More about superconformal field theory. Hamiltonian formalism

The Hamiltonian formalism for theN=1,d=4 superconformal system is given. The first-order formalism is found by starting from the canonical covariant one. As the conformal supergravity is a higher-derivative theory, to analyze the second-order Hamiltonian formalism the Ostrogradski transformation is introduced to define canonical momenta.     

Perturbation theory with higher derivative couplings

The formalism of partial differential equations with respect to coupling constants is used to develop a covariant perturbation theory for the interpolating fields and theS matrix when the coupling terms in the Larangian density involve arbitrary (first and higher) derivatives. Through the notion of pure noncovariant contractions, the free-fieldT and the (covariant)T ^* products can be related to each other, allowing us to avoid the Hamiltonian density altogether when dealing with theS matrix. The important ingredients in our approach are (1) the adiabatic switching on and off of the interactions in the infinite past and future, respectively, and (2) the vanishing of four-dimensional delta functions and their derivatives at zero space-time points. The latter ingredient is a prerequisite that our formalism and the canonical formalism be consistent with each other, and on the other hand, it is supported by the dimensional regularization. Corresponding to any Lagrangian, the generalized interaction Hamiltonian density is defined from the covariantS matrix with the help of the pure noncovariant contractions. This interaction Hamiltonian density reduces to the usual one when the Lagrangian density depends on just first derivatives and when the usual canonical formalism can be applied.     

Gravitational collapse with charge and small asymmetries. II. Interacting electromagnetic and gravitational perturbations

Paper I analyzed the evolution of nonspherical scalar-field perturbations of an electrically charged, collapsing star; this paper treats coupled electromagnetic and gravitational perturbations. It employs the results of recent detailed work in which coupled perturbations were studied in a gauge-invariant manner by using the Hamiltonian (Moncrief s) approach and the Newman-Penrose formalism, and the relations between the fundamental quantities of these two methods were obtained.It is shown that scalar-field perturbations are a prototype for coupled perturbations. The collapse produces a Reissner-Nordström black hole, and the perturbations are radiated away completely. Alll-pole parts of the perturbations of the metric and the electromagnetic field decay according to power laws; in the extreme case (e ² =M ²), the interaction causes the quadrupole perturbations to die out more slowly than the dipole perturbations.     

Second-Order Five-Dimensional Chern–Simons Gravity

Continuing our previous discussion of the canonical covariant formalism (Zandron, O. S. (in press). International Journal of Theoretical Physics), the second-order canonical fünfbein formalism of the topological five-dimensional Chern–Simons gravity is constructed. Since this gravity model naturally contains a Gauss–Bonnet term quadratic in curvature, the second-order formalism requires the implementation of the Ostrogradski transformation in order to introduce canonical momenta. This is due to the presence of second time-derivatives of the fünfbein field. By performing the space–time decomposition of the manifold M ⁵, the set of first-class constraints that determines all the Hamiltonian gauge symmetries can be found. The total Hamiltonian as generator of time evolution is constructed, and the apparent gauge degrees of freedom are unambiguously removed, leaving only the physical ones.     

The Application of Operator Formalism for Describing Experiments in Multipole Spin Systems under Multipulse Actions

An operator formalism is developed which allows multi-pulse actions on scalar-coupled spin systems with quadrupole nuclei to be analyzed. The formulas describing the evolution of angular-momentum operators under the action of a spin-spin interaction Hamiltonian are derived. The calculation of an NMR-spectrum for scalar-coupled multipole spin systems of the AX (A = 1/2, X = 1, 3/2) and AMX (M = 1, X = 1) types is presented and the polarization transfer processes under multi-pulse action are examined as examples of application of the operator formalism. A series of pulse sequences is proposed which allows individual longitudinal spin orders to be observed.     

