Showing papers on "Canonical transformation published in 2001"
TL;DR: For phase-amplitude problems, the original Fourier transform is found to be more important than the amplitude for the FRFT/LCT, and the WDF is used to explain why fractional/canonical convolution can be used for space-variant pattern recognition.
Abstract: The fractional Fourier transform (FRFT) is a useful tool for signal processing. It is the generalization of the Fourier transform. Many fractional operations, such as fractional convolution, fractional correlation, and the fractional Hilbert transform, are defined from it. In fact, the FRFT can be further generalized into the linear canonical transform (LCT), and we can also use the LCT to define several canonical operations. In this paper, we discuss the relations between the operations described above and some important time-frequency distributions (TFDs), such as the Wigner distribution function (WDF), the ambiguity function (AF), the signal correlation function, and the spectrum correlation function. First, we systematically review the previous works in brief. Then, some new relations are derived and listed in tables. Then, we use these relations to analyze the applications of the FRPT/LCT to fractional/canonical filter design, fractional/canonical Hilbert transform, beam shaping, and then we analyze the phase-amplitude problems of the FRFT/LCT. For phase-amplitude problems, we find, as with the original Fourier transform, that in most cases, the phase is more important than the amplitude for the FRFT/LCT. We also use the WDF to explain why fractional/canonical convolution can be used for space-variant pattern recognition.
266 citations
TL;DR: In this article, generalized canonical transformations for generalized Hamiltonian systems are introduced, which convert a generalized Hamiltonians system into another one, and preserve the original structure of the original one.
265 citations
TL;DR: In this article, the canonical forms into which any pure three-qubit state can be cast are analyzed and the minimal forms, i.e., the ones with the minimal number of product states built from local bases, are also presented and lead to a complete classification of pure threequbit states.
Abstract: In this paper we analyse the canonical forms into which any pure three-qubit state can be cast. The minimal forms, i.e. the ones with the minimal number of product states built from local bases, are also presented and lead to a complete classification of pure three-qubit states. This classification is related to the values of the polynomial invariants under local unitary transformations by a one-to-one correspondence.
135 citations
TL;DR: In this paper, the nonlinear supersymmetry of one-dimensional systems is investigated in the context of the quantum anomaly problem, and the most general one-parametric Calogero-like solution with the second order supercharges is found.
113 citations
TL;DR: This paper is concerned with the stabilization of nonholonomic systems in port-controlled Hamiltonian formulae based on time-varying generalized canonical transformations which modify the kinetic energy of the original system without changing the generalized Hamiltonian structure with passivity.
56 citations
TL;DR: In this article, it was shown that from the three roots of the implicit cubic equation defining the bound-state energy eigenvalues, there is always only one that leads to a meaningful physical state.
Abstract: We study a quantum mechanical potential introduced previously as a conditionally exactly solvable (CES) model. Besides an analysis following its original introduction in terms of the point canonical transformation, we also present an alternative supersymmetric construction of it. We demonstrate that from the three roots of the implicit cubic equation defining the bound-state energy eigenvalues, there is always only one that leads to a meaningful physical state. Finally we demonstrate that the present CES interaction is, in fact, an exactly solvable Natanzon-class potential.
56 citations
TL;DR: In this article, a new approach to numerical modelling of water-wave evolution based on the Zakharov integrodifferential equation is presented, which is known to follow from the exact equations of potential water waves by symmetry-preserving truncation at a certain order in wave steepness.
