Showing papers on "Canonical transformation published in 1977"

A canonical analysis of multiple time series

Abstract: : This paper proposes a canonical transformation of a k dimensional stationary autoregressive process. The components of the transformed process are ordered from least predictable to most predictable. The least predictable components are often nearly white noise and the most predictable can be nearly nonstationary. Transformed variables which are white noise can reflect relationships which may be associated with or point to economic or physical laws. A 5-variate example is given.

Kinetic exchange interaction in a narrow S-band

Abstract: A new canonical transformation is applied to the Hubbard Hamiltonian under strong correlation with arbitrary n to obtain an effective Hamiltonian which consists of both the itinerant property of electrons in split sub-bands and the Heisenberg-type spin-spin coupling. It reduces to that for a Heisenberg antiferromagnet as t/U to 0, but the intrasub-band electron hopping becomes important as t/U increases. This effective Hamiltonian is suitable for investigating the antiferro-paramagnetic transition.

Quantization of the linearly damped harmonic oscillator

Abstract: A previously developed novel theory for the formal canonical quantization of classically dissipating systems will be the starting point for a detailed discussion of the quantum statistical aspects of the simple linearly damped harmonic oscillator. The formalism essentially involves complex classical canonical coordinates and momenta, and thus quite naturally leads to the possibility of creation and annihilation operators. Furthermore, the occurrence of quantal noise operators appears to be of principal importance for the conservation of the fundamental commutator in the course of time, as will be expressed in a simple fluctuation-dissipation relation. Making a canonical transformation back to the real, Cartesian Hermitian position and momentum an "effective" Hermitian Hamiltonian will be derived, with which a transformation is made from the Heisenberg frame to the Schr\"odinger frame where the density operator equation will be computed. This will make it obvious that no proper Schr\"odinger equation exists for the dissipative subsystem on its own, thus reflecting an incomplete knowledge. The master equation will then be translated into its Wigner representation. The intimate connection between the diffusion coefficients in the resulting Fokker-Planck equation and the uncertainty relation will be demonstrated in a clear fashion.

Generalized invariants for the time-dependent harmonic oscillator

Abstract: A generalized class of invariants, I (t), for the three‐dimensional, time‐dependent harmonic oscillator is presented in both classical and quantum mechanics. For convenience a simple notation for types of harmonic oscillator is introduced. Two interpretations, one in terms of angular momentum and the other employing a canonical transformation, are offered for I (t). An invariant symmetric tensor, Imn(t), is constructed and shown to reduce to Fradkin’s invariant tensor for time‐independent systems. The usual SU(3) (compact) or SU(2,1) (noncompact) is shown to be a noninvariance group for the time‐dependent oscillator with S{U(2) ⊗U(1) } as the invariance subgroup. Extensions to anisotropic systems and the singular quadratic perturbation problem are discussed.

Exciton dispersion in semiconductors with degenerate bands

Abstract: The problem of energy versus momentum dispersion is considered for excitons in semiconductors of the diamond and zincblende families, which are characterized by a fourfold degeneracy of the upper valence band (in the strong spin-orbit limit). This degeneracy results in the well-known impossibility of a complete decoupling of relative and translational dynamics of the electron-hole pair. It is shown, however, how the canonical transformation to the new coordinates can be optimized to reduce and simplify the coupling terms, and to obtain, on the basis of simple physical arguments, the description of the motion that is most convenient for practical calculations in a particular semiconductor. Results of such calculations are presented for several materials and are favorably compared with experimental information obtained by modulation spectroscopy. The striking deviations from the "hydrogenic" parabolic dispersion law that occur in some materials are briefly discussed.

Kustaanheimo-Stiefel Regularization and nonclassical canonical transformations.

Abstract: This paper presents a method for constructing a transformation of coordinates in phase space yielding canonical equations. The general transformation, which does not necessarily yield a canonical system, has an arbitrary function in its right-hand members. The cases when the transformed equations may be made canonical by appropriate choice of this function are established. The method is illustrated by means of the Kustaanheimo-Stiefel Regularization and applied to the circular restricted three body problem.