Generalized Hamiltonian formalism of (2+1)-dimensional non-linear \sigma-model in polynomial formulation

We investigate the canonical structure of the (2+1)-dimensional non-linear model in a polynomial formulation. A current density defined in the non-linear model is a vector field, which satisfies a formal flatness (or pure gauge) condition. It is the polynomial formulation in which the vector field is regarded as a dynamic variable on which the flatness condition is imposed as a constraint condition by introducing a Lagrange multiplier field. The model so formulated has gauge symmetry under a transformation of the Lagrange multiplier field. We construct the generalized Hamiltonian formalism of the model explicitly by using the Dirac method for constrained systems. We derive three types of the pre-gauge-fixing Hamiltonian systems: In the first system the current algebra is realized as the fundamental Dirac Brackets. The second one manifests the similar canonical structure as the Chern-Simons or BF theories. In the last one there appears an interesting interaction as the dynamic variables are coupled to their conjugate momenta via the covariant derivative. Received: 29 September 1998 / Published online: 14 January 1999     

On the theories of the interacting perturbations of the Reissner-Nordström black hole

A complete account of the Hamiltonian approach to the coupled perturbations of the Reissner-Nordström black hole, initiated by Moncrief, is given. All Hamiltonian equations are expressed explicitly in suitable forms; the metric and electromagnetic field perturbations are found in terms of Moncrief's gauge invariant canonical variables in the Regge-Wheeler gauge. The basic (both tetrad and coordinate) gauge invariant scalars occurring in the perturbation studies based on the Newman-Penrose formalism are then related to Moncrief's variables. The strikingly simple relations obtained enable us to show that the fundamental pair of decoupled equations, derived recently within the Newman-Penrose formalism by Chandrasekhar, can be cast into gauge invariant form, and that it can be obtained from Moncrief's formalism.It is demonstrated how the fundamental equations, supplemented by another combination of the Newman — Penrose equations, generalize the Bardeen-Press equations for uncoupled electromagnetic and gravitational perturbations of the Schwarzschild black hole.The odd and the even parityl=1 perturbations are also considered in detail. In the Appendix the relations to Zerilli's work on coupled perturbations of the Reissner-Nordström black hole are given.     

Hamiltonian analysis of linearized extension of Hořava–Lifshitz gravity

We investigate the Hamiltonian structure of linearized extended Ho?ava–Lifshitz gravity in a flat cosmological background following the Faddeev–Jackiw's Hamiltonian reduction formalism. The Hamiltonian structure of extended Ho?ava–Lifshitz gravity is similar to that of the projectable version of original Ho?ava–Lifshitz gravity, in which there is one primary constraint and so there are two physical degrees of freedom. In the infrared (IR) limit, however, there is one propagating degree of freedom in the general cosmological background, and that is coupled to the scalar graviton mode. We find that extra scalar graviton mode in an inflationary background can be decoupled from the matter field in the IR limit. But it is necessary to go beyond linear order in order to draw any conclusion of the strong coupling problem.     

Analytic calculation of field-strength correlators

Field correlators are expressed using background-field formalism through the gluelump Green’s functions. The latter are obtained in the path-integral and Hamiltonian formalism. As a result, the behavior of field correlators is obtained at small and large distances for both perturbative and nonperturbative parts. The latter decay exponentially at large distances and are finite at x = 0, in agreement with OPE and lattice data. The text was submitted by the author in English.     

Supersymmetric anyon model coupled to the electromagnetic field

We construct a supersymmetric gauge model describing the electromagnetic interaction of anyons. This is done by means of the supersymmetric generalization of theU(1) ×U(1) gauge theory. The model contains the statisticalU(1) gauge field endowed with a Chern-Simons mass term and the electromagnetic field, both with the corresponding superpartners, coupled to matter fields. This constrained system is analyzed from the Hamiltonian point of view and the canonical quantization is found. The path-integral method is used to develop the perturbative formalism. We define suitable propagators and vertices and give the diagrammatics and the Feynman rules.     

