transformation of Hamiltonian equations is symplectic, and any area-preserving transformation in 2D is a symplectomorphism. Based on these, a discrete symplectic map and its continuous symplectic analog are derived for forward magnetic field line trajectories in natural canonical coordinates. The unperturbed axisymmetric Hamiltonian for magnetic field lines is constructed from the experimental data in the DIII-D [J. L. Luxon and L. E. Davis, Fusion Technol. 8, 441 (1985)]. The equilibrium Hamiltonian is a highly accurate, analytic, and realistic representation of the magnetic geometry of the DIII-D. These symplectic mathematical maps are used to calculate the magnetic footprint on the inboard collector plate in the DIII-D. Internal statistical topological noise and field errors are irreducible and ubiquitous in magnetic confinement schemes for fusion. It is important to know the stochasticity and magnetic footprint from noise and error fields. The estimates of the spectrum and mode amplitude...

Quantizing preserving Noether symmetries

Abstract: A procedure which obviates the constraint imposed by the conflict between consistent quantization and the invariance of the Hamiltonian description under nonlinear canonical transformation is proposed. This new quantization scheme preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schr\"odinger's equation. As an example, the quantization of the `goldfish' many-body problem extensively studied by Calogero et al. is presented.

Comment on `Hamiltonian formulation for the theory of gravity and canonical transformations in extended phase space' by T P Shestakova

Abstract: We argue that the conclusion, `we cannot consider the Dirac approach as fundamental and undoubted', made in the paper by Shestakova (Class. Quantum Grav. 28 055009, 2011), is based upon an incomplete and flawed analysis of the simple model presented in section 3 of the article. We re-examine the analysis of this model and find that it does not support the author's conclusion. For the theory of gravity neither the equivalence of the effective action nor its Hamiltonian formulation is given by the author, therefore, we only provide a brief commentary.

Canonical Forms of Linear Nonstationary Observation Systems with Quasidifferentiable Coefficients with Respect to Various Transformation Groups

Abstract: We obtain conditions for the existence and suggest a method for the construction of canonical forms of linear differential observation systems with the use of arbitrary linear transformation groups on the basis of the technique of quasidifferentiation with respect to lower triangular matrices.

An alternative approach to Schrödinger equations with a spatially varying mass

Abstract: Extending the point canonical transformation approach in a manner distinct from the previous ones, we propose a unified approach of generating potentials of all classes having non-constant masses.

Hydrodynamics, particle relabelling and relativity

Abstract: Using the wave equation as an example, it is shown how to extend the hydrodynamic Lagrangian-picture method of constructing field evolution using a continuum of trajectories to second-order theories. The wave equation is represented through Eulerian-picture models that are distinguished by their Lorentz transformation properties. Introducing the idea of the relativity of the particle label, it is demonstrated how the corresponding trajectory models are compatible with the relativity principle. It is also shown how the Eulerian variational formulation may be obtained by canonical transformation from the Lagrangian picture, and how symmetries in the Lagrangian picture may be used to generate Eulerian conserved charges.

The method of variation on parameters for integration of a generalized Birkhoffian system

Abstract: This paper focuses on studying the integration method of a generalized Birkhoffian system. The method of variation on parameters for the dynamical equations of a generalized Birkhoffian system is presented. The procedure for solving the problem can be divided into two steps: the first step, a system of auxiliary equations is constructed and its general solution is given; the second step, the parameters are varied, and the solution of the problem is obtained by using the properties of generalized canonical transformation. The method of variation on parameters for the generalized Birkhoffian system is of universal significance, and we take a nonholonomic system and a nonconservative system as examples to illustrate the application of the results of this paper.

An alternative approach to Schr\"odinger equations with a spatially varying mass

Abstract: Extending the point canonical transformation approach in a manner distinct from the previous ones, we propose a unified approach of generating potentials of all classes having non-constant masses.

The Key to Three-View Geometry

Abstract: In this article we describe a set of canonical transformations of the image spaces that make the description of three-view geometry very simple. The transformations depend on the three-view geometry and the canonically transformed trifocal tensor $\mathcal {T}'$ takes the form of a sparse array where 17 elements in well-defined positions are zero, it has a linear relation to the camera matrices and to two of the fundamental matrices, a third order relation to the third fundamental matrix, a second order relation to the other two trifocal tensors, and first order relations to the 10 three-view all-point matching constraints. In this canonical form, it is also simple to determine if the corresponding camera configuration is degenerate or co-linear. An important property of the three canonical transformations of the images spaces is that they are in SO(3). The 9 parameters needed to determine these transformations and the 9 parameters that determine the elements of $\mathcal {T}'$ together provide a minimal parameterization of the tensor. It does not have problems with multiple maps or multiple solutions that other parameterizations have, and is therefore simple to use. It also provides an implicit representation of the trifocal internal constraints: the sparse canonical representation of the trifocal tensor can be determined if and only if it is consistent with its internal constraints. In the non-ideal case, the canonical transformation can be determined by solving a minimization problem and a simple algorithm for determining the solution is provided. This allows us to extend the standard linear method for estimation of the trifocal tensor to include a constraint enforcement as a final step, similar to the constraint enforcement of the fundamental matrix. Experimental evaluation of this extended linear estimation method shows that it significantly reduces the geometric error of the resulting tensor, but on average the algebraic estimation method is even better. For a small percentage of cases, however, the extended linear method gives a smaller geometric error, implying that it can be used as a complement to the algebraic method for these cases.