Showing papers on "Canonical transformation published in 2005"

New approach to (quasi-)exactly solvable Schrödinger equations with a position-dependent effective mass

Abstract: By using the point canonical transformation approach in a manner distinct from previous ones, we generate some new exactly solvable or quasi-exactly solvable potentials for the one-dimensional Schrodinger equation with a position-dependent effective mass. In the latter case, SUSYQM techniques provide us with some additional new potentials.

Direct algorithm for pole placement by state-derivative feedback for multi-inputlinear systems - nonsingular case

Abstract: This paper deals with the direct solution of the pole placement problem by state-derivative feedback for multi- input linear systems. The paper describes the solution of this pole placement problem for any controllable system with nonsingular system matrix and nonzero desired poles. Then closed-loop poles can be placed in order to achieve the desired system performance. The solving procedure results into a formula similar to Ackermann one. Its derivation is based on the transformation of linear multi-input systems into Frobenius canonical form by coordinate transformation, then solving the pole placement problem by state derivative feedback and transforming the solution into original coordinates. The procedure is demonstrated on examples. In the present work, both time- invariant and time-varying systems are treated.

Adaptive Geometric Integrators for Hamiltonian Problems with Approximate Scale Invariance

Abstract: We consider adaptive geometric integrators for the numerical integration of Hamiltonian systems with greatly varying time scales. A time regularization is considered using either the Sundman or the Poincare transformation. In the latter case, this gives a new Hamiltonian which is usually separable, but with one of the parts not always exactly solvable. This system can be numerically integrated with a splitting scheme where each part can be computed using a symplectic implicit or explicit method, preserving the qualitative properties of the exact solution. In this case, a backward error analysis for the numerical integration is presented. For a one-dimensional near singular problem, this analysis reveals a strong dependence of the performance of the method with the choice of the monitor function g, which is also observed when using other symmetric nonsymplectic integrators. We also show how this dependence greatly increases with the order of the numerical integrator used. The optimal choice corresponds to the function g, which nearly preserves the scaling invariance of the system. Numerical examples supporting this result are presented. In some cases a canonical transformation can also be considered, making the system more regular or easy to compute, and this is also illustrated with some examples.

On the Whitham hierarchy: dressing scheme, string equations and additional symmetrie

Abstract: A new description of the universal Whitham hierarchy in terms of a factorization problem in the Lie group of canonical transformations is provided. This scheme allows us to give a natural description of dressing transformations, string equations and additional symmetries for the Whitham hierarchy. We show how to dress any given solution and prove that any solution of the hierarchy may be undressed, and therefore comes from a factorization of a canonical transformation. A particulary important function, related to the $\tau$-function, appears as a potential of the hierarchy. We introduce a class of string equations which extends and contains previous classes of string equations considered by Krichever and by Takasaki and Takebe. The scheme is also applied for an convenient derivation of additional symmetries. Moreover, new functional symmetries of the Zakharov extension of the Benney gas equations are given and the action of additional symmetries over the potential in terms of linear PDEs is characterized.

Consistent discretization and canonical classical and quantum Regge calculus

Abstract: We apply the ``consistent discretization'' technique to the Regge action for (Euclidean and Lorentzian) general relativity in arbitrary number of dimensions. The result is a well defined canonical theory that is free of constraints and where the dynamics is implemented as a canonical transformation. This provides a framework for the discussion of topology change in canonical quantum gravity. In the Lorentzian case, the framework appears to be naturally free of the ``spikes'' that plague traditional formulations. It also provides a well defined recipe for determining the measure of the path integral.

The Lagrangian origin of MHV rules

Abstract: We construct a canonical transformation that takes the usual Yang-Mills action into one whose Feynman diagram expansion generates the MHV rules. The off-shell continuation appears as a natural consequence of using light-front quantisation surfaces. The construction extends to include massless fermions.

The Untangling Transformation

Abstract: It is always possible to find canonical forms for quadratic Hamiltonians. In cases in which the eigenvalues of the associated linear system are simple, either real or pure imaginary, the canonical form introduces action-angle coordinates that are most useful for the application of perturbation theory; this is the case in which the quadratic Hamiltonian is the first term in the expansion of a nonlinear Hamiltonian around an equilibrium. The general theory is rather involved, and it may be worthwhile to find shortcuts in simple situations. We present here such a shortcut for a quadratic Hamiltonian of 2 degrees of freedom with a symmetry that is often present in celestial mechanics. The canonical transformation proposed has the added advantage of having a clear geometric interpretation and of being a generalization of the so-called reducing transformation that has been useful in several problems.