Calculated effects of interface grading in GaAs----Ga1−xAlxAs quantum wells

Effects of interface grading on energy levels of electrons in GaAs---Ga_1−xAl_xAs quantum wells have been estimated using both a tight-binding formalism and an effective-mass Hamiltonian of the BenDaniel-Duke form. Graded interfaces a few atomic layers thich have only a small effect on energy levels in both schemes. Self-consistent calculations for electrons in a relatively wide (40 nm) quantum well show how the lowest levels change from those characteristic of the empty well to those characteristic of two weakly coupled heterojunctions as the electron density is increased.     

The Hamiltonian formulation of regular rth-order Lagrangian field theories

A Hamiltonian formulation of regular rth-order Lagrangian field theories over an m-dimensional manifold is presented in terms of the Hamilton-Cartan formalism. It is demonstrated that a uniquely determined Cartan m-form may be associated to an rth-order Lagrangian by imposing conditions of congruence modulo a suitably defined system of contact m-forms. A geometric regularity condition is given and it is shown that, for a regular Lagrangian, the momenta defined by the Hamilton-Cartan formalism, together with the coordinates on the (r−1)st-order jet bundle, are a minimal set of local coordinates needed to express the Euler-Lagrange equations. When r is greater than one, the number of variables required is strictly less than the dimension of the (2r−1)st order jet bundle. It is shown that, in these coordinates, the Euler-Lagrange equations take the first-order Hamiltonian form given by de Donder. It is also shown that the geometrically natural generalization of the Hamilton-Jacobi procedure for finding extremals is equivalent to de Donder's Hamilton-Jacobi equation. Research supported by the Natural Sciences and Engineering Research Council.     

Study of Bohr–Mottelson with minimal length effect for Q-deformed modified Eckart potential and Bohr–Mottelson with Q-deformed quantum for three-dimensional harmonic oscillator potential

Bohr–Mottelson Hamiltonian on the γ-rigid regime for Q-deformed modified Eckart and three-dimensional harmonic oscillator potentials in the β-collective shape variable was investigated in the presence of minimal length formalism and Q-deformed of the radial momentum part. By introducing new wave function and using the Q-deformed potential concept in Bohr–Mottelson Hamiltonian in the minimal length formalism, the un-normalized wave function and energy spectra equation were obtained by using the hypergeometric method. Meanwhile, the Bohr–Mottelson Hamiltonian in the presence of the quadratic spatial deformation to the momentum in collective shape variable was investigated using transformation of a new variable such as the Schrodinger-like equation with shape invariant potential. The energy equation and un-normalized wave function were obtained using the hypergeometric method. The results showed that the Bohr–Mottelson equations with different energy potentials and different deformation forms in the radial momentum reduced to the similar Schrodinger-like equation with the modified Poschl–Teller potential.     

Exponential Transformations for the Harmonic Chain with a Molecular Defect

The exponential transformation, developed in an earlier paper [1], is applied to the Hamiltonian of a linear harmonic chain with a molecular defect. The resulting eigenvalue equation is solved for the localized frequency. A discussion of the renormalized in-band frequencies shows that in good approximation the entire Hamiltonian is diagonalized by a single transformation. This is of great advantage, since in the classical Lifshitz formalism each single frequency has to be evaluated separately. Furthermore, a simpler transformation is discussed, which is derived from an U-matrix formalism. Numerical results of the two transformations are given for a chain with 999 lattice points and compared with the exact values from the classical Lifshitz formalism.     

Ab initio ro-vibrational Hamiltonian in irreducible tensor formalism: a method for computing energy levels from potential energy surfaces for symmetric-top molecules

A theoretical approach to study ro-vibrational molecular states from a full nuclear Hamiltonian expressed in terms of normal-mode irreducible tensor operators is presented for the first time. Each term of the Hamiltonian expansion can thus be cast in the tensor form in a systematic way using the formalism of ladder operators. Pyramidal XY₃ molecules appear to be good candidates to validate this approach which allows taking advantage of the symmetry properties when doubly degenerate vibrational modes are considered. Examples of applications will be given for PH₃ where variational calculations have been carried out from our recent potential energy surface [Nikitin et al., J. Chem. Phys. 130, 244312 (2009)].     

