The classical limit of temperature Green's function expansions

Rules are obtained for calculating the classical limit of Green's function diagrammatic expansions. The classical cluster expansion is derived by calculating the classical limit of the exact Green's function. Other operators of interest in linear response theory may be calculated in the classical limit. The retarded real-time spin density correlation function, proportional to the magnetic susceptibility, is shown to be exactly proportional to the density in this limit. The relation of this work to other approaches is discussed.     

Asymptotic expansions in limits of large momenta and masses

Asymptotic expansions of renormalized Feynman amplitudes in limits of large momenta and/or masses are proved. The corresponding asymptotic operator expansions for theS-matrix, composite operators and their time-ordered products are presented. Coefficient functions of these expansions are homogeneous within a regularization of dimensional or analytic type. Furthermore, they are explicitly expressed in terms of renormalized Feynman amplitudes (at the diagrammatic level) and certain Green functions (at the operator level).     

A general dispersion relation for spin-phonon excitations in paramagnets

The advantage of using standard basis operators in evaluating the semi-invariants occurring in the diagrammatic theory of spin-phonon modes is discussed. The dispersion relation of the coupled-mode excitations is derived for paramagnetic ions of arbitrary spin having a general static spin hamiltonian.     

Power Series Expansions for Vacuum Projection Operator of Paraparticles

Some power series expansions for the vacuum projection operator of paraboson and parafermion are derived. A recursion method is essential to these derivations of the power series expansion in terms of creation and annihilation operators of the paraparticles.     

Thermodynamic calculations in relativistic finite-temperature quantum field theories

A Minkowski space formalism of finite-temperature quantum field theory is used to compute static thermodynamic quantities in the one- and two-loop approximation in an elegant and straightforward way using a generalization of Weinberg's tadpole method of calculating effective potentials. Systematic diagrammatic techniques for low- and high-temperature expansions are developed. Renormalizability by zero-temperature symmetric counterterms is proven for all orders in the loop expansion and demonstrated explicitly to two loops. Many useful computational techniques applicable to general finite temperature calculations are explained.     

Higher-order approximations for the particle-particle propagator

A new type of approximations for many-body Green's functions proposed recently is applied to the particle-particle (pp) propagator for anN-particle fermion system. The new approach which is referred to as the algebraic diagrammatic construction (ADC) is based on an exact resummation of the perturbation series for the pp-propagator in terms of a simple algebraic form introducing energy-independent effective interaction matrix elements and transition amplitudes. These effective quantities are represented by perturbation expansions and can be determined consistently through a given ordern of perturbation theory by comparing the algebraic form with the diagrammatic perturbation series of the pp-propagator through ordern. By this procedure one obtaines a systematic set of approximation schemes (ADC(n)) that represent infinite partial summations for the pp-propagator being complete throughnth order of perturbation theory. The explicit ADC equations forn=1 and 2 are presented and discussed. Comparison is made with the particle-particle random phase approximation (RPA). It is demonstrated that the second-order ADC scheme constitutes an essential step beyond the RPA which is consistent only through first order.     

Operator product expansions in conformally covariant quantum field theory

Operator products in quantum field theory on two-dimensional Minkowski space are expanded into a series of local operators by means of the tensor product decomposition theorem for representations of the conformal group. The Thirring model is used as an explicit example. Two types of expansions result. If the operator product acts on the vacuum state, we obtain strictly covariant expansions. In general however, each term in the expansion is only semicovariant.     

Regularization of diagrammatic series with zero convergence radius

The divergence of perturbative expansions which occurs for the vast majority of macroscopic systems and follows from Dyson's collapse argument prevents the direct use of Feynman's diagrammatic technique for controllable studies of strongly interacting systems. We show how the problem of divergence can be solved by replacing the original model with a convergent sequence of successive approximations which have a convergent perturbative series while maintaining the diagrammatic structure. As an instructive model, we consider the zero-dimensional |ψ|? theory.     

A multi-reference exponential expansion for the electronic wave function

The expansion structure of a CI vector as combination of excitations from a model-space reference determinant is investigated. It is shown that between the linear and the exponential expansions there is a relation which is similar to the single-reference case, if the internal excitations are adsorbed into the reference vector. Moreover, expansions with respect to different determinants are related by a set of linear equations. By using these two properties, a State-Specific Coupled-Cluster formalism is proposed. Received 2 November 2000 / Received in final form 1st March 2002 Published online 28 June 2002     

Exact series expansions, recurrence relations, properties and integrals of the generalized exponential integral functions

Generalized exponential integral functions (GEIF) are encountered in multi-dimensional thermal radiative transfer problems in the integral equation kernels. Several series expansions for the first-order generalized exponential integral function, along with a series expansion for the general nth order GEIF, are derived. The convergence issues of these series expansions are investigated numerically as well as theoretically, and a recurrence relation which does not require derivatives of the GEIF is developed. The exact series expansions of the two dimensional cylindrical and/or two-dimensional planar integral kernels as well as their spatial moments have been explicitly derived and compared with numerical values.     

Contractions of expansions of fermion operators

General formulae relating the contractions of expansions of fermion operators in the finite-dimensional case to the expansions of contractions of these operators are derived and discussed.     

