Showing papers on "Canonical transformation published in 1990"
TL;DR: In this paper, a previously derived expression for the energy of arbitrary perturbations about arbitrary Vlasov-Maxwell equilibria is transformed into a very compact form by a canonical transformation method based on Lie group theory, which is simpler than the one used before and provides better physical insight.
Abstract: A previously derived expression [Phys Rev A 40, 3898 (1989)] for the energy of arbitrary perturbations about arbitrary Vlasov–Maxwell equilibria is transformed into a very compact form The new form is also obtained by a canonical transformation method for solving Vlasov’s equation, which is based on Lie group theory This method is simpler than the one used before and provides better physical insight Finally, a procedure is presented for determining the existence of negative‐energy modes In this context the question of why there is an accessibility constraint for the particles, but not for the fields, is discussed
80 citations
TL;DR: The phase space picture of quantum mechanics and some examples illustrating it are presented in this article, where the uncertainty relation is stated in terms of an area element in phase space, whose minimum size is Planck's constant.
Abstract: The phase‐space picture of quantum mechanics and some examples illustrating it are presented. Since the position and momentum are c numbers in this picture, it is possible to introduce the concept of phase space in quantum mechanics. The uncertainty relation is stated in terms of an area element in phase space, whose minimum size is Planck’s constant. Area‐preserving canonical transformations in phase space are therefore uncertainty‐preserving transformations. The wave‐packet spread, coherent‐state representation, and squeezed states of light are discussed as illustrative examples.
42 citations
TL;DR: It is shown that canonical transformations have a particularly useful form when describing the evolution of quantum dissipative systems in phase space using the Wigner function.
Abstract: We show that the master equation describing the dissipative evolution of a bosonic quantum system coupled to a phase-sensitive reservoir may be reduced to a standard form of dissipation for a thermal reservoir by canonical transformations. The solution to the master equation for complex phase-sensitive interaction induced, for example, by broadband squeezed light, may then be obtained by a transformation from the known solutions for the simpler thermal case. We show that canonical transformations have a particularly useful form when describing the evolution of quantum dissipative systems in phase space using the Wigner function. We illustrate these ideas with two phase-sensitive dissipative problems: that of the correlated-emission laser and of decay induced by a broadband squeezed vacuum field.
40 citations
Journal Article•
36 citations
TL;DR: In this article, two types of canonical forms for certain classes of functional differential equations are introduced, which preserve oscillatory or non-oscillatory behaviour of solutions. And they are also suitable for studying both-side solutions of equivalent functional-differential equations.
Abstract: Functional-differential equations, especially linear ones, are considered with respect to global pointwise transformations. Two types of canonical forms for certain classes of these equations are introduced. These transformations and the corresponding canonical forms preserve oscillatory or non-oscillatory behaviour of solutions. They are also suitable for studying both-side solutions of equivalent functional-differential equations.
20 citations
TL;DR: In this article, the multidimensional bound optical polaron problem is studied using various approximation schemes viz. the Lee, Low, Pines canonical transformation method and its modified versions as developed by Gross and by Huybrechts, the Landau-Pekar variational method, and the Feynman-Haken path integral formalism.
20 citations
TL;DR: In this article, the authors generalize the approximate constant of motion and obtain a family of operators depending on arbitrary parameters, which can be used to separate the Hamiltonian into collective, internal, and coupling terms.
Abstract: The approximate constant of motion introduced previously for the description of a charged system in a homogeneous magnetic field is interpreted physically as the kinetic momentum of the collective motion. The algebra satisfied by this operator and by the exact constants of motion describes the behavior in a magnetic field of a single particle possessing the total charge of the system. With this algebra, we generalize the approximate constant of motion and obtain a family of operators depending on arbitrary parameters. Canonical transformations based on these new operators separate the Hamiltonian into collective, internal, and coupling terms. These terms take the same form for charged and neutral systems although the collective energy represents different physical behaviors. The coupling between the internal and collective motions is small for some choices of the parameters if at least one particle is much heavier than the other ones.
16 citations
TL;DR: The complex canonical transformation introduced by Ashtekar (1987) in general relativity is extended to simple supergravity by as discussed by the authors, which is a variant of the canonical transformation used in classical general relativity.
Abstract: The complex canonical transformation introduced by Ashtekar (1987) in general relativity is extended to simple supergravity.