Oscillations of charged particles in an external magnetic field about steady motion

We develop a Hamiltonian formalism that can be used to study the particle dynamics near stable equilibria. The construction of an original canonical transformation allowed us to prove the conservation of the linear momentum P₃, which permitted the expansion of the Hamiltonian about a fixed point. The definition of the rotational variable h whose Poisson algebra properties played the essential role in the diagonalization of the quadratic Hamiltonian yielding two uncoupled oscillators with definite frequencies and amplitudes. It is through applying this variable near a fixed point that come to light Heisenberg's and Harmonic Oscillator equations of motion of the particles, leading thus the association of the fixed point trajectories with arbitrary trajectories in its immediate neighborhood. The present formalism succeeded to treat the problem of free-electron laser dynamics and may be applied to similar cases. Received 20 October 2001     

The physical role of gravitational and gauge degrees of freedom in general relativity — I: Dynamical synchronization and generalized inertial effects

This is the first of a couple of papers in which the peculiar capabilities of the Hamiltonian approach to general relativity are exploited to get both new results concerning specific technical issues, and new insights about old foundational problems of the theory. The first paper includes: (1) a critical analysis of the various concepts of symmetry related to the Einstein-Hilbert Lagrangian viewpoint on the one hand, and to the Hamiltonian viewpoint, on the other. This analysis leads, in particular, to a re-interpretation of active diffeomorphisms as passive and metric-dependent dynamical symmetries of Einstein's equations, a re-interpretation which enables to disclose the (not widely known)) connection of a subgroup of them to Hamiltonian gauge transformations on-shell; (2) a re-visitation of the canonical reduction of the ADM formulation of general relativity, with particular emphasis on the geometro-dynamical effects of the gauge-fixing procedure, which amounts to the definition of a global non-inertial, space-time laboratory. This analysis discloses the peculiar dynamical nature that the traditional definition of distant simultaneity and clock-synchronization assume in general relativity, as well as the gauge relatedness of the “conventions” which generalize the classical Einstein's convention. (3) a clarification of the physical role of Dirac and gauge variables, as their being related to tidal-like and generalized inertial effects, respectively. This clarification is mainly due to the fact that, unlike the standard formulations of the equivalence principle, the Hamiltonian formalism allows to define a generalized notion of “force” in general relativity in a natural way.     

Coupling of one and two quasiparticles to a Bohr-Mottelson core in195,196,197,198Hg

A new model for coupling the motion of particles to that of a quadrupole collective core with rotations andβ andγ vibrations is proposed. The Hamiltonian describing the core is obtained by quantising the classical Hamiltonian associated with the quadrupole degrees of freedom. The inertial parameters and the deformation energy surface are determined microscopically. The spherical shell model particles interacting among themselves by pairing are coupled to the core by aλ ²-pole (λ=0, 2, 4) potential. The theory is applied to^195–198Hg. The predicted results agree very well the experimental data. A comparison of the present model to the other formalism is also given.     

Hamiltonian Dynamics for Einstein’s Action in <Emphasis Type="Italic">G</Emphasis>→0 Limit

The Hamiltonian analysis for the Einstein’s action in G→0 limit is performed. Considering the original configuration space without involve the usual ADM variables we show that the version G→0 for Einstein’s action is devoid of physical degrees of freedom. In addition, we will identify the relevant symmetries of the theory such as the extended action, the extended Hamiltonian, the gauge transformations and the algebra of the constraints. As complement part of this work, we develop the covariant canonical formalism where will be constructed a closed and gauge invariant symplectic form. In particular, using the geometric form we will obtain by means of other way the same symmetries that we found using the Hamiltonian analysis.