Showing papers on "Canonical transformation published in 2016"

One-Dimensional Quasi-Exactly Solvable Schr\"odinger Equations

Abstract: Quasi-Exactly Solvable Schrodinger Equations occupy an intermediate place between exactly-solvable (e.g. the harmonic oscillator and Coulomb problems etc) and non-solvable ones. Their major property is an explicit knowledge of several eigenstates while the remaining ones are unknown. Many of these problems are of the anharmonic oscillator type with a special type of anharmonicity. The Hamiltonians of quasi-exactly-solvable problems are characterized by the existence of a hidden algebraic structure but do not have any hidden symmetry properties. In particular, all known one-dimensional (quasi)-exactly-solvable problems possess a hidden $\mathfrak{sl}(2,\bf{R})-$ Lie algebra. They are equivalent to the $\mathfrak{sl}(2,\bf{R})$ Euler-Arnold quantum top in a constant magnetic field. Quasi-Exactly Solvable problems are highly non-trivial, they shed light on delicate analytic properties of the Schrodinger Equations in coupling constant. The Lie-algebraic formalism allows us to make a link between the Schrodinger Equations and finite-difference equations on uniform and/or exponential lattices, it implies that the spectra is preserved. This link takes the form of quantum canonical transformation. The corresponding isospectral spectral problems for finite-difference operators are described. The underlying Fock space formalism giving rise to this correspondence is uncovered. For a quite general class of perturbations of unperturbed problems with the hidden Lie algebra property we can construct an algebraic perturbation theory, where the wavefunction corrections are of polynomial nature, thus, can be found by algebraic means.

2-adic integral canonical models

Abstract: We use Lau’s classification of 2-divisible groups using Dieudonne displays to construct integral canonical models for Shimura varieties of abelian type at 2-adic places where the level is hyperspecial.

Low energy ghosts and the Jeans’ instability

Abstract: We show that a massless canonical scalar field minimally coupled to general relativity can become a tachyonic ghost at low energies around a background in which the scalar’s gradient is spacelike. By performing a canonical transformation we demonstrate that this low energy ghost can be recast, at the level of the action, in a form of a fluid that undergoes a Jeans-like instability affecting only modes with large wavelength. This illustrates that low energy tachyonic ghosts do not lead to a catastrophic quantum vacuum instability, unlike the usual high-energy ghost degrees of freedom.

Canonical invariance of spatially covariant scalar-tensor theory

Abstract: We investigate the invariant canonical transformation of a spatially covariant scalar-tensor theory of gravity, called the XG theory. Under the invariant canonical transformation, the forms of the action or the Hamiltonian and the primary constraints are preserved, but the action or the Hamiltonian is not invariant. We derive the Hamiltonian in a nonperturbative manner and perform the Hamiltonian analysis for full theory. We confirm that the theory has at most 3 degrees of freedom as long as the theory has the spatial diffeormorphism symmetry. Then, we derive the invariant canonical transformation by using the infinitesimal transformation. The invariant metric transformation of the XG theory contains a vector product as well as the disformal transformation. The vector product and the disformal factor can depend on the higher order derivatives of the scalar field and the metric. Furthermore, we discover additional canonical transformation which leaves the theory invariant. Using the invariant transformation, we study the relation between the Horndeski theory and a beyond Horndeski theory, called the GLPV theory, and find that we cannot obtain general GLPV theory from the Horndeski theory through the invariant canonical transformation we found.

Topological membranes, current algebras and H-flux-R-flux duality based on Courant algebroids

Abstract: We construct a topological sigma model and a current algebra based on a Courant algebroid structure on a Poisson manifold. In order to construct models, we reformulate the Poisson Courant algebroid by supergeometric construction on a QP-manifold. A new duality of Courant algebroids which transforms H-flux and R-flux is proposed, where the transformation is interpreted as a canonical transformation of a graded symplectic manifold.