Generalized Ponderomotive Forces and Three-Wave Interaction

Abstract: The subject of this article is a unified Hamiltonian approach to the theory of nonlinear interactions among waves and particles. The unifying feature of the approach is a generalization of the concept of “ponderomotive force”. The formulation can be said to retain the conceptual simplicity of the familiar ponderomotivepotential method [l–3], but to remove the approximations [4]. The essence of the approach is to replace the usual method of time-averaging by the performance of a canonical transformation. The transformation is designed to eliminate the terms in the Hamiltonian of a particle which are linear in the wave potentials, replacing them with bilinear terms at combination frequencies. The new entity (the “oscillation center”) thus has no first order uttering motion. The transformation formalism leads to explicit expressions for the required nonlinear currents, which can be decomposed into the current of oscillation centers and the “polarization” corrections [4]. The oscillation-center representation is thus quite analogous to the more familiar guiding-center representation in strong magnetic fields.

Canonical transformation to the free particle

Abstract: It is shown how to find some canonical transformations without solving the Hamilton–Jacobi equation. The canonical transformation from the harmonic oscillator to the free particle is constructed and used as an example of transformations which cannot be maintained when going from classical to quantum systems. The group of canonical transformations which can be carried over to quantum mechanics is discussed.

Generalized ponderomotive forces and three-wave interaction

Abstract: The subject of this article is a unified Hamiltonian approach to the theory of nonlinear interactions among waves and particles. The unifying feature of the approach is a generalization of the concept of “ponderomotive force”. The formulation can be said to retain the conceptual simplicity of the familiar ponderomotivepotential method [l–3], but to remove the approximations [4]. The essence of the approach is to replace the usual method of time-averaging by the performance of a canonical transformation. The transformation is designed to eliminate the terms in the Hamiltonian of a particle which are linear in the wave potentials, replacing them with bilinear terms at combination frequencies. The new entity (the “oscillation center”) thus has no first order uttering motion. The transformation formalism leads to explicit expressions for the required nonlinear currents, which can be decomposed into the current of oscillation centers and the “polarization” corrections [4]. The oscillation-center representation is thus quite analogous to the more familiar guiding-center representation in strong magnetic fields.

Time dependent canonical transformations and the symmetry‐equals‐invariant theorem

Abstract: Expressions for the remainder function of a time dependent infinitesimally generated canonical transformation have recently been found by Dewar, who considered the action of the transformation operators on Liouville’s equation. Here an alternate proof of the remainder function expression is given, based on the transformations of particle trajectories. Then, using this expression, a proof of the symmetry‐equals‐invariant theorem is given.

A canonical transformation theory of the generalised Dicke model

Abstract: The Hamiltonian of the generalised Dicke model is transformed to describe collective excitations above a ground state. When the wavelength of the field modes is considerably less than the system dimensions the condition for the existence of a phase transition in a lattice configuration differs from the long-wave result. The effect of atomic separation on certain ground state properties is examined and a short description of the excitation spectrum is given. For a dielectric crystal it is shown that the super-radiant phase transition is impossible in principle for up to 48 modes.

A Practical Method to Derive Canonical Transformations in a Closed Form and it’s Application to an Antiresonant Electron-Phonon-System

Abstract: A practical method for the calculation of canonical exponential transformations in closed analytic forms is presented. An application to an antiresonant electron-phonon system is given. In particular the optical zero phonon line is calculated, which reflects the resonant nature of the system