Abstract: We develop a new approach to numerical modelling of water-wave evolution based on the Zakharov integrodifferential equation and outline its areas of application.The Zakharov equation is known to follow from the exact equations of potential water waves by the symmetry-preserving truncation at a certain order in wave steepness. This equation, being formulated in terms of nonlinear normal variables, has long been recognized as an indispensable tool for theoretical analysis of surface wave dynamics. However, its potential as the basis for the numerical modelling of wave evolution has not been adequately explored. We partly fill this gap by presenting a new algorithm for the numerical simulation of the evolution of surface waves, based on the Hamiltonian form of the Zakharov equation taking account of quintet interactions. Time integration is performed either by a symplectic scheme, devised as a canonical transformation of a given order on a timestep, or by the conventional Runge–Kutta algorithm. In the latter case, non-conservative effects, small enough to preserve the Hamiltonian structure of the equation to the required order, can be taken into account. The bulky coefficients of the equation are computed only once, by a preprocessing routine, and stored in a convenient way in order to make the subsequent operations vectorized.The advantages of the present method over conventional numerical models are most apparent when the triplet interactions are not important. Then, due to the removal of non-resonant interactions by means of a canonical transformation, there are incomparably fewer interactions to consider and the integration can be carried out on the slow time scale (O(e2), where e is a small parameter characterizing wave slope), leading to a substantial gain in computational efficiency. For instance, a simulation of the long-term evolution of 103 normal modes requires only moderate computational resources; a corresponding simulation in physical space would involve millions of degrees of freedom and much smaller integration timestep.A number of examples aimed at problems of independent physical interest, where the use of other existing methods would have been difficult or impossible, illustrates various aspects of the implementation of the approach. The specific problems include establishing the range of validity of the deterministic description of water wave evolution, the emergence of sporadic horseshoe patterns on the water surface, and the study of the coupled evolution of a steep wave and low-intensity broad-band noise.
55 citations
TL;DR: In this article, the authors define the rest-frame instant form of tetrad gravity restricted to Christodoulou-Klainermann spacetimes and solve the multitemporal equations associated with the rotation and space diffeomorphism constraints.
Abstract: We define the {\it rest-frame instant form} of tetrad gravity restricted to Christodoulou-Klainermann spacetimes. After a study of the Hamiltonian group of gauge transformations generated by the 14 first class constraints of the theory, we define and solve the multitemporal equations associated with the rotation and space diffeomorphism constraints, finding how the cotriads and their momenta depend on the corresponding gauge variables. This allows to find quasi-Shanmugadhasan canonical transformation to the class of 3-orthogonal gauges and to find the Dirac observables for superspace in these gauges.
The construction of the explicit form of the transformation and of the solution of the rotation and supermomentum constraints is reduced to solve a system of elliptic linear and quasi-linear partial differential equations. We then show that the superhamiltonian constraint becomes the Lichnerowicz equation for the conformal factor of the 3-metric and that the last gauge variable is the momentum conjugated to the conformal factor. The gauge transformations generated by the superhamiltonian constraint perform the transitions among the allowed foliations of spacetime, so that the theory is independent from its 3+1 splittings. In the special 3-orthogonal gauge defined by the vanishing of the conformal factor momentum we determine the final Dirac observables for the gravitational field even if we are not able to solve the Lichnerowicz equation. The final Hamiltonian is the weak ADM energy restricted to this completely fixed gauge.
54 citations
TL;DR: For a time-dependent classical quadratic oscillator, this paper introduced pairs of real and complex invariants that realize explicitly a canonical transformation from the phase space to the invariant space, in which the action-phase variables are defined.
Abstract: For a time-dependent classical quadratic oscillator we introduce pairs of real and complex invariants that are linear in position and momentum. Each pair of invariants realize explicitly a canonical transformation from the phase space to the invariant space, in which the action-phase variables are defined. We find the action operator for the time-dependent quantum oscillator via the classical-quantum correspondence. Candidate phase operators conjugate to the action operator are discussed, but no satisfactory ones are found.
50 citations
TL;DR: In this paper, the authors define a space having a canonical symplectic structure where they appear as conjugate variables and the associated intensive variables γ i, the partial derivatives of the entropy S = Sq1,..., qn ≡q0, in the form γi = - pi/p0 where p0 behaves as a gauge factor.
Abstract: Denoting by qii = 1,..., n the set of extensive variables which characterize the state of a thermodynamic system, we write the associated intensive variables γ i, the partial derivatives of the entropy S = Sq1,..., qn ≡q0, in the form γ i = - pi/p0 where p0 behaves as a gauge factor. When regarded as independent, the variables qi, pii = 0,..., n define a space having a canonical symplectic structure where they appear as conjugate. A thermodynamic system is represented by a n + 1-dimensional gauge-invariant Lagrangian submanifold of. Any thermodynamic process, even dissipative, taking place on is represented by a Hamiltonian trajectory in, governed by a Hamiltonian function which is zero on. A mapping between the equations of state of different systems is likewise represented by a canonical transformation in. Moreover a Riemannian metric arises naturally from statistical mechanics for any thermodynamic system, with the differentials dqi as contravariant components of an infinitesimal shift and the dpi's as covariant ones. Illustrative examples are given.