Quantum oscillator and kepler-coulomb problems in curved spaces: Deformed shape invariance, point canonical transformations, and rational extensions

Abstract: The quantum oscillator and Kepler-Coulomb problems in d-dimensional spaces with constant curvature are analyzed from several viewpoints. In a deformed supersymmetric framework, the corresponding nonlinear potentials are shown to exhibit a deformed shape invariance property. By using the point canonical transformation method, the two deformed Schrodinger equations are mapped onto conventional ones corresponding to some shape-invariant potentials, whose rational extensions are well known. The inverse point canonical transformations then provide some rational extensions of the oscillator and Kepler-Coulomb potentials in curved space. The oscillator on the sphere and the Kepler-Coulomb potential in a hyperbolic space are studied in detail and their extensions are proved to be consistent with already known ones in Euclidean space. The partnership between nonextended and extended potentials is interpreted in a deformed supersymmetric framework. Those extended potentials that are isospectral to some nonextended ones are shown to display deformed shape invariance, which in the Kepler-Coulomb case is enlarged by also translating the degree of the polynomial arising in the rational part denominator.

Repeatability of measurements: Non-Hermitian observables and quantum Coriolis force

Abstract: A noncommuting measurement transfers, via the apparatus, information encoded in a system's state to the external ``observer.'' Classical measurements determine properties of physical objects. In the quantum realm, the very same notion restricts the recording process to orthogonal states as only those are distinguishable by measurements. Therefore, even a possibility to describe physical reality by means of non-Hermitian operators should volens nolens be excluded as their eigenstates are not orthogonal. Here, we show that non-Hermitian operators with real spectra can be treated within the standard framework of quantum mechanics. Furthermore, we propose a quantum canonical transformation that maps Hermitian systems onto non-Hermitian ones. Similar to classical inertial forces this map is accompanied by an energetic cost, pinning the system on the unitary path.

The Hamilton–Jacobi Equation

Abstract: We already know that canonical transformations are useful for solving mechanical problems. We now want to look for a canonical transformation that transforms the 2N coordinates (q i , p i ) to 2N constant values (Q i , P i ), e.g., to the 2N initial values \((q_{i}^{0},p_{i}^{0})\) at time t = 0. Then the problem would be solved, q = q(q0, p0, t), p = p(q0, p0, t).

The Linear Canonical Transformation: Definition and Properties

Abstract: In this chapter we introduce the class of linear canonical transformations, which includes as particular cases the Fourier transformation (and its generalization: the fractional Fourier transformation), the Fresnel transformation, and magnifier, rotation and shearing operations. The basic properties of these transformations—such as cascadability, scaling, shift, phase modulation, coordinate multiplication and differentiation—are considered. We demonstrate that any linear canonical transformation is associated with affine transformations in phase space, defined by time-frequency or position-momentum coordinates. The affine transformation is described by a symplectic matrix, which defines the parameters of the transformation kernel. This alternative matrix description of linear canonical transformations is widely used along the chapter and allows simplifying the classification of such transformations, their eigenfunction identification, the interpretation of the related Wigner distribution and ambiguity function transformations, among many other tasks. Special attention is paid to the consideration of one- and two-dimensional linear canonical transformations, which are more often used in signal processing, optics and mechanics. Analytic expressions for the transforms of some selected functions are provided.

Design of robust H∞ fault detection filter for uncertain time-delay systems using canonical form approach

Abstract: In this paper, the problem of fault detection for an uncertain time-delay system is considered. Using a canonical form transformation, original time-delay system is transformed into a delay-free system. The fault detection scheme is devised using H ∞ model matching approach for the delay-free system and then implemented on original system. The so-called canonical transformation is found to be less conservative and offers reduced complexity in comparison with bicausal transformation, which can also be employed for the aforementioned purpose. An algorithm is proposed for design of such a fault detection system. Simulation results show the effectiveness of proposed scheme.

Development of Linear Canonical Transforms: A Historical Sketch

Abstract: Linear canonical transformations (LCTs) were introduced almost simultaneously during the early 1970s by Stuart A. Collins Jr. in paraxial optics, and independently by Marcos Moshinsky and Christiane Quesne in quantum mechanics, to understand the conservation of information and of uncertainty under linear maps of phase space. Only in the 1990s did both sources begin to be referred jointly in the growing literature, which has expanded into a field common to applied optics, mathematical physics, and analogic and digital signal analysis. In this introductory chapter we recapitulate the construction of the LCT integral transforms, detailing their Lie-algebraic relation with second-order differential operators, which is the origin of the metaplectic phase. Radial and hyperbolic LCTs are reviewed as unitary integral representations of the two-dimensional symplectic group, with complex extension to a semigroup for systems with loss or gain. Some of the more recent developments on discrete and finite analogues of LCTs are commented with their concomitant problems, whose solutions and alternatives are contained the body of this book.

