Showing papers on "Canonical transformation published in 1972"

A Bubble‐Time Canonical Formalism for Geometrodynamics

Abstract: A functional differential version of the ADM canonical formalism is proposed. The existence of a canonical transformation separating canonical variables into internal coordinates, energy‐momentum densities, and two pairs of true dynamical variables is assumed. The evolution of dynamical variables is governed by functional differential Hamilton's equations. They satisfy certain integrability conditions ensuring the internal path independence of dynamical evolution. The change of dynamical variables along any spacelike hypersurface is given by their Lie derivatives. This allows an elimination of 3∞3 components of the Hamilton equation, leading to a functional differential Hamilton equation based on a single bubble time. The Hamilton‐Jacobi theory is built along the same lines. The formalism is illustrated in the mini‐phase‐space of the cylindrical Einstein‐Rosen wave.

Perturbation Methods in Non-Linear Systems

Abstract: I. Canonical Transformation Theory and Generalizations.- 1. Introduction.- 2. Canonical Transformations.- 3. Hamilton-Jacobi Equation. Generalizations.- 4. Lie Series and Lie Transforms.- Lie's Theorem (1888).- 5. Lie Transform Depending on a Parameter.- 6. Equivalence Relations.- 7. General Transformations Induced by Lie Series.- Vector Transformation.- Notes.- References.- II. Perturbation Methods for Hamiltonian Systems. Generalizations.- 1. Introduction.- 2. Convergence of a Classical Method of Iteration.- 3. Secular Terms. Lindstedt's Device.- 4. Poincare's Method (Lindstedt's Method).- 5. Fast and Slow Variables.- 6. Generalization of the Averaging Procedure, Birkoff's Normalization and Adelphic Integrals.- 7. The Solution of Poincare's Problem in Poisson's Parentheses. Elimination of Secular Terms from Adelphic Integrals.- 8. Perturbation Techniques Based on Lie Transforms.- 9. Perturbation Methods of Non-Hamiltonian Systems Based on Lie Transforms.- Van der Pol Equation.- Notes.- References.- III. Perturbations of Integrable Systemsl.- 1. Motion of an Integrable System.- 2. Perturbations of an Integral System.- 3. Degenerate Systems.- 4. Perturbed Linear Oscillations.- 5. Linear Periodic Perturbations.- Notes.- References.- IV. Perturbations of Area Preserving Mappings.- 1. Preliminary Considerations.- 2. Regions of Motion. Perturbation of a Truncated Birkoff's Normal Form.- 3.Moser's Theorem.- 4. System with n Degree of Freedom.- 5. Degenerate Systems.- Notes.- References.- V. Resonance.- 1. Introduction.- 2. Motion in the Neighborhood of an Equilibrium Point.- 3. Solution by Formal Series28l.- 4. Equivalence with the Problem of Perturbation of a Linear System.- 5. Nonlinear Resonance.- 6. Asymptotic Expansion to Any Order.- 7. Extended Theory and the Ideal Resonance Problem.- 8. Several Degrees of Freedom.- 9. Coupling of Two Harmonic Oscillators.- Non-resonance Case.- Resonance Case.- Notes.- References.- Appendix. Remarks, Some Open Questions and Research Topics.- References.

Canonical Transformations and the Radial Oscillator and Coulomb Problems

Abstract: In a previous paper a discussion was given of linear canonical transformations and their unitary representation. We wish to extend this analysis to nonlinear canonical transformations, particularly those that are relevant to physically interesting many‐body problems. As a first step in this direction we discuss the nonlinear canonical transformations associated with the radial oscillator and Coulomb problems in which the corresponding Hamiltonian has a centrifugal force of arbitrary strength. By embedding the radial oscillator problem in a higher dimensional configuration space, we obtain its dynamical group of canonical transformations as well as its unitary representation, from the Sp(2) group of linear transformations and its representation in the higher‐dimensional space. The results of the Coulomb problem can be derived from those of the oscillator with the help of the well‐known canonical transformation that maps the first problem on the second in two‐dimensional configuration space. Finally, we make use of these nonlinear canonical transformations, to derive the matrix elements of powers of r in the oscillator and Coulomb problems from a group theoretical standpoint.

Theory of Semiclassical Transition Probabilities (S Matrix) for Inelastic and Reactive Collisions. III. Uniformization Using Exact Trajectories

Abstract: A canonical transformation of coordinates in Part I is made using exact trajectories. The transformation tends to uniformize all coordinates including that for the radial motion, thus removing the singularities in the simple semiclassical exponential wavefunction in typical cases. The new coordinates are ``time'' and certain constants of the motion. A symmetrical choice for the transformation then yields an integral expression for the S matrix satisfying the principle of microscopic reversibility. Topics discussed include semiclassical unitary transformations and time‐reversal properties of action‐angle variables and of semi‐classical wavefunctions. Applications and numerical tests of the integral expression for Smn are in progress.

Theory of Semiclassical Transition Probabilities (S Matrix) for Inelastic and Reactive Collisions. Uniformization with Elastic Collision Trajectories

Abstract: A canonical transformation is described for uniformizing the coordinates used in Paper I of this series. For comparison with the results of Paper III, which based a uniformization on exact trajectories, the present article describes one based on elastic collision trajectories. The question of invariance of S‐matrix elements with respect to semiclassical unitary transformations is also discussed.