47 citations
TL;DR: An exact invariant is derived for n-degree-of-freedom Hamiltonian systems with general time-dependent potentials and it is shown that the invariant can be interpreted as the time integral of an energy balance equation.
Abstract: An exact invariant is derived for n-degree-of-freedom Hamiltonian systems with general time-dependent potentials. The invariant is worked out in two equivalent ways. In the first approach, we define a special Ansatz for the invariant and determine its time-dependent coefficients. In the second approach, we perform a two-step canonical transformation of the initially time-dependent Hamiltonian to a time-independent one. The invariant is found to contain a function of time ${f}_{2}(t),$ defined as a solution of a linear third-order differential equation whose coefficients depend in general on the explicitly known configuration space trajectory that follows from the system's time evolution. It is shown that the invariant can be interpreted as the time integral of an energy balance equation. Our result is applied to a one-dimensional, time-dependent, damped non-linear oscillator, and to a three-dimensional system of Coulomb-interacting particles that are confined in a time-dependent quadratic external potential. We finally show that our results can be used to assess the accuracy of numerical simulations of time-dependent Hamiltonian systems.
TL;DR: In this article, the authors introduce a canonical transformation of the fundamental single-mode field operators a and b that generalizes the linear Bogoliubov transformation familiar in the construction of the harmonic oscillator squeezed states, and examine the structure and properties of the quantum states defined as eigenvectors of the transformed annihilation operator b.
Abstract: We introduce a linear, canonical transformation of the fundamental single-mode field operators a and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ that generalizes the linear Bogoliubov transformation familiar in the construction of the harmonic oscillator squeezed states. This generalization is obtained by adding a nonlinear function of any of the fundamental quadrature operators ${X}_{1}$ and ${X}_{2}$ to the linear transformation, thus making the original Bogoliubov transformation quadrature dependent. Remarkably, the conditions of canonicity do not impose any constraint on the form of the nonlinear function, and lead to a set of nontrivial algebraic relations between the c-number coefficients of the transformation. We examine in detail the structure and the properties of the quantum states defined as eigenvectors of the transformed annihilation operator b. These eigenvectors define a class of multiphoton squeezed states. The structure of the uncertainty products and of the quasiprobability distributions in phase space shows that besides coherence properties, these states exhibit a squeezing and a deformation (cooling) of the phase-space trajectories, both of which strongly depend on the form of the nonlinear function. The presence of the extra nonlinear term in the phase of the wave functions has also relevant consequences on photon statistics and correlation properties. The nonquadratic structure of the associated Hamiltonians suggests that these states be generated in connection with multiphoton processes in media with higher nonlinearities. We give a detailed description of the quadratic nonlinear transformation, which defines four-photon squeezed states. In particular, the behaviors of the second-order correlation function ${g}^{(2)}(0)$ and of the fourth-order correlation function ${g}^{(4)}(0)$ are studied. The former exhibits super-Poissonian statistics, while the latter indicates photon bunching in the four-photon emissions.
TL;DR: The canonical formalism of N = 1 supergravity with the real Ashtekar variables is considered in this article, where the canonical structure of the theory is examined and the canonical constraints in terms of these variables are given.
Abstract: The canonical formalism of N = 1 supergravity with the real Ashtekar variables are considered. This canonical formalism is derived from that of the usual N = 1 supergravity by performing the Barbero-type canonical transformation with a free parameter β. We also introduce new variables to simplify the Dirac brackets of the canonical variables and give the canonical constraints in terms of these variables. We examine the canonical structure of the theory.
TL;DR: In this article, the reparametrization-invariant generating functional for the unitary and causal perturbation theory in general relativity in a finite space-time is obtained.