Quantum oscillator and Kepler-Coulomb problems in curved spaces: deformed shape invariance, point canonical transformations, and rational extensions

Abstract: The quantum oscillator and Kepler-Coulomb problems in $d$-dimensional spaces with constant curvature are analyzed from several viewpoints. In a deformed supersymmetric framework, the corresponding nonlinear potentials are shown to exhibit a deformed shape invariance property. By using the point canonical transformation method, the two deformed Schrodinger equations are mapped onto conventional ones corresponding to some shape-invariant potentials, whose rational extensions are well known. The inverse point canonical transformations then provide some rational extensions of the oscillator and Kepler-Coulomb potentials in curved space. The oscillator on the sphere and the Kepler-Coulomb potential in a hyperbolic space are studied in detail and their extensions are proved to be consistent with already known ones in Euclidean space. The partnership between nonextended and extended potentials is interpreted in a deformed supersymmetric framework. Those extended potentials that are isospectral to some nonextended ones are shown to display deformed shape invariance, which in the Kepler-Coulomb case is enlarged by also translating the degree of the polynomial arising in the rational part denominator.

Characteristics with singularities and the boundary values of the asymptotic solution of the Cauchy problem for a degenerate wave equation

Abstract: We consider the Cauchy problem with spatially localized initial data for the twodimensional wave equation degenerating on the boundary of the domain. This problem arises, in particular, in the theory of tsunami wave run-up on a shallow beach. Earlier, S. Yu. Dobrokhotov, V. E. Nazaikinskii, and B. Tirozzi developed a method for constructing asymptotic solutions of this problem. The method is based on a modified Maslov canonical operator and on characteristics (trajectories) unbounded in the momentum variables; such characteristics are nonstandard from the viewpoint of the theory of partial differential equations. In a neighborhood of the velocity degeneration line, which is a caustic of a special form, the canonical operator is defined via the Hankel transform, which arises when applying Fock’s quantization procedure to the canonical transformation regularizing the above-mentioned nonstandard characteristics in a neighborhood of the velocity degeneration line (the boundary of the domain). It is shown in the present paper that the restriction of the asymptotic solutions to the boundary is determined by the standard canonical operator, which simplifies the asymptotic formulas for the solution on the boundary dramatically; for initial perturbations of special form, the solutions can be expressed via simple algebraic functions.

Physical background for parameters of the quantum Rabi model

Abstract: We investigate the applicability of the two major approximations which are most commonly employed in the study of the quantum Rabi model, namely the description of a resonant cavity mode as a single-mode quantized field and the use of the rotating wave approximation. Starting from the Hamiltonian of a two-level system interacting with a multi-mode quantized field, we perform the canonical transformation of the field operators. This allows one to partition the Hamiltonian of the system into two parts. The first part is the interaction of the two-level system with a single collective field mode, while the second one describes the interaction with field fluctuations. The first part is usually associated with the resonant cavity mode. This division enables us to determine the applicability condition of the single-mode approximation. In addition we identify simple approximate relations for the description of the eigenstates, eigenfunctions and the time evolution of the quantum Rabi model beyond the rotating wave approximation.

Morse, Lennard-Jones, and Kratzer Potentials: A Canonical Perspective with Applications

Abstract: Canonical approaches are applied to classic Morse, Lennard-Jones, and Kratzer potentials. Using the canonical transformation generated for the Morse potential as a reference, inverse transformations allow the accurate generation of the Born–Oppenheimer potential for the H2+ ion, neutral covalently bound H2, van der Waals bound Ar2, and the hydrogen bonded one-dimensional dissociative coordinate in a water dimer. Similar transformations are also generated using the Lennard-Jones and Kratzer potentials as references. Following application of inverse transformations, vibrational eigenvalues generated from the Born–Oppenheimer potentials give significantly improved quantitative comparison with values determined from the original accurately known potentials. In addition, an algorithmic strategy based upon a canonical transformation to the dimensionless form applied to the force distribution associated with a potential is presented. The resulting canonical force distribution is employed to construct an algorith...