Mapping of the position-dependent mass Schrödinger equation under point canonical transformation

Abstract: In this paper, the three-dimensional radial position-dependent mass Schrodinger equation is exactly solved through mapping this wave equation into the constant mass Schrodinger equation with Coulomb potential by means of point canonical transformation. The wavefunctions here can be given in terms of confluent hypergeometric functions.

Quantization of Simple Parametrized Systems

Abstract: I study the canonical formulation and quantization of some simple parametrized systems, including the non-relativistic parametrized particle and the relativistic parametrized particle. Using Dirac's formalism I construct for each case the classical reduced phase space and study the dependence on the gauge fixing used. Two separate features of these systems can make this construction difficult: the actions are not invariant at the boundaries, and the constraints may have disconnected solution spaces. The relativistic particle is affected by both, while the non-relativistic particle displays only by the first. Analyzing the role of canonical transformations in the reduced phase space, I show that a change of gauge fixing is equivalent to a canonical transformation. In the relativistic case, quantization of one branch of the constraint at the time is applied and I analyze the electromagenetic backgrounds in which it is possible to quantize simultaneously both branches and still obtain a covariant unitary quantum theory. To preserve unitarity and space-time covariance, second quantization is needed unless there is no electric field. I motivate a definition of the inner product in all these cases and derive the Klein-Gordon inner product for the relativistic case. I construct phase space path integral representations for amplitudes for the BFV and the Faddeev path integrals, from which the path integrals in coordinate space (Faddeev-Popov and geometric path integrals) are derived.

On the Transformations of Hamiltonian Gauge Algebra Under Rotations of Constraints

Abstract: By explicit calculation of the effect of a ghost-dependent canonical transformation of BRST-charge, we derive the corresponding transformation law for structure coefficients of Hamiltonian gauge algebra under rotation of constraints. We show the transformation law to deviate from the behavior (expected naively) characteristic to a genuine connection.

Quantization of simple parametrized systems

Abstract: I study the canonical formulation and quantization of some simple parametrized systems, including the non-relativistic parametrized particle and the relativistic parametrized particle. Using Dirac's formalism I construct for each case the classical reduced phase space and study the dependence on the gauge fixing used. Two separate features of these systems can make this construction difficult: the actions are not invariant at the boundaries, and the constraints may have disconnected solution spaces. The relativistic particle is affected by both, while the non-relativistic particle displays only by the first. Analyzing the role of canonical transformations in the reduced phase space, I show that a change of gauge fixing is equivalent to a canonical transformation. In the relativistic case, quantization of one branch of the constraint at the time is applied and I analyze the electromagenetic backgrounds in which it is possible to quantize simultaneously both branches and still obtain a covariant unitary quantum theory. To preserve unitarity and space-time covariance, second quantization is needed unless there is no electric field. I motivate a definition of the inner product in all these cases and derive the Klein-Gordon inner product for the relativistic case. I construct phase space path integral representations for amplitudes for the BFV and the Faddeev path integrals, from which the path integrals in coordinate space (Faddeev-Popov and geometric path integrals) are derived.

Function group approach to unconstrained hamiltonian yang-mills theory

Abstract: Starting from the temporal gauge Hamiltonian for classical pure Yang–Mills theory with the gauge group SU(2) a canonical transformation is initiated by parametrizing the Gauss law generators with three new canonical variables. The construction of the remaining variables of the new set proceeds through a number of intermediate variables in several steps, which are suggested by the Poisson bracket relations and the gauge transformation properties of these variables. The unconstrained Hamiltonian is obtained from the original one by expressing it in the new variables and then setting the Gauss law generators to zero. This Hamiltonian turns out to be local and it decomposes into a finite Laurent series in powers of the coupling constant.