Abstract: The reparametrization-invariant generating functional for the unitary and causal perturbation theory in general relativity in a finite space–time is obtained. The region of validity of the Faddeev–Popov–DeWitt functional is studied. It is shown that the invariant content of general relativity as a constrained system can be covered by two "equivalent" unconstrained systems: the "dynamic" (with "dynamic" evolution parameter as the metric scale factor) and "geometric" (given by the Levi–Civita type canonical transformation to the action-angle variables where the energy constraint converts into a new momentum). "Big Bang," the Hubble evolution, and creation of matter fields by the "geometric" vacuum are described by the inverted Levi–Civita transformation of the geomeric system into the dynamic one. The particular case of the Levi–Civita transformations are the Bogoliubov ones of the particle variables (diagonalizing the dynamic Hamiltonian) to the quasiparticles (diagonalizing the equations of motion). The choice of initial conditions for the "Big Bang" in the form of the Bogoliubov (squeezed) vacuum reproduces all stages of the evolution of the Friedmann–Robertson–Walker universe in their conformal (Hoyle–Narlikar) versions.
TL;DR: In this article, a non-singular canonical transformation to Kruskal coordinates has been derived from the usual ADM metric-extrinsic curvature variables on the phase space of Schwarzschild black holes.
Abstract: We derive a transformation from the usual ADM metric-extrinsic curvature variables on the phase space of Schwarzschild black holes to new canonical variables which have the interpretation of Kruskal coordinates. We explicitly show that this transformation is non-singular, even at the horizon. The constraints of the theory simplify in terms of the new canonical variables and are equivalent to the vanishing of the canonical momenta. Our work is based on earlier seminal work by Kucha\ifmmode \check{r}\else \v{r}\fi{} in which he reconstructed curvature coordinates and a mass function from spherically symmetric canonical data. The key feature in our construction of a nonsingular canonical transformation to Kruskal variables is the scaling of the curvature coordinate variables by the mass function rather than by the mass at left spatial infinity.
TL;DR: In this article, a new integration technique, the symplectic method, is introduced and applied for the tracing of a charged particle motion in a dipole magnetic field, which is an integral technique for the Hamilton's system by the repetition of canonical transformation, and has been tested in celestial mechanics.
Abstract: A new integration technique, symplectic method, is introduced and applied for the tracing of a charged particle motion in a dipole magnetic field. This method is an integral technique for the Hamilton's system by the repetition of canonical transformation, and has been tested in celestial mechanics in recent years. Compared with the standard Runge-Kutta method (4th order), the numerical error accumulation is much smaller for the symplectic method. An effective potential is introduced and used to discuss an abrupt transition between trapping and un-trapping regions.
TL;DR: In this article, the boundedness of all the solutions for the 2π-periodic equation x + ax+−bx− = f(t) is considered. But the bounded solutions are not bounded in the sense that a and b are positive constants.
Abstract: In this paper we consider the boundedness of all the solutions for the equation x″ + ax+−bx− = f(t) is a smooth 2π-periodic function, a and b are positive constants (a≠b).
TL;DR: In this paper, the generalized periodic Anderson model was investigated from the point of view of the possible appearance of coupled electron pairs in the local model, where the renormalized energy spectrum was divided into low and high energy parts separated by an interval of the order of the Coulomb electron-repulsion energy.
Abstract: We investigate the generalized periodic Anderson model describing two groups of strongly correlated (d- and f-) electrons with local hybridization of states and d-electron hopping between lattice sites from the standpoint of the possible appearance of coupled electron pairs in it. The atomic limit of this model admits an exact solution based on the canonical transformation method. The renormalized energy spectrum of the local model is divided into low- and high-energy parts separated by an interval of the order of the Coulomb electron-repulsion energy. The projection of the Hamiltonian on the states in the low-energy part of the spectrum leads to pair-interaction terms appearing for electrons belonging to d- and f-orbitals and to their possible tunneling between these orbitals. In this case, the terms in the Hamiltonian that are due to ion energies and electron hopping are strongly correlated and can be realized only between states that are not twice occupied. The resulting Hamiltonian no longer involves strong couplings, which are suppressed by quantum fluctuations of state hybridization. After linearizing this Hamiltonian in the mean-field approximation, we find the quasiparticle energy spectrum and outline a method for attaining self-consistency of the order parameters of the superconducting phase. For simplicity, we perform all calculations for a symmetric Anderson model in which the energies of twice occupied d- and f-orbitals are assumed to be the same.