On the question of symmetries in non-relativistic diffeomorphism invariant theories

Abstract: A novel algorithm is provided to couple a Galilean invariant model with curved spatial background by taking nonrelativistic limit of a unique minimally coupled relativistic theory, which ensures Galilean symmetry in the flat limit and canonical transformation of the original fields. That the twin requirements are fulfilled is ensured by a new field, the existence of which was demonstrated recently from Galilean gauge theory. The ambiguities and anomalies concerning the recovery of Galilean symmetry in the flat limit of spatial non relativistic diffeomorphic theories, reported in the literature, are focused and resolved from a new angle.

Optimal low-thrust transfers between coplanar orbits with small eccentricities

Abstract: In this paper, a first-order analytical solution, which includes the short periodic terms, for the problem of optimal time-fixed low-thrust limited-power transfers (no rendezvous), in an inverse-square force field, between coplanar orbits with small eccentricities is obtained through canonical transformation theory. Short periodic terms are eliminated from the maximum Hamiltonian, expressed in non-singular orbital elements, through an infinitesimal canonical transformation built through the Hori method. Closed-form analytical solutions are obtained for the average canonical system by solving the Hamilton–Jacobi equation through the separation of variables technique. For long duration maneuvers, the existence of conjugate points is investigated through the Jacobi condition.

Restoration of four-dimensional diffeomorphism covariance in canonical general relativity: An intrinsic Hamilton–Jacobi approach

Abstract: Classical background independence is reflected in Lagrangian general relativity through covariance under the full diffeomorphism group. We show how this independence can be maintained in a Hamilton–Jacobi approach that does not accord special privilege to any geometric structure. Intrinsic space–time curvature-based coordinates grant equal status to all geometric backgrounds. They play an essential role as a starting point for inequivalent semiclassical quantizations. The scheme calls into question Wheeler’s geometrodynamical approach and the associated Wheeler–DeWitt equation in which 3-metrics are featured geometrical objects. The formalism deals with variables that are manifestly invariant under the full diffeomorphism group. Yet, perhaps paradoxically, the liberty in selecting intrinsic coordinates is precisely as broad as is the original diffeomorphism freedom. We show how various ideas from the past five decades concerning the true degrees of freedom of general relativity can be interpreted in light of this new constrained Hamiltonian description. In particular, we show how the Kuchař multi-fingered time approach can be understood as a means of introducing full four-dimensional diffeomorphism invariants. Every choice of new phase space variables yields new Einstein–Hamilton–Jacobi constraining relations, and corresponding intrinsic Schrodinger equations. We show how to implement this freedom by canonical transformation of the intrinsic Hamiltonian. We also reinterpret and rectify significant work by Dittrich on the construction of “Dirac observables.”

The functional integral formulation of the Schrieffer-Wolff transformation

Abstract: We revisit the Schrieffer–Wolff transformation and present a path integral version of this important canonical transformation. The equivalence between the low-energy sector of the Anderson model in the so-called local moment regime and the spin-isotropic Kondo model is usually established via a canonical transformation performed on the Hamiltonian, followed by a projection. Here we present a path integral formulation of the Schrieffer–Wolff transformation which relates the functional integral form of the partition function of the Anderson model to that of its effective low-energy model. The resulting functional integral assumes the form of a spin path integral and includes a geometric phase factor, i.e. a Berry phase. Our approach stresses the underlying symmetries of the model and allows for a straightforward generalization of the transformation to more involved models. It thus not only sheds new light on a classic problem, it also offers a systematic route of obtaining effective low-energy models and higher order corrections. This is demonstrated by obtaining the effective low-energy model of a quantum dot attached to two ferromagnetic leads.

A canonical approach to forces in molecules

Abstract: In previous studies, we introduced a generalized formulation for canonical transformations and spectra to investigate the concept of canonical potentials strictly within the Born–Oppenheimer approximation. Data for the most accurate available ground electronic state pairwise intramolecular potentials in H2+, H2, HeH+, and LiH were used to rigorously establish such conclusions. Now, a canonical transformation is derived for the molecular force, F(R), with H2+ as molecular reference. These transformations are demonstrated to be inherently canonical to high accuracy but distinctly different from those corresponding to the respective potentials of H2, HeH+, and LiH. In this paper, we establish the canonical nature of the molecular force which is key to fundamental generalization of canonical approaches to molecular bonding. As further examples Mg2, benzene dimer and to water dimer are also considered within the radial limit as applications of the current methodology.