14 citations
TL;DR: In this paper, Canonical transformations are used to build realisations of SL(2, R) in terms of the basic quantum mechanical operators Q and P. The results were used to construct solvable potentials in the framework of one-dimensional quantum mechanics.
Abstract: Canonical transformations are used to build realisations of SL(2, R) in terms of the basic quantum mechanical operators Q and P. The results are used to construct solvable potentials in the framework of one-dimensional quantum mechanics.
11 citations
TL;DR: In this paper, a canonical transformation technique through which new independent traits are introduced is presented and it is shown that the number of independent transformed traits is reduced by thenumber of restrictions imposed.
Abstract: neous equations if there are many traits and a large number of animals to be evaluated. In this paper, a canonical transformation technique through which new independent traits are introduced is presented. Thus only equations of relatively low order for each transformed trait have to be solved. Furthermore, it is shown that the number of independent transformed traits is reduced by the number of restrictions imposed. The technique is applicable when a multiple-trait animal model is assumed.
11 citations
TL;DR: In this paper, a deformation approach is proposed for the quantization of non-integrable Hamiltonian systems about a point of stable equilibrium, which resolves problems concerning the ordering of operators and the commutation of quantization with canonical transformation.
Abstract: A deformation approach resolves problems concerning the ordering of operators and the commutation of quantization with canonical transformation. This completes the programme of Robnik (1984) to develop a purely algebraic method for the quantization of non-integrable Hamiltonian systems about a point of stable equilibrium.
TL;DR: In this article, a canonical set of classical observables for the open Nambu string is determined by determining a canonical transformation to a new set of variables, of which the constraints of the string are a subset.
Abstract: As a prosecution of a preceeding paper, a canonical set of classical observables for the open Nambu string is determined. As a by-product, a local involutory set of constants of motion is found. This result is obtained by determining a canonical transformation to a new set of variables, of which the constraints of the string are a subset. The Poincare algebra is analyzed in terms of the new variables.
Journal Article•
TL;DR: In this paper, the authors introduced a class of Hamiltonian scattering systems which can be reduced to the normal form by means of a global canonical transformation (P,Q) = A(p,q), p,q∈Rn, defined through asymptotic properties of the trajectories.
TL;DR: For a given singular Lagrangian containing higher-order time derivatives, a dynamically equivalent Lagrangians with only first order time derivatives is constructed in this paper, and a Hamiltonian structure for this first-order Lagrangia is then found with the use of the Dirac theory of constraints.
Abstract: For a given, in general, singular Lagrangian containing higher‐order time derivatives, a dynamically equivalent Lagrangian with only first‐order time derivatives is constructed. A Hamiltonian structure for this first‐order Lagrangian is then found with the use of the Dirac theory of constraints. It is shown that in the case of a nonsingular higher‐order Lagrangian, the Ostrogradsky dynamics is derived in this way. Further, it is shown that ambiguities characteristic of higher‐order Lagrangian systems do not appear when using this construction. In particular, it is shown that the addition of a total time derivative term to the higher‐order Lagrangian can only induce a time‐independent canonical transformation, even in the case of a singular Lagrangian.
TL;DR: The Schrieffer-Wolff transformation of the large negative-U Hubbard model into the t-J model has been studied in detail in this paper, where the authors discuss similarities to and differences from the canonical transformation from the large positive-U Hamiltonian into the TJ model.
Abstract: The Schrieffer-Wolff transformation of the large-negative-U Hubbard model is performed in detail. Similarities to and differences from the canonical transformation of the large-positive-U Hamiltonian into the t-J model are discussed.
TL;DR: In this paper, an always maximum variational principle has been proposed for finding approximate solutions for several problems, and the error of such approximate solutions is estimated, where the corresponding functional is difference between primal and dual functionals for the same problem.
Abstract: Motion of mechanical system with n degrees of freedom is analysed in a special phase space. The space is obtained by a canonical transformation. In that phase space we have constructed an always maximum variational principle. The corresponding functional is difference between primal and dual functionals for the same problem. The principle has always a weak maximum on the exact solution of the equations of motion in the new phase space. The variational principle is used for finding approximate solutions for several problems. Also, the error of such approximate solutions is estimated.
TL;DR: In this article, a variational wave function was proposed to describe the ground-state properties of the attractive Hubbard model for positive U. The results are in agreement with the canonical transformation which is known to relate the attractive and repulsive Hubbard models for all electronic densities.