The Canonical Transformation Theory and its application to the Reaction Path Hamiltonian

Abstract: The reaction path Hamiltonian is formulated using the canonical transformation theory of classical mechanics. A method for checking the accuracy of the reaction path approach is presented, being based on the definition of a functional depending on the system Lagrangian. It is shown that the difference between the classical exact and the reaction path Hamiltonian dynamics can be expressed in terms of the potential energy sampled during the corresponding trajectories.

Calculating intensities using effective Hamiltonians in terms of Coriolis-adapted normal modes.

Abstract: The calculation of rovibrational transition energies and intensities is often hampered by the fact that vibrational states are strongly coupled by Coriolis terms. Because it invalidates the use of perturbation theory for the purpose of decoupling these states, the coupling makes it difficult to analyze spectra and to extract information from them. One either ignores the problem and hopes that the effect of the coupling is minimal or one is forced to diagonalize effective rovibrational matrices (rather than diagonalizing effective rotational matrices). In this paper we apply a procedure, based on a quantum mechanical canonical transformation for deriving decoupled effective rotational Hamiltonians. In previous papers we have used this technique to compute energy levels. In this paper we show that it can also be applied to determine intensities. The ideas are applied to the ethylene molecule.

The quantal Poincaré-Cartan integral invariantfor singular higher-order Lagrangian in field theories

Abstract: Based on the phase-space generating functional of the Green function for a system with a regular/singular higher-order Lagrangian, the quantal Poincare-Cartan integral invariant (QPCII) for the higher-order Lagrangian in field theories is derived. It is shown that this QPCII is equivalent to the quantal canonical equations. For the case in which the Jacobian of the transformation may not be equal to unity, the QPCII can still be derived. This case is different from the quantal first Noether theorem. The relations between QPCII and a canonical transformation and those between QPCII and the Hamilton-Jacobi equation at the quantum level are also discussed.

Gauge transformations are not canonical transformations

Abstract: In classical mechanics, we can describe the dynamics of a given system using either the Lagrangian formalism or the Hamiltonian formalism, the choice of either one being determined by whether one wants to deal with a second degree differential equation or a pair of first degree ones. For the former approach, we know that the Euler-Lagrange equation of motion remains invariant under additive total derivative with respect to time of any function of coordinates and time in the Lagrangian function, whereas the latter one is invariant under canonical transformations. In this short paper we address the question whether the transformation that leaves the Euler-Lagrange equation of motion invariant is also a canonical transformation and show that it is not.

Spectral functions of the spinless Holstein model

Abstract: An analytical approach to the one-dimensional spinless Holstein model is proposed, which is valid at finite charge-carrier concentrations. Spectral functions of charge carriers are computed on the basis of self-energy calculations. A generalization of the Lang-Firsov canonical transformation method is shown to provide an interpolation scheme between the extreme weak- and strong-coupling cases. The transformation depends on a variationally determined parameterthat characterizes the charge distribution across the polaron volume. The relation between the spectral functions of polarons and electrons, the latter corresponding to the photoemission spectrum, is derived. Particular attention is paid to the distinction between the coherent and incoherent parts of the spectra, and their evolution as a function of band filling and model parameters. Results are discussed and compared with recent numerical calculations for the many-polaron problem.

Alternative description of the linear canonical integral transformation

Abstract: First-optical systems (or ABCD-systems) with a singular submatrix B are considered. Starting with the Iwasawa-type decomposition of a first-order optical system as a cascade of a lens, a magnifier, and an ortho-symplectic system (a system that is both symplectic and orthogonal), a further decomposition of the ortho-symplectic system in the form of a separable fractional Fourier transformer embedded in between two spatial-coordinate rotators is proposed. The resulting decomposition of the entire firstorder optical system leads to a representation of the linear canonical integral transformation, which is valid also in the case of a singular submatrix B. Some examples of ABCDsystems with a singular submatrix B are given.

Canonical Transformations and Hamiltonian Evolutionary Systems

Abstract: In many Lagrangian field theories one has a Poisson bracket defined on the space of local functionals. We find necessary and sufficient conditions for a transformation on the space of local functionals to be canonical in three different cases. These three cases depend on the specific dimensions of the vector bundle of the theory and the associated Hamiltonian differential operator. We also show how a canonical transformation transforms a Hamiltonian evolutionary system and its conservation laws. Finally we illustrate these ideas with three examples.