Plebanski-like action for general relativity and anti-self-dual gravity

Abstract: We present a new $BF$-type action for complex general relativity with or without a cosmological constant resembling Plebanski's action, which depends on an SO(3,$\mathbb{C}$) connection, a set of 2-forms, a symmetric matrix, and a 4-form. However, it differs from the Plebanski formulation in the way that the symmetric matrix enters into the action. The advantage of this fact is twofold. First, as compared to Plebanski's action, the symmetric matrix can now be integrated out, which leads to a pure $BF$-type action principle for general relativity; the canonical analysis of the new action then shows that it has the same phase space of the Ashtekar formalism up to a canonical transformation induced by a topological term. Second, a particular choice of the parameters involved in the formulation produces a $BF$-type action principle describing conformally anti-self-dual gravity. Therefore, the new action unifies both general relativity and anti-self-dual gravity.

Computing Fourier integral operators with caustics

Abstract: Fourier integral operators (FIOs) have widespread applications in imaging, inverse problems, and PDEs An implementation of a generic algorithm for computing FIOs associated with canonical graphs is presented, based on a recent paper of de Hoop et al Given the canonical transformation and principal symbol of the operator, a preprocessing step reduces application of an FIO approximately to multiplications, pushforwards and forward and inverse discrete Fourier transforms, which can be computed in time for an n-dimensional FIO The same preprocessed data also allows computation of the inverse and transpose of the FIO, with identical runtime Examples demonstrate the algorithm's output, and easily extendible MATLAB/C++ source code is available from the author

Soliton-like solution in quantum electrodynamics

Abstract: A novel soliton-like solution in quantum electrodynamics is obtained via a self-consistent field method. By writing the Hamiltonian of quantum electrodynamics in the Coulomb gauge, we separate out a classical component in the density operator of the electron-positron field. Then, by modeling the state vector in analogy with the theory of superconductivity, we minimize the functional for the energy of the system. This results in the equations of the self-consistent field, where the solutions are associated with the collective excitation of the electron-positron field---the soliton-like solution. In addition, the canonical transformation of the variables allowed us to separate out the total momentum of the system and, consequently, to find the relativistic energy dispersion relation for the moving soliton.

The linear canonical transformations in classical optics

Abstract: In this chapter we consider the application of the linear canonical transformations (LCTs) for the description of light propagation through optical systems. It is shown that the paraxial approximation of ray and wave optics leads to matrix and integral forms of the two-dimensional LCTs. The LCT description of the first-order optical systems consisting of basic optical elements: lenses, mirrors, homogeneous and quadratic refractive index medium intervals and their compositions is discussed. The applications of these systems for the characterization of the completely and partially coherent monochromatic light are considered. For this purpose the phase space beam representation in the form of the Wigner distribution (WD), which reveals local beam coherence properties, is used. The phase space tomography method of the WD reconstruction is discussed. The physical meaning and application of the second-order WD moments for global beam analysis, classification, and comparison are reviewed. At the similar way optical systems used for manipulation and characterization of optical pulses are described by the one-dimensional LCTs.

Gauge theory by canonical transformations

Abstract: Electromagnetism, the strong and the weak interactions are commonly formulated as gauge theories in a Lagrangian description. In this paper, we present an alternative formal derivation of U(1)-gauge theory in a manifestly covariant Hamilton formalism. We make use of canonical transformations as our guiding tool to formalize the gauging procedure. The introduction of the gauge field, its transformation behavior and a dynamical gauge field Lagrangian/Hamiltonian are unavoidable consequences of this formalism, whereas the form of the free gauge Lagrangian/Hamiltonian depends on the selection of the gauge dependence of the canonically conjugate gauge fields.

Some reference formulas for the generating functions of canonical transformations

Abstract: We study some properties of the canonical transformations in classical mechanics and quantum field theory and give a number of practical formulas concerning their generating functions. First, we give a diagrammatic formula for the perturbative expansion of the composition law around the identity map. Then we propose a standard way to express the generating function of a canonical transformation by means of a certain “componential” map, which obeys the Baker–Campbell–Hausdorff formula. We derive the diagrammatic interpretation of the componential map, work out its relation with the solution of the Hamilton–Jacobi equation and derive its time-ordered version. Finally, we generalize the results to the Batalin–Vilkovisky formalism, where the conjugate variables may have both bosonic and fermionic statistics, and describe applications to quantum field theory.