Abstract: A variational wave function to describe the ground-state properties of the attractive Hubbard model is presented. The function is complementary to the Gutzwiller wave function for positive U. The results are in agreement with the canonical transformation which is known to relate the attractive and repulsive Hubbard models for all electronic densities.
TL;DR: In this paper, the authors investigated the question of general covariance of the continuous path integral representation for supersymmetric quantum mechanics and showed that the perturbation expansion is covariant.
15 Oct 1990-Nuclear Instruments & Methods in Physics Research Section A-accelerators Spectrometers Detectors and Associated Equipment
TL;DR: In this paper, the design of electrostatic-accelerator free electron lasers requires an optical treatment of the accelerating electron beam that includes the effects of finite emittance, space charge and acceleration.
Abstract: The design of electrostatic-accelerator free electron lasers requires an optical treatment of the accelerating electron beam that includes the effects of finite emittance, space charge and acceleration. Use of a canonical transformation allows the problem to be treated in a new coordinate system where the Twiss parametrization of Courant and Snyder may be employed.
01 Jul 1990
TL;DR: In this paper, a canonical transformation to new action-angle variables (J, {Psi}), such that action J is nearly constant while the angle π advances almost linearly with the time, is presented.
Abstract: In various applications of nonlinear mechanics, especially in accelerator design, it would be useful to set bounds on the motion for finite but very long times. Such bounds can be sought with the help of a canonical transformation to new action-angle variables (J, {Psi}), such that action J is nearly constant while the angle {Psi} advances almost linearly with the time. By examining the change in J during a time T{sub 0} from many initial conditions in the open domain {Omega} of phase space, one can estimate the change in J during a much larger time T, on any orbit starting in a smaller open domain {Omega}{sub 0} {contained in} {Omega}. A numerical realization of this idea is described. The canonical transformations, equivalent to close approximations to invariant tori, are constructed by an effective new method in which surfaces are fitted to orbit data. In a first application to a model sextupole lattice in a region of strong nonlinearity, we predict stability of betatron motion in two degrees of freedom for a time comparable to the storage time in a proton storage ring (10{sup 8} turns). 10 refs., 6 figs., 1 tab.
11 Jul 1990
TL;DR: These algorithms directly give a linear time isomorphism algorithm for partial 3-trees and they are given as linear time algorithms constructing canonical representations of partial 2- and 3-Trees.
Abstract: We give linear time algorithms constructing canonical representations of partial 2-trees (series parallel graphs) and partial 3-trees. These algorithms directly give a linear time isomorphism algorithm for partial 3-trees.
01 Jan 1990
TL;DR: In this article, a criterion of global equivalence of the Halphen and Laguerre-Forsyth forms was derived and sufficient and necessary conditions that guarantee that an ordinary linear differential equation of an arbitrary order can be globally transformed (i.e. on its whole interval of definition).
Abstract: Two types of canonical forms of ordinary linear differential equations were known, the so-called Halphen and Laguerre—Forsyth forms. Already in 1910 G. D. Birkhoff showed that the Laguerre—Forsyth form is local for the third order equations. The same is true for an arbitrary order as well as for the Halphen canonical forms. Recently global canonical forms of linear differential equations were intruduced and a criterion of global equivalence of those equations was derived. On the basis of this criterion we establish in this paper sufficient and necessary conditions that guarantee that an ordinary linear differential equation of an arbitrary order can be globally transformed (i.e. on its whole interval of definition) into the Halphen or the Laguerre—Forsyth canonical form.
01 Jan 1990
TL;DR: In this article, it was shown that the integration of a dynamical system can be effected by transforming it into another dynamical systems with fewer degrees of freedom by the use of cyclic coordinates.
Abstract: This chapter is divided into two parts: “Canonical Transformations” and “Hamilton—Jacobi Theory”. Concerning the subject of canonical transformations, we have seen in our study of cyclic coordinates that the integration of a dynamical system can generally be effected by transforming it into another dynamical system with fewer degrees of freedom by the use of cyclic coordinates. We also saw that, in the Hamiltonian formulation, the Hamiltonian does not contain the cyclic coordinates and the corresponding conjugate momenta are conserved. Such transformations that decrease the number of degrees of freedom of the system and leave Hamilton’s canonical equations of motion invariant, are called canonical transformations. We shall investigate these ideas in the context of the general theory that underlies the solutions of dynamical systems and is the basis for the Hamilton—Jacobi theory. This theory, to be discussed in Part II, gives the foundation for the modern theory of partial differential equations (PDEs) as applied to wave propagation phenomena.