TL;DR: In this paper, it was shown that quantum canonical transformations which are used to establish the equivalence or duality of different quantum systems and which have non-linear classical counterparts will be non-unitary or, more generally, linear isomorphisms but not isometries.
01 Oct 2001
TL;DR: In this paper, a nonlinear canonical transformation is used to transform the rapidly oscillating terms in the original Hamiltonian to a new Hamiltonian that contains slowly varying terms only. And a stationary solution to the transformed Vlasov equation has been obtained.
Abstract: A Hamiltonian approach to the solution of the Vlasov-Poisson equations has been developed. Based on a nonlinear canonical transformation, the rapidly oscillating terms in the original Hamiltonian are transformed away, yielding a new Hamiltonian that contains slowly varying terms only. The formalism has been applied to the coherent beam-beam interaction, and a stationary solution to the transformed Vlasov equation has been obtained.
TL;DR: For integrable systems with several coupling constants, the corresponding Hamiltonians satisfy Whitham equations and after quantization (of the original system) become operators satisfying the zero-curvature condition in the space of coupling constants.
Abstract: Variation of coupling constants of integrable system can be considered as canonical transformation or, infinitesimally, a Hamiltonian flow in the space of such systems. Any function $T(\vec p, \vec q)$ generates a one-parametric family of integrable systems in vicinity of a single system: this gives an idea of how many integrable systems there are in the space of coupling constants. Inverse flow is generated by a dual "Hamiltonian", $\widetilde T(\vec p, \vec q)$ associated with the dual integrable system. In vicinity of a self-dual point the duality transformation just interchanges momenta and coordinates in such a "Hamiltonian": $\widetilde T(\vec p, \vec q) = T(\vec q, \vec p)$. For integrable system with several coupling constants the corresponding "Hamiltonians" $T_i(\vec p, \vec q)$ satisfy Whitham equations and after quantization (of the original system) become operators satisfying the zero-curvature condition in the space of coupling constants: [ d/dg_a - T_a(p,q), d/dg_b - T_b(p,q) ] = 0. Some explicit formulas are given for harmonic oscillator and for Calogero-Ruijsenaars-Dell system.
18 Jun 2001
TL;DR: In this paper, a nonlinear canonical transformation is used to transform the rapidly oscillating terms in the original Hamiltonian to a new Hamiltonian that contains slowly varying terms only. And a stationary solution to the transformed Vlasov equation has been obtained.
Abstract: A Hamiltonian approach to the solution of the Vlasov-Poisson equations has been developed. Based on a nonlinear canonical transformation, the rapidly oscillating terms in the original Hamiltonian are transformed away, yielding a new Hamiltonian that contains slowly varying terms only. The formalism has been applied to the coherent beam-beam interaction, and a stationary solution to the transformed Vlasov equation has been obtained.
TL;DR: In this paper, exact solutions of Dirac equation at zero kinetic energy for radial power-law relativistic potentials were obtained by point canonical transformation of the exactly solvable problem of the three dimensional oscillator.
Abstract: We obtain exact solutions of Dirac equation at zero kinetic energy for radial power-law relativistic potentials. It turns out that these are the relativistic extension of a subclass of exact solutions of Schrodinger equation with two-term power-law potentials at zero energy. The latter is solved by point canonical transformation of the exactly solvable problem of the three dimensional oscillator. The wavefunction solutions are written in terms of the confluent hypergeometric functions and almost always square integrable. For most cases these solutions support bound states at zero energy. Some exceptional unbounded states are normalizable for non-zero angular momentum. Using a generalized definition, degeneracy of the nonrelativistic states is demonstrated and the associated degenerate observable is defined.
Journal Article•
TL;DR: A path integral formalism for Feshbach-Villars equation by using the fermionic Schwinger model for Pauli matrices which describe an isocharge symmetry is presented in this paper.
