Showing papers on "Canonical transformation published in 2008"
TL;DR: This paper extends the uncertainty principle for real signals in the fractional Fourier domains to the linear canonical transform domains, giving us the tighter lower bound on the product of the spreads of the signal in two specific LCT domains than the existing lower bounds.
Abstract: The linear canonical transform (LCT) is a generalization of the fractional Fourier transform (FRFT) having applications in several areas of signal processing and optics. In this paper, we extend the uncertainty principle for real signals in the fractional Fourier domains to the linear canonical transform domains, giving us the tighter lower bound on the product of the spreads of the signal in two specific LCT domains than the existing lower bounds in the LCT domains. It is seen that this lower bound can be achieved by a Gaussian signal. The effect of time-shifting and scaling the signal on the uncertainty principle is also discussed. It is shown here that a signal bandlimited in one LCT domain can be bandlimited in some other LCT domains also. The exceptions to the uncertainty principle in the LCT domains arising out of this are also discussed.
94 citations
TL;DR: The Hamiltonian formulation of a plasma four-field fluid model that describes collisionless reconnection is presented in this article, which is noncanonical with a corresponding Lie-Poisson bracket.
Abstract: The Hamiltonian formulation of a plasma four-field fluid model that describes collisionless reconnection is presented. The formulation is noncanonical with a corresponding Lie‐Poisson bracket. The bracket is used to obtain new independent families of invariants, so-called Casimir invariants, three of which are directly related to Lagrangian invariants of the system. The Casimirs are used to obtain a variational principle for equilibrium equations that generalize the Grad‐Shafranov equation to include flow. Dipole and homogeneous equilibria are constructed. The linear dynamics of the latter is treated in detail in a Hamiltonian context: canonically conjugate variables are obtained; the dispersion relation is analyzed and exact thresholds for spectral stability are obtained; the canonical transformation to normal form is described; an unambiguous definition of negative energy modes is given; and thresholdssufficientforenergy-Casimirstabilityareobtained. TheHamiltonian formulationisalsousedtoobtainanexpressionforthecollisionlessconductivity and it is further used to describe the linear growth and nonlinear saturation of the collisionless tearing mode. (Some figures in this article are in colour only in the electronic version)
77 citations
TL;DR: In this paper, a consistent, local coordinate formulation of covariant Hamiltonian field theory is presented, where the covariant canonical transformation theory offers more general means for defining mappings that preserve the form of the field equations than the usual Lagrangian description.
Abstract: A consistent, local coordinate formulation of covariant Hamiltonian field theory is presented. Whereas the covariant canonical field equations are equivalent to the Euler–Lagrange field equations, the covariant canonical transformation theory offers more general means for defining mappings that preserve the form of the field equations than the usual Lagrangian description. It is proven that Poisson brackets, Lagrange brackets, and canonical 2-forms exist that are invariant under canonical transformations of the fields. The technique to derive transformation rules for the fields from generating functions is demonstrated by means of various examples. In particular, it is shown that the infinitesimal canonical transformation furnishes the most general form of Noether's theorem. Furthermore, we specify the generating function of an infinitesimal space-time step that conforms to the field equations.
57 citations
TL;DR: The Hamiltonian formulation of a plasma four-field fluid model that describes collisionless reconnection is presented in this paper, which is noncanonical with a corresponding Lie-Poisson bracket.
Abstract: The Hamiltonian formulation of a plasma four-field fluid model that describes collisionless reconnection is presented. The formulation is noncanonical with a corresponding Lie-Poisson bracket. The bracket is used to obtain new independent families of invariants, so-called Casimir invariants, three of which are directly related to Lagrangian invariants of the system. The Casimirs are used to obtain a variational principle for equilibrium equations that generalize the Grad-Shafranov equation to include flow. Dipole and homogeneous equilibria are constructed. The linear dynamics of the latter is treated in detail in a Hamiltonian context: canonically conjugate variables are obtained; the dispersion relation is analyzed and exact thresholds for spectral stability are obtained; the canonical transformation to normal form is described; an unambiguous definition of negative energy modes is given; and thresholds sufficient for energy-Casimir stability are obtained. The Hamiltonian formulation also is used to obtain an expression for the collisionless conductivity and it is further used to describe the linear growth and nonlinear saturation of the collisionless tearing mode.
