Hamiltonian Perturbation Research Articles

The method of minimized iterations in multi-reference (effective hamiltonian) perturbation theory

A method is presented for performing partial summations in multi-reference perturbation theory that is stable in the presence of intruder states. The method is invariant under translation and thus can be used to evaluate the folded terms as well. Several numerical examples are presented which demonstrate the efficiency of the method.

An INDO MO Study on the Ground State and the Cationic States of Cyclopentadienyl Nickel Nitrosyl

The electronic structure of cyclopentadienyl nickel nitrosyl (1) in the ground state as well as the cationic states of 1 are investigated by means of a semiempirical INDO Hamiltonian and many body perturbation theory. It is demonstrated that the nature of the NiNO coupling is largely covalent while the interaction between the 3d center and the cyclopentadienyl ligand is predominantly of ionic type. The ground state MO sequence of the Ni 3d orbitals is 4e2(3dx²-y²/3dxy) below 7e1 (3dXz/3dyz) and 15a1(3dz 2). The sequence of the ionization potentials is 8e1 (Cp - π) < 15a1<4e2<7e1. The ionization energies have been determined by means of the Green’s function formalism; the self-energy part has been calculated by a second order and a renormalized approximation. Both procedures predict the correct sequence of ionization events.

Classical Hamiltonian perturbation theory without secular terms or small denominators

We consider perturbation theory in ϵ for the classical Hamiltonian H = H 0 + ϵH 1, where H 0 gives rise to a known motion and ϵ is small. First we demonstrate how the usual secular terms and small denominators arise from a straightforward expansion in ϵ and argue that they are artifacts of the method. Then we present an alternative perturbation theory based on an analysis of the operator ( s − L) −1, where s is a complex number and L is the Liouville operator corresponding to H. This perturbation series contains neither secular terms nor small denominators. In the case of almost multiply periodic systems we show, to lowest non-trivial order in ϵ, how our series reproduces the standard results both in the resonant and nonresonant regions — all in one analytic formula. As a final exercise we demonstrate that energy is conserved at order ϵ n+1 when the accuracy of the theory is order ϵ n .

Oscillation centres and mode coupling in non-uniform Vlasov plasma

The general coupling coefficient for three electromagnetic linear modes of an inhomogeneous and relativistic plasma is derived from the oscillation-centre viewpoint. A concise and manifestly symmetric formula is obtained; it is cast in terms of Poisson brackets of the single-particle perturbation Hamiltonian and its convective time-integral along unperturbed orbits. The simplicity of the compact expression obtained is shown to lead to a new insight into the essence of three-wave coupling and of the Manley–Rowe relations governing such interactions. Thus, the interaction Hamiltonian of the three waves is identified as simply the trilinear contribution to the single-particle (new) Hamiltonian, summed over all non-resonant particles. The relation between this work and the Lie-transform approach to Hamiltonian perturbation theory is discussed.

Improved strong-coupling expansions and matrix Padé approximants for lattice theories

A generalization of Hamiltonian perturbation theory is presented. The method solves a low-energy sector of the theory exactly while systematically accounting for higher-energy states perturbatively. It produces an expansion for the mass matrix of the low-energy sector. In applications to lattice theories, the mass matrix can be extrapolated to the continuum limit using matrix Pad\'e approximants. The method is illustrated for a lattice potential model and fast convergence to the continuum limit is verified numerically.

A Method of Transfer Maps for Linear and Nonlinear Beam Elements

Transfer matrix methods are widely used to calculate properties of particle orbits as they pass through linear beam elements such as drif t spaces, bending magnets, and quadrupoles. A new method of has been developed to also include nonlinear transformations that result from nonlinear beam elements such as sextupoles, octupoles, etc. The method of transfer maps therefore provides a complete theory of beam transport through both linear and nonlinear elements. In particular, it is possible to use transfer maps in the context of circular machines to study tune shifts, structure resonances, stop band widths, emittance growth rates, etc. Consequently, the method of transfer maps provides an alternative to the method of Hamiltonian perturbation theory usually employed for this purpose.