Abstract: In this paper we have set up a path integral formalism for Feshbach-Villars equation by using the fermionic Schwinger model for Pauli matrices which describe an isocharge symmetry. This choice is made in analogy with spin model and the coherent state representation is then used. We have also given a general method of treating the problem of vanishing scalar potential by reducing it the to non-relativistic case and then, via Foldy-Wouthuysen canonical transformation, an explicit solution is constructed. The free case and constant magnetic field interaction are explicitly exposed. In each cases the propagators are evaluated and the energy spectrum and the corresponding wave functions are deduced. P.A.C.S.03.65 Ca.Formalism P.A.C.S.03.65 Db.Functional analytical methods P.A.C.S.03.65 Pm.Relativistic wave equations P.A.C.S.03.65 Ge.Solutions of wave equations: bound states.
01 Jan 2001
TL;DR: A classical Hamiltonian system is called time-reversal invariant if from any given solution x(t), p(t) of Hamilton's equations an independent solution x, t, p, t is obtained with t −t and some operation relating x and p to the original coordinates x and momenta p as mentioned in this paper.
Abstract: A classical Hamiltonian system is called time-reversal invariant if from any given solution x(t), p(t) of Hamilton’s equations an independent solution x′(t′), p′(t′), is obtained with t′ = −t and some operation relating x′ and p′ to the original coordinates x and momenta p.
TL;DR: In this article, the canonical transformation employed in polaron theory at intermediate coupling is generalized to the electron phonon interaction of an entire band of electrons, and the overall lattice-electron correlation is obtained with a variational calculation, balancing the extra elastic energy stored in the lattice against the negative energy arising from BCS pair correlation.
Abstract: The canonical transformation employed in polaron theory at intermediate coupling is generalized to the electron phonon interaction of an entire band of electrons. Interelectron interactions result. The overall lattice-electron correlation is obtained with a variational calculation, balancing the extra elastic energy stored in the lattice against the negative energy arising from BCS pair correlation. The requisite phonon softening to attain any given value of Tc in metallic LaBaCuO and YBaCuO are estimated. The implicit fluctuations in the lattice normal coordinates are compared with results yielded by the EXAFS measurements of Oyanagi et al.
TL;DR: In this paper, a direct quantization scheme for a harmonic oscillator whose damping coefficient is time-dependent was proposed by adopting the Gaussian-type propagator and Feynman's path integral method.
Abstract: By means of canonical transformation,a direct quantization scheme is found for a harmonic oscillator whose damping coefficient is time-dependent.By adopting the Gaussian-type propagator and Feynman's path integral method,the exact wave function is derived.Some discussions made.
31 Dec 2001
Posted Content•
TL;DR: The existence of such a canonical transformation is guaranteed by a theorem due to Maskawa and Nakajima as mentioned in this paper, who also showed that the angular momentum of a particle moving on a circle is quantized using the star product formalism.
Abstract: The noncommutative star product of phase space functions is, by construction, associative for both non-degenerate and degenerate case (involving only second class constraints) as has been shown by Berezin, Batalin and Tyutin. However, for the latter case, the manifest associativity is lost if an arbitrary coordinate system is used but can be restored by using an unconstrained canonical set.
The existence of such a canonical transformation is guaranteed by a theorem due to Maskawa and Nakajima. In terms of these new variables, the Kontsevich series for the star product reduces to an exponential series which is manifestly associative. We also show, using the star product formalism, that the angular momentum of a particle moving on a circle is quantized.
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TL;DR: In this article, the Hamiltonian analysis of Polyakov action is reviewed putting emphasis in two topics: Dirac observables and gauge conditions, and two relatives of string theory whose actions are fully gauge-invariant under the gauge symmetry involved when the spatial slice is closed are built.
Abstract: The Hamiltonian analysis of Polyakov action is reviewed putting emphasis in two topics: Dirac observables and gauge conditions. In the case of the closed string it is computed the change of its action induced by the gauge transformation coming from the first class constraints. As expected, the Hamiltonian action is not gauge invariant due to the Hamiltonian constraint quadratic in the momenta. However, it is possible to add a boundary term to the original action to build a fully gauge-invariant action at first order. In addition, two relatives of string theory whose actions are fully gauge-invariant under the gauge symmetry involved when the spatial slice is closed are built. The first one is pure diffeomorphism in the sense it has no Hamiltonian constraint and thus bosonic string theory becomes a sub-sector of its space of solutions. The second one is associated with the tensionless bosonic string, its boundary term induces a canonical transformation and the fully gauge-invariant action written in terms of the new canonical variables becomes linear in the momenta.