57 citations
TL;DR: In this paper, a detailed derivation of Cachazo-Svrcek-Witten-style Feynman rules for Yang-Mills gauge theory coupled to a massive coloured scalar is presented.
Abstract: This article provides two complementary detailed derivations of Cachazo-Svrcek-Witten-style Feynman rules for Yang-Mills gauge theory coupled to a massive coloured scalar as presented in earlier work. These proceed through a direct canonical transformation method on space-time and through a gauge transformation in an action constructed on twistor space. It is shown explicitly that the field transformations are identical in both cases. Some simple tree-level examples of our rules are given and we comment on the application of them to the calculation of the rational part of one-loop pure glue amplitudes. A possible direct quantum completion of pure glue CSW rules based on dimensional regularisation motivated by these results is sketched. Finally, it is shown how to derive CSW rules for effective Higgs-gluon and Higgs-matter couplings proposed in the literature directly from the action. This derivation yields additional towers of vertices which generate a subset of the contributions to effective multi-Higgs scattering amplitudes.
49 citations
TL;DR: In this article, a complete exposition of the rest-frame instant form of dynamics for arbitrary isolated systems (particles, fields, strings, fluids) admitting a Lagrangian description is given.
Abstract: A complete exposition of the rest-frame instant form of dynamics for arbitrary isolated systems (particles, fields, strings, fluids)admitting a Lagrangian description is given. The starting point is the parametrized Minkowski theory describing the system in arbitrary admissible non-inertial frames in Minkowski space-time, which allows one to define the energy-momentum tensor of the system and to show the independence of the description from the clock synchronization convention and from the choice of the 3-coordinates. In the inertial rest frame the isolated system is seen as a decoupled non-covariant canonical external center of mass carrying a pole-dipole structure (the invariant mass $M$ and the rest spin ${\vec {\bar S}}$ of the system) and an external realization of the Poincare' group. Then an isolated system of positive-energy charged scalar articles plus an arbitrary electro-magnetic field in the radiation gauge is investigated as a classical background for defining relativistic atomic physics. The electric charges of the particles are Grassmann-valued to regularize the self-energies. The rest-frame conditions and their gauge-fixings (needed for the elimination of the internal 3-center of mass) are explicitly given. It is shown that there is a canonical transformation which allows one to describe the isolated system as a set of Coulomb-dressed charged particles interacting through a Coulomb plus Darwin potential plus a free transverse radiation field: these two subsystems are not mutually interacting and are interconnected only by the rest-frame conditions and the elimination of the internal 3-center of mass. Therefore in this framework with a fixed number of particles there is a way out from the Haag theorem,at least at the classical level.
48 citations
TL;DR: In this paper, the authors prove existence of a ground state and resonances in the standard model of the non-relativistic quantum electro-dynamics (QED).
Abstract: We prove existence of a ground state and resonances in the standard model of the non-relativistic quantum electro-dynamics (QED). To this end we introduce a new canonical transformation of QED Hamiltonians and use the spectral renormalization group technique with a new choice of Banach spaces.
42 citations
TL;DR: In this paper, a consistent, local coordinate formulation of covariant Hamiltonian field theory is presented, where the covariant canonical transformation theory offers more general means for defining mappings that preserve the form of the field equations than the usual Lagrangian description.
Abstract: A consistent, local coordinate formulation of covariant Hamiltonian field theory is presented. Whereas the covariant canonical field equations are equivalent to the Euler-Lagrange field equations, the covariant canonical transformation theory offers more general means for defining mappings that preserve the form of the field equations than the usual Lagrangian description. It is proved that Poisson brackets, Lagrange brackets, and canonical 2-forms exist that are invariant under canonical transformations of the fields. The technique to derive transformation rules for the fields from generating functions is demonstrated by means of various examples. In particular, it is shown that the infinitesimal canonical transformation furnishes the most general form of Noether's theorem. We furthermore specify the generating function of an infinitesimal space-time step that conforms to the field equations.