Nonlinear canonical transformations in the coupling between degenerate high and low energy excitations

Abstract: In microscopic many-body physics the coupling between the motion of fast particles (electrons) and slow particles (nuclei) is universal. The standard Born-Oppenheimer decoupling procedure breaks down, if the energy separation in the “fast” system is of the same order as the elementary excitation in the “slow” system. In this case “dynamical resonance” effects are to be expected. In the present investigation a model system of a coupling between a doubly degenerate high energy excitation and doubly degenerate low energy oscillator is handled by a non-linear canonical transformation which is shown to be quasi-exact in the sense that it diagonalizes the Hamiltonian in both extremal coupling cases. The transformation has some flexibility, so that the diagonalization regions can be enlarged. It is employed to calculate the “zero-phonon” optical response, which indeed displays aresonance effect. Likewise, another nonlinear transformation is devised, which only in the strong coupling limit yields diagonalization. This latter transformation in a natural way leads to the conventional semi-classical approaches to the dynamical Jahn-Teller problem. The results gotten with it are identical with those from our transformation in the strong coupling limit. On the basis of our results some remarks are made concerning the possible impact of the breakdown of the adiabatic approximation in other regions.

Canonical transformations and quadratic hamiltonians

Abstract: It is shown that there exist symmetry transformations in phase space that preserve Hamilton’s canonical equations of motion for one Hamiltonian, but not for all. Examples of these « canonoid » transformations are given and their relation to canonical transformations is developed. It is demonstrated that a sufficient condition for a canonoid transformation to be canonical is that it preserve Hamilton’s equations for all Hamiltonians quadratic in theq’s andp’s.

On the canonical transformation in the theory of exciton–phonon interaction

Abstract: A canonical transformation which eliminates the trilinear exciton–phonon interaction and introduces new elementary excitations — “dressed” excitons — is proposed; it holds true both for the narrow and for broad exciton bands The energy spectrum of the dressed excitons is investigated in detail for a one-dimensional model of Frenkel excitons interacting with optical phonons Es wird eine kanonische Transformation vorgeschlagen, die die trilineare Exziton–Phonon-wechselwirkung eliminiert und neue Elementaranregungen, „bekleidete” Exzitonen, einfuhrt Dies gilt sowohl fur schmale als auch fur breite Exzitonenbanden Das Energiespektrum der bekleideten Exzitonen wird ausfuhrlich fur ein eindimensionales Modell von Frenkel-Exzitonen, die mit optischen Phononen wechselwirken, untersucht

The Stokes' Shift in Impurity Spectral Lines. II. Relationship to the Adiabatic Approximation Approach

Abstract: The derivation of the Stokes' shift has been performed without making the adiabatic approximation. A canonical transformation is used to show the relationship between that method and the more usual one in which the adiabatic approximation is made.

Necessary and Sufficient Conditions for a Canonical Transformation

Abstract: Necessary and sufficient conditions are given for a transformation to be canonical. It is shown that if the transformation is to be canonical for all Hamiltonians, the condition is equivalent to the usual fundamental Poisson bracket conditions.

Canonical Formalism in Special Relativity

Abstract: A covariant hamiltonian description was introduced in the dynamics of charges and electromagnetic interaction. By a canonical transformation this hamiltonian formalism was transformed to obtain the Dirac generators for any form of relativistic dynamics, as coefficients of a first degree polynomial in the ten translation and rotation velocities of the Poincare transformation. The Currie's world line conditions were generalized to any form of the dynamics. The explicit relation between the covariant field variables and the more usual 3-dimensional Fourier variables was derived.

Simplification of the constraints in the dirac formalism for general relativity

Abstract: In any classical theory in canonical form, the Poisson bracket relations between the constraints are preserved under canonical transformations. We show that in the Dirac formalism for general relativity this condition places certain limits on the degree to which one can simplify the form of the constraints. It implies, for instance, that the constraints cannot all be written as canonical momenta. Furthermore, it is not even possible to reduce them all to purely algebraic functions of the momenta by means of a canonical tansformation which preserves the original configuration space subspace of phase space.

Schwinger's Variational Principle in Quantum Mechanics with Velocity Dependent Potential. I

Abstract: Schwinger's variational principle is formulated for the quantum system which corresponds to the one-dimensional classical system described by the Lagrangian L 0 (i:, x) = (M/2)F 1 (x) i:2 - v (x). It is sufficient for the purpose of deriving the laws of quantum mechanics to con­ sider only c-number variations of coordinate and time. The Euler-Lagrange equation, the canonical equations of motion and the canonical commutation relation are derived from the principle. All resulting relations are consistent with one another. Further, it is shown that an arbitrary point transformation leaves the forms of the fundamental equations invariant and is suitable to be called a canonical transformation. The appropriate choice of the Lag­ rangian operator is essential in our formulation.

The Model Hamiltonian in Superconductivity Theory

Abstract: A system of fermions with attraction described by the model Hamiltonian in superconductivity theory with separable interaction is considered. Asymptotically exact estimates (as V → ∞) for the minimal eigenvalue of the Hamiltonian, correlation functions, and Green’s functions are obtained.