Lie-Operator Approach to Mode Coupling in Nonuniform Plasma

Hamiltonian perturbation theory based on recent Lie transform techniques is applied to the Hamiltonian of a single particle in nonuniform Vlasov plasma. A simple relation is derived between the field-plasma interaction energy and the transformed single-particle Hamiltonian. This relation implies as special cases a general formula for ponderomotive force in terms of the linear Vlasov susceptibility, and a symmetric Poisson-bracket formula for the general three-mode coupling coefficient.

SU(2)-flavor Schwinger model on the lattice

Hamiltonian lattice perturbation methods are used to study the (1 + 1)-dimensional, SU(2)-flavor Abelian gauge model. For a model with coupling constant $g$ and fermion mass $m$, two distinct vacuum regions characterized by the magnitude of $\frac{m}{g}$ are found. Expansions about the Ising-type vacuum corresponding to large $\frac{m}{g}$ are carried out to order $\frac{1}{{g}^{8}{a}^{8}}$ ($a=\mathrm{lattice}\mathrm{spacing}$) and improved using Pad\'e approximants. The results compare favorably qualitatively and quantitatively with recent studies of the weak-coupling limit of the continuum theory. Expansions to order $\frac{1}{{g}^{4}{a}^{4}}$ about the Heisenberg antiferromagnetlike vacuum corresponding to small $\frac{m}{g}$ agree qualitatively with known results for the strong-coupling limit of the continuum theory.

The swinging spring ? Approximate analyses for low and very high energy, II

The two families of periodic solutions emanating from the lower equilibrium of a planar spring-pendulum system are analyzed both at and near resonance. Hamiltonian perturbation theory is used to obtain approximate formulas for the characteristic and the period of the motion. As the energy is increased to very high levels, a circulatory and an almost rectilinear periodic motion persist. Expressions for the trace of a stability matrix are determined for both these solutions. Comparisons are made throughout with accurate results from numerical integration.

Infinite-Momentum Limit of a Spinor Field Theory. II

A self-contained theory of Dirac fermions interacting with (pseudo-) scalar bosons at infinite momentum has been constructed. The fields are coupled with a $\overline{\ensuremath{\psi}}\ensuremath{\Gamma}\ensuremath{\psi}\ensuremath{\varphi}$ interaction where $\ensuremath{\Gamma}=1$ or ${\ensuremath{\gamma}}^{5}$, depending on the parity of the boson. The theory is designed to give the correct relativistic dynamics when the space of intermediate states is restricted to those states which can be interpreted as the infinite-momentum limit of sets of particles with finite momentum, that is, those states which correspond to the particle configurations observed in nature. The intermediate states which were eliminated are those containing particles moving away from the infinite-momentum observer. The dynamical effect of these states is incorporated into a counterterm in the Hamiltonian. The effect of this counterterm is to produce a set of four-point vertices in the old-fashioned Hamiltonian perturbation expansion. The perturbation-theory rules for this model were described in an earlier paper. A general formalism for handling interactions at infinite momentum is presented here. It is used to derive the counterterm for our model theory. A complete set of generators for the Poincar\'e group in a basis appropriate to the infinite-momentum limit is demonstrated. These generators satisfy the commutation algebra of the Poincar\'e group, proving the relativistic invariance of the theory and its internal consistency.

A new approach to renormalization in quantum field theory

A new representation of the creation and annihilation operators is built, which realizes the momentum cut-off of the causal functions of the theory and establishes a connection between the regularizations in both the Hamiltonian method and perturbation theory.

An effective Hamiltonian and time-independent perturbation theory

Using a formalism set up by Bloch an effective Hamiltonian is introduced which, when expanded in powers of a perturbation V, generalizes the Rayleigh-Schr?dinger perturbation theory. This effective Hamiltonian provides a simple way of handling degeneracies and the orthonormalization of the perturbed wave functions. As an illustration the spin Hamiltonian introduced by Pryce in paramagnetic resonance is derived up to third-order terms, and the electron-lattice interaction is discussed showing that the Born-Oppenheimer approximation follows from firstorder terms in the case of non-degenerate electronic ground state.

Modified long-period behavior due to tesseral harmonics.

Von Zeipel method and Hamiltonian perturbation mechanics used for orbits at resonance with tesseral harmonics of geopotential

Renormalization and Statistical Mechanics in Many-Particle Systems. I. Hamiltonian Perturbation Method

Renormalization and Statistical Mechanics in Many-Particle Systems. I. Hamiltonian Perturbation Method