36 citations
TL;DR: In this paper, the point canonical transformation (PCT) was employed to solve the D-dimensional Schr\"{o}dinger equation with position-dependent effective mass (PDEM) function for two molecular pseudoharmonic and modified Kratzer (Mie-type) potentials.
Abstract: We employ the point canonical transformation (PCT) to solve the D-dimensional Schr\"{o}dinger equation with position-dependent effective mass (PDEM) function for two molecular pseudoharmonic and modified Kratzer (Mie-type) potentials. In mapping the transformed exactly solvable D-dimensional ($D\geq 2$) Schr\"{o}dinger equation with constant mass into the effective mass equation by employing a proper transformation, the exact bound state solutions including the energy eigenvalues and corresponding wave functions are derived. The well-known pseudoharmonic and modified Kratzer exact eigenstates of various dimensionality is manifested.
28 citations
TL;DR: In this paper, the exact solutions of Schrodinger equation characterized by position-dependent effective mass via point canonical transformations are considered, and the energy of the bound states and corresponding wavefunctions are determined exactly.
Abstract: We deal with the exact solutions of Schrodinger equation characterized by position-dependent effective mass via point canonical transformations. The Morse, Poschl-Teller, and Hulthen type potentials are considered, respectively. With the choice of position-dependent mass forms, exactly solvable target potentials are constructed. Their energy of the bound states and corresponding wavefunctions are determined exactly.
TL;DR: In this article, the exact solution of the position-dependent effective mass Schrodinger equations with PT symmetry was obtained for the Rosen-Morse-type potentials, and the energy spectra of the bound states and corresponding wave functions for the PT-symmetric potentials were given in the exact closed forms.
Abstract: We use the method of point canonical transformations and choose the Rosen-Morse-type potential as the reference potential to study exact solutions of the position-dependent effective mass Schrodinger equations. Choosing three position-dependent mass distributions, we construct seven exactly solvable target potentials with PT symmetry. The energy spectra of the bound states and corresponding wavefunctions for the PT-symmetric potentials are given in the exact closed forms. We also discuss the isospectrality of different Schrodinger equations with the same mass distribution or different mass distributions for different PT-symmetric potentials.
TL;DR: In this article, the bound-state solutions and the su(1,1) description of the d-dimensional radial harmonic oscillator, the Morse and Coulomb Schrodinger equations are reviewed in a unified way using the point canonical transformation method.
Abstract: The bound-state solutions and the su(1,1) description of the d-dimensional radial harmonic oscillator, the Morse, and the D-dimensional radial Coulomb Schrodinger equations are reviewed in a unified way using the point canonical transformation method. It is established that the spectrum generating su(1,1) algebra for the first problem is converted into a potential algebra for the remaining two. This analysis is then extended to Schrodinger equations containing some position-dependent mass. The deformed su(1,1) construction recently achieved for a d-dimensional radial harmonic oscillator is easily extended to the Morse and Coulomb potentials. In the last two cases, the equivalence between the resulting deformed su(1,1) potential algebra approach and a previous deformed shape invariance one generalizes to a position-dependent mass background a well-known relationship in the context of constant mass.
TL;DR: In this paper, an analytic equilibrium generating function (EGF) is constructed in natural canonical coordinates (ψ,θ) from experimental data from a Grad-Shafranov equilibrium solver for a tokamak.
Abstract: A new approach to integration of magnetic field lines in divertor tokamaks is proposed. In this approach, an analytic equilibrium generating function (EGF) is constructed in natural canonical coordinates (ψ,θ) from experimental data from a Grad–Shafranov equilibrium solver for a tokamak. ψ is the toroidal magnetic flux and θ is the poloidal angle. Natural canonical coordinates (ψ,θ,φ) can be transformed to physical position (R,Z,φ) using a canonical transformation. (R,Z,φ) are cylindrical coordinates. Another canonical transformation is used to construct a symplectic map for integration of magnetic field lines. Trajectories of field lines calculated from this symplectic map in natural canonical coordinates can be transformed to trajectories in real physical space. Unlike in magnetic coordinates [O. Kerwin, A. Punjabi, and H. Ali, Phys. Plasmas 15, 072504 (2008)], the symplectic map in natural canonical coordinates can integrate trajectories across the separatrix surface, and at the same time, give traject...