One-component formulation of Reggeon field theory

Abstract: We present a nonperturbative formulation of the anti-Hermitian cubic Reggeon field theory (RFT) in terms of a single field chi. We analyze the structure of RFT as ..cap alpha../sub 0/ is increased above 1 and clarify the relation between the perturbative vacuum and the classical stationary points. A canonical transformation is performed so that the new Hamiltonian depends on the sign of ..delta../sub 0/ equivalent 1 - ..cap alpha../sub 0/ only through a potential of the Landau-Ginzburg type. Our one-component theory is normal-ordered with respect to the original Pomeron field without tadpoles, and it allows a path-integral formalism with undistorted contours. For ..delta../sub 0/ /g/sub 0/ large and ..delta../sub 0/ < 0, we formulate two different and yet equivalent analog models. We unambiguously derive an analog model in terms of a single classical spin at each rapidity--impact-parameter site. Through the use of an asymmetrical transfer matrix, we obtain a kinklike ground-state configuration for the D = 0 model. Alternatively, by going on a lattice for the impact-parameter space only, we arrive at a quantum lattice-spin model. We explicitly demonstrate that the quantum spin model at D = 0 is equivalent to the classical lattice spin model.

A2 Term, Renormalization of Matter-Photon Interaction and Coherent States in Matter-Photon Systems

Abstract: A non-relativistic quantum theory of matter-photon interaction is formulated, within the framework of the dipole approximation, by employing a canonical transformation to diago­ nalize the A' term plus the free photon Hamiltonian. The matter-photon interaction is there­ by expressed rigorously in a renormalized A·p form. Using this form for the Dicke model of two-level atoms interacting with a single-mode radiation field, we examine a possibility of the occurrence of stationary coherent states which could arise for a photon and atomic polarization as the ground state of the system, deducing a condition for such coherent states. The condition, expressed as an inequality which imposes a stable nontrivial Bloch angle of the uniform atomic polarization, is identical with that for the occurrence of a second order phase transition, and indicates that the non-existence of a second order phase transition pointed out by Rza2ewski et al. io; in fact a consequence of the present treatment of the A'­ term renormalization. However, a fulfillment of the condition is recovered, if exchange­ type atom-atom interaction is taken into account in the matter system. We also present a method to construct an effective Hamiltonian of atom-atom interaction which is equivalent to the above Dicke model in the thermodynamic limit.

Equivalent transformation of optimal control problems

Abstract: The equivalent transformation of the optimal control problems is proposed. The equivalent transformation is the expanded canonical transformation involving the transformation of the control. The generating function plays an important role in the equivalent transformation. The quadratic generating function is discussed. In this case the equivalent transformation is linear.

General Ward-Like Relations in Canonical Field Theory

Abstract: Infinitesimal canonical transformation of a field operator 1s written in terms of the chronological product of the field operator and the generating current. This relation is then generalized to the infinitesimal transformation of the chronological (and retarded) products of any number of field operators. It is shown that the general relations thus obtained con­ tain identities known as the generalized Ward relations in various forms. The general relations given in integral form are converted into differential form. The relation to the similar identities obtained by the path integral method is discussed and use of the general relations is suggested. The argument of a previous paper is, thus, refined and extended in three aspects: (i) constraint variables are treated systematically, (ii) it is pointed out that the general relations hold true for the retarded products, (iii) relation to the path integral method is discussed.

A Canonical Transformation for Equation of the Form φxt=h(φ)

Abstract: For evolution equation φ x t = h (φ), a canonical transformation which keeps the Hamiltonian form invariant is investigated. It is shown that the canonical transformation of this type exists, if h (φ) is linear or h φφ (φ)=µ 2 h (φ), where µ is a constant, and that a restricted Backlund transformation proposed by McLaughlin and Scott is a canonical transformation. For such nonlinearities, for example the sine-Gordon type h (φ)=sin µφ, the canonical transformations are explicitly constructed and are used to display nonlinear interactions. A relationship between the condition h φφ (φ)=µ 2 h (φ) and the existence of an infinite number of conservation laws are discussed from the view point of the theory of the canonical transformation. These results will be clarify the dynamical significance of the Backlund transformation and the soliton phenomena.