TL;DR: In this article, it was shown that the canonical transformation g:( phi 1, pi 1) to (phi 2, pi 2) leads to a Backlund transformation H(phi 1/pi 1)H*( phi 2/pi 2), which gives some new solutions for the equations delta alpha delta alpha phi =dFi( phI/d phi, i=1,2,
Abstract: Let the Hamiltonian of the field phi be H It is shown that the canonical transformation g:( phi 1, pi 1) to ( phi 2, pi 2) leads to a Backlund transformation H( phi 1, pi 1)H*( phi 2, pi 2) and the latter gives some new solutions for the equations delta alpha delta alpha phi =dFi( phi )/d phi , i=1,2,
TL;DR: In this paper, a generalised Luttinger model is proposed and studied in the spinless and spin-dependent cases, and the solution of the back-scattering and Umklapp scattering problems are obtained using a new strategy.
Abstract: A generalised Luttinger model is proposed and studied in the spinless and spin-dependent cases. The solution of the back-scattering and Umklapp scattering problems are obtained using a new strategy. For the standard Luttinger model the results already known are obtained again.
TL;DR: In this paper, the quasi-particle creation and annihilation operators introduced in the BCS formalism for the sake of convenience in the calculations, are taken as a function of a parameter (σ) in terms of the probability of the states being occupied or unoccupied.
Abstract: The quasi-particle creation and annihilation operators introduced in the BCS formalism for the sake of convenience in the calculations, are taken as a function of a parameter (σ). This parameter is also a function of an arbitrary parameter in terms of the probability of the states being occupied or unoccupied. The canonical transformation of the general Hamiltonian of the nucleus has been performed in terms of this new parameter. Using the Hamiltonian obtained, the matrix elements of the odd-mass nucleus are calculated. The energy spectra are determined with the values 0.25; 0.50; 0.75; 1.00; 1.25; 1.50; 1.75; 2.00 of the parameterz which is function of σ. The single-particle energies appearing in the matrix elements are taken separately for the 0.20; 0.25; 0.30 values of the deformation parameter of the nucleus. The single-particle energies are calculated by diagonalization of the Lamm Hamiltonian with the asymptotic base wave function. With this procedure it is observed that the energy values can be found in a rational manner by a suitable choice of all the parameters.
01 Jan 1990
TL;DR: The Lagrangian functions are particularly useful for studying the role symmetries and invariances of a given system play in its description as discussed by the authors, and they are central to the formulation of canonical mechanics, as developed by Hamilton and Jacobi.
Abstract: Canonical mechanics is a central part of general mechanics, where one goes beyond the somewhat narrow framework of Newtonian mechanics with position coordinates in the three-dimensional space, towards a more general formulation of mechanical systems belonging to a much larger class. This is the first step of abstraction, leaving behind ballistics, satellite orbits, inclined planes, and pendulum-clocks; it leads to a new kind of description that turns out to be useful in areas of physics far beyond mechanics. Through d’Alembert’s principle we discover the concept of the Lagrangian function and the framework of Lagrangian mechanics that is built onto it. Lagrangian functions are particularly useful for studying the role symmetries and invariances of a given system play in its description. By means of the Legendre transformation we are then led to the Hamiltonian function, which is central to the formulation of canonical mechanics, as developed by Hamilton and Jacobi.
01 Jan 1990
TL;DR: For a many-fermion system, Bogoliubov's Principle of Compensation of Dangerous Diagrams (PCDD) is derived from the variational principle that the number of quasiparticles in the system is a minimum as discussed by the authors.
Abstract: For a many-fermion system, Bogoliubov's Principle of Compensation of Dangerous Diagrams (PCDD) to determine the coefficients in a canonical transformation to quasiparticles is derived from the variational principle that the number of quasiparticles in the system is a minimum The PCDD states that the sum of the diagrams going from a two-quasiparticle state to the vacuum is zero When the PCDD is used with the quasiparticle self energy, both in first order of finite-temperature perturbation theory, the finite- temperature Hartree-Fock-Bogoliubov theory is obtained Corrections to the quasiparticle energy can be calculated systematically by going to second or higher orders in both the PCDD and self energy, and solving the equations self consistently