TL;DR: In this article, the authors evaluate the quality of the approximation of periodic orbits in the logarithmic potential constructed using perturbation theory based on Hamiltonian normal forms and show that with a normal form truncated at the lowest order incorporating the relevant resonance, it is possible to construct accurate solutions both for normal modes and periodic orbit in general position.
Abstract: Context. We investigate periodic orbits in galactic potentials by developing analytical methods. Aims. We evaluate the quality of the approximation of periodic orbits in the logarithmic potential constructed using perturbation theory based on Hamiltonian normal forms. Methods. The solutions of the equations of motion corresponding to periodic orbits are obtained as series expansions computed by inverting the normalizing canonical transformation. To improve the convergence of the series, a resummation based on a continued fraction may be performed. This method is analogous to the Prendergast method, which searches for approximate rational solutions. Results. It is shown that with a normal form truncated at the lowest order incorporating the relevant resonance it is possible to construct accurate solutions both for normal modes and periodic orbits in general position.
Posted Content•
TL;DR: In this article, a detailed investigation of the action for pure Yang-Mills theory which L. Mason formulated in twistor space is carried out, and explicit solutions are found in each case and connections with earlier work are examined.
Abstract: This thesis carries out a detailed investigation of the action for pure Yang- Mills theory which L. Mason formulated in twistor space. The rich structure of twistor space results in greater gauge freedom compared to the theory in ordinary space-time. One particular gauge choice, the CSW gauge, allows simplifications to be made at both the classical and quantum level.
The equations of motion have an interesting form in the CSW gauge, which suggests a possible solution procedure. This is explored in three special cases. Explicit solutions are found in each case and connections with earlier work are examined. The equations are then reformulated in Minkowski space, in order to deal with an initial-value, rather than boundary-value, problem. An interesting form of the Yang-Mills equation is obtained, for which we propose an iteration procedure.
The quantum theory is also simplified by adopting the CSW gauge. The Feynman rules are derived and are shown to reproduce the MHV diagram formalism straightforwardly, once LSZ reduction is taken into account. The three-point amplitude missing in the MHV formalism can be recovered in our theory. Finally, relations to Mansfield's canonical transformation approach are elucidated.
TL;DR: In this paper, a complete first-order analytical solution, which includes the short periodic terms, for the problem of optimal low-thrust limited-power transfers between arbitrary elliptic coplanar orbits in a Newtonian central gravity field is obtained through canonical transformation theory.
Abstract: A complete first-order analytical solution, which includes the short periodic terms, for the problem of optimal low-thrust limited-power transfers between arbitrary elliptic coplanar orbits in a Newtonian central gravity field is obtained through canonical transformation theory. The optimization problem is formulated as a Mayer problem of optimal control theory with Cartesian elements—position and velocity vectors—as state variables. After applying the Pontryagin maximum principle and determining the maximum Hamiltonian, classical orbital elements are introduced through a Mathieu transformation. The short periodic terms are then eliminated from the maximum Hamiltonian through an infinitesimal canonical transformation built through Hori method. Closed-form analytical solutions are obtained for the average canonical system by solving the Hamilton-Jacobi equation through separation of variables technique. For transfers between close orbits a simplified solution is straightforwardly derived by linearizing the new Hamiltonian and the generating function obtained through Hori method.
TL;DR: In this paper, the Coulomb and Kratzer potentials can be mapped to the Morse potential by using the method of point canonical transformation, and the shape-invariant algebra for Coulomb, Krzer, and Morse potentials is SU(1, 1), while the shape invariant algebra of Poschl-Teller type I and Hulthen potential is SU (2).
Abstract: In this paper by using the method of point canonical transformation we find that the Coulomb and Kratzer potentials can be mapped to the Morse potential. Then we show that the Poschl-Teller potential type I belongs to the same subclass of shape invariant potentials as Hulthen potential. Also we show that the shape-invariant algebra for Coulomb, Kratzer, and Morse potentials is SU(1,1), while the shape-invariant algebra for Poschl-Teller type I and Hulthen is SU(2).