Convergence of Cluster and Virial expansions for Repulsive Classical Gases

We study the convergence of cluster and virial expansions for systems of particles subject to positive two-body interactions. Our results strengthen and generalize existing lower bounds on the radii of convergence and on the value of the pressure. Our treatment of the cluster coefficients is based on expressing the truncated weights in terms of trees and partition schemes, and generalize to soft repulsions previous approaches for models with hard exclusions. Our main theorem holds in a very general framework that does not require translation invariance and is applicable to models in general measure spaces. Our virial results, stated only for homogeneous single-space systems, rely on an approach due to Ramawadh and Tate. The virial coefficients are computed using Lagrange inversion techniques but only at the level of formal power series, thereby yielding diagrammatic expressions in terms of trees, rather than the doubly connected diagrams traditionally used. We obtain a new criterion that strengthens, for repulsive interactions, the best criterion previously available (proposed by Groeneveld and proven by Ramawadh and Tate). We illustrate our results with a few applications showing noticeable improvements in the lower bound of convergence radii.

     

Semi-microscopic optical potential calculation by the nuclear matter approach

In this paper the semi-microscopic nuclear matter approach has been introduced to calculate the microscopic optical potential. The first- and second-order mass operators in asymmetric nuclear matter have been derived with Skyrme effective interactions and the real and imaginary parts of the optical potential for finite nuclei have been obtained by applying a local density approximation. Five Skyrme interactions II–VI have been used and compared with the experimental data to determine how well these Skyrme interaction function for our purposes. Our results obtained in this simple way are to some extent comparable with those obtained with the “nuclear matter” and “nuclear structure” approaches without adjusting the parameters of the Skyrme interactions.     

Kinetic theory of hard spheres

Kinetic equations for the hard-sphere system are derived by diagrammatic techniques. A linear equation is obtained for the one-particle-one particle equilibrium time correlation function and a nonlinear equation for the one-particle distribution function in nonequilibrium. Both equations are nonlocal, noninstantaneous, and extremely complicated. They are valid for general density, since statistical correlations are taken into account systematically. This method derives several known and new results from a unified point of view. Simple approximations lead to the Boltzmann equation for low densities and to a modified form of the Enskog equation for higher densities.     

Magnetization and spin-spin energy diffusion in the XY model: a diagrammatic approach

It is shown that the diagrammatic cluster expansion technique for equilibrium averages of spin operators may be straightforwardly extended to the calculation of time-dependent correlation functions of spin operators. We use this technique to calculate exactly the first two non-vanishing moments of the spin-spin and energy-energy correlation functions of the XY model with arbitrary couplings, in the long-wavelength, infinite temperature limit appropriate for spin diffusion. These moments are then used to estimate the magnetization and spin-spin energy diffusion coefficients of the model using a phenomenological theory of Redfield. Qualitative agreement is obtained with recent experiments measuring diffusion of dipolar energy in calcium fluoride.     

An alternative analytic treatment of the Gutzwiller ansatz

An alternative analytic approach is formulated for treating Gutzwiller's variational ansatz. In contrast to the diagrammatic treatment recently published by Metzner and Vollhardt, our approach is purely algebraic and mainly based on expansions in terms of cluster integrals. The present theory is carried out for arbitrary dimension and band filling, but is applied here only to the Hubbard interaction term. In one dimension we find exact agreement between our results and those obtained by Metzner and Vollhardt.     

Diagrammatic analysis of correlations in polymer fluids: Cluster diagrams via Edwards’ field theory

Edwards’ functional integral approach to the statistical mechanics of polymer liquids is amenable to a diagrammatic analysis in which free energies and correlation functions are expanded as infinite sums of Feynman diagrams. This analysis is shown to lead naturally to a perturbative cluster expansion that is closely related to the Mayer cluster expansion developed for molecular liquids by Chandler and co-workers. Expansion of the functional integral representation of the grand-canonical partition function yields a perturbation theory in which all quantities of interest are expressed as functionals of a monomer-monomer pair potential, as functionals of intramolecular correlation functions of non-interacting molecules, and as functions of molecular activities. In different variants of the theory, the pair potential may be either a bare or a screened potential. A series of topological reductions yields a renormalized diagrammatic expansion in which collective correlation functions are instead expressed diagrammatically as functionals of the true single-molecule correlation functions in the interacting fluid, and as functions of molecular number density. Similar renormalized expansions are also obtained for a collective Ornstein-Zernicke direct correlation function, and for intramolecular correlation functions. A concise discussion is given of the corresponding Mayer cluster expansion, and of the relationship between the Mayer and perturbative cluster expansions for liquids of flexible molecules. The application of the perturbative cluster expansion to coarse-grained models of dense multi-component polymer liquids is discussed, and a justification is given for the use of a loop expansion. As an example, the formalism is used to derive a new expression for the wave-number dependent direct correlation function and recover known expressions for the intramolecular two-point correlation function to first-order in a renormalized loop expansion for coarse-grained models of binary homopolymer blends and diblock copolymer melts.     

Translationally invariant calculations of form factors, nucleon densities and momentum distributions for finite nuclei with short-range correlations included

Relying upon our previous treatment of the density matrices for nuclei (in general, nonrelativistic self-bound finite systems) we are studying a combined effect of center-of-mass motion and short-range nucleon-nucleon correlations on the nucleon density and momentum distributions in light nuclei (⁴He and ¹⁶O). Their intrinsic ground-state wave functions are constructed in the so-called fixed center-of-mass approximation, starting with mean-field Slater determinants modified by some correlator (e.g., after Jastrow or Villars). We develop the formalism based upon the Cartesian or boson representation, in which the coordinate and momentum operators are linear combinations of the creation and annihilation operators for oscillatory quanta in the three different space directions, and get the own “Tassie-Barker” factors for each distribution and point out other model-independent results. After this separation of the center-of-mass motion effects we propose additional analytic means in order to simplify the subsequent calculations (e.g., within the Jastrow approach or the unitary correlation operator method). The charge form factors, densities and momentum distributions of ⁴He and ¹⁶O evaluated by using the well-known cluster expansions are compared with data, our exact (numerical) results and microscopic calculations.