TL;DR: In this article, the authors evaluate the quality of the approximation of periodic orbits in the logarithmic potential constructed using perturbation theory based on Hamiltonian normal forms, and show that with a normal form truncated at the lowest order incorporating the relevant resonance, it is possible to construct quite accurate solutions both for normal modes and periodic orbit in general position.
Abstract: Analytic methods to investigate periodic orbits in galactic potentials. To evaluate the quality of the approximation of periodic orbits in the logarithmic potential constructed using perturbation theory based on Hamiltonian normal forms. The solutions of the equations of motion corresponding to periodic orbits are obtained as series expansions computed by inverting the normalizing canonical transformation. To improve the convergence of the series a resummation based on a continued fraction may be performed. This method is analogous to that looking for approximate rational solutions (Prendergast method). It is shown that with a normal form truncated at the lowest order incorporating the relevant resonance it is possible to construct quite accurate solutions both for normal modes and periodic orbits in general position.
TL;DR: In this paper, a correspondence between the stationary coherent states associated with the commensurate anisotropic two-dimensional harmonic oscillator and the classical Lissajous orbits was shown.
Abstract: We demonstrate a formally exact quantum?classical correspondence between the stationary coherent states associated with the commensurate anisotropic two-dimensional harmonic oscillator and the classical Lissajous orbits. Our derivation draws upon the earlier work of Louck et al (1973 J. Math. Phys. 14 692) wherein they have provided a non-bijective canonical transformation that maps, within a degenerate eigenspace, the commensurate anisotropic oscillator on to the isotropic oscillator. This mapping leads, in a natural manner, to a Schwinger realization of SU(2) in terms of the canonically transformed creation and annihilation operators. Through the corresponding coherent states built over a degenerate eigenspace, we directly effect the classical limit via the expectation values of the underlying generators. Our work completely accounts for the fact that the SU(2) coherent state in general corresponds to an ensemble of Lissajous orbits.
Journal Article•
TL;DR: In this article, the fractional Hamilton-Jacobi equation has been derived for dynamical systems involving the Caputo derivative, and a fractional Poisson-bracket is introduced.
Abstract: In the present paper fractional Hamilton-Jacobi equation has been derived for dynamical systems involving the Caputo derivative. Fractional Poisson-bracket is introduced. Further Hamilton’s canonical equations are formulated and quantum wave equation corresponds to the fractional Hamilton-Jacobi equation is suggested. Illustrative examples have been worked out to explain the formalism.
TL;DR: In this article, an area-preserving map for the magnetic field line trajectories is obtained in magnetic coordinates from the Hamiltonian equations of motion for the lines and a canonical transformation.
Abstract: A highly accurate calculation of the magnetic field line Hamiltonian in DIII-D [J. L. Luxon and L. E. Davis, Fusion Technol. 8, 441 (1985)] is made from piecewise analytic equilibrium fit data for shot 115467 3000ms. The safety factor calculated from this Hamiltonian has a logarithmic singularity at an ideal separatrix. The logarithmic region inside the ideal separatrix contains 2.5% of toroidal flux inside the separatrix. The logarithmic region is symmetric about the separatrix. An area-preserving map for the field line trajectories is obtained in magnetic coordinates from the Hamiltonian equations of motion for the lines and a canonical transformation. This map is used to calculate trajectories of magnetic field lines in DIII-D. The field line Hamiltonian in DIII-D is used as the generating function for the map and to calculate stochastic broadening from field-errors and spatial noise near the separatrix. A very negligible amount (0.03%) of magnetic flux is lost from inside the separatrix due to these n...
TL;DR: In this paper, the authors present a new energy-shaping methodology for tracking control based on the introduction of virtual non-homogeneous fields where a desired energy is defined to compensate for the actual energy of the robot manipulator while a virtual field forces the system to track a general reference trajectory.
TL;DR: The present results demonstrate that the general real solutions may involve either exp, sin, cos, sinh or cosh under certain conditions depending on the type of the constants in the canonical transformation.
TL;DR: In this article, the similarity of massive CSW scalar vertices and quark vertices can be understood using a kind of light-cone SUSY transformation presented in this paper.
Abstract: The similarity of massive CSW scalar vertices and quark vertices can be understood using a kind of light-cone SUSY transformation presented in this paper. We also show that the canonical transformation generating the MHV-SQCD Lagrangian, can be fixed by applying this light-cone SUSY transformation to the canonical transformation for MHV-QCD obtained in paper arxiv:0805.0239. Most of the massive CSW vertices for SQCD can also be pinned down in this way.
TL;DR: In this paper, a general point canonical transformation is applied by using a free parameter to obtain the energy eigenvalues of the bound states and corresponding wave functions for target potentials as a function of the free parameter.
Abstract: PT-symmetric solutions of Schrodinger equation are obtained for the Scarf and generalized harmonic oscillator potentials with the position-dependent mass. A general point canonical transformation is applied by using a free parameter. Three different forms of mass distributions are used. A set of the energy eigenvalues of the bound states and corresponding wave functions for target potentials are obtained as a function of the free parameter.
TL;DR: In this article, a quasi-exactly solvable two-mode bosonic realization of SU(2) was constructed by using canonical transformation and a differential equation, which is solved by quantum Hamilton-Jacobi formalism.
Abstract: We have constructed the quasi-exactly-solvable two-mode bosonic realization of SU(2). Two-mode boson Hamiltonian is defined through a differential equation which is solved by quantum Hamilton-Jacobi formalism. The squeezed states of two-mode boson systems are characterized through canonical transformation. The illustrated concept of squeezed boson systems has been applied two-mode bosonic Hamiltonian which is a squeezed one and is determined through a differential equation. This differential equation is solved and energy eigenvalues are found approximately.
TL;DR: In this paper, the nonrelativistic limit of the linear wave equation for zero and unity spin bosons of mass m in the Duffin-Kemmer-Petiau representation is investigated by means of a unitary transformation, analogous to the Foldy-Wouthuysen canonical transformation for a relativistic electron.
Abstract: The nonrelativistic limit of the linear wave equation for zero and unity spin bosons of mass m in the Duffin–Kemmer–Petiau representation is investigated by means of a unitary transformation, analogous to the Foldy–Wouthuysen canonical transformation for a relativistic electron. The interacting case is also analyzed, by considering a power series expansion of the transformed Hamiltonian, thus demonstrating that all features of particle dynamics can be recovered if corrections of order 1/m2 are taken into account through a recursive iteration procedure.
TL;DR: In this article, the similarity of massive CSW scalar vertices and quark vertices can be understood using a kind of light-cone SUSY transformation presented in this paper.
Abstract: The similarity of massive CSW scalar vertices and quark vertices can be understood using a kind of light-cone SUSY transformation presented in this paper. We also show that the canonical transformation generating the MHV-SQCD lagrangian, can be fixed by applying this light-cone SUSY transformation to the canonical transformation for MHV-QCD obtained in paper arxiv:0805.0239. Most of the massive CSW vertices for SQCD can also be pinned down in this way.
TL;DR: In this paper, a detailed derivation of Cachazo-Svrcek-Witten-style Feynman rules for Yang-Mills gauge theory coupled to a massive coloured scalar is presented.
Abstract: This article provides two complementary detailed derivations of Cachazo-Svrcek-Witten-style Feynman rules for Yang-Mills gauge theory coupled to a massive coloured scalar as presented in earlier work. These proceed through a direct canonical transformation method on space-time and through a gauge transformation in an action constructed on twistor space. It is shown explicitly that the field transformations are identical in both cases. Some simple tree-level examples of our rules are given and we comment on the application of them to the calculation of the rational part of one-loop pure glue amplitudes. A possible direct quantum completion of pure glue CSW rules based on dimensional regularisation motivated by these results is sketched. Finally, it is shown how to derive CSW rules for effective Higgs-gluon and Higgs-matter couplings proposed in the literature directly from the action. This derivation yields additional towers of vertices which generate a subset of the contributions to effective multi-Higgs scattering amplitudes.