Perturbation theory for QED bound states

A new method is presented for deriving a systematic perturbative expansion for QED bound states, which does not rely upon solving any new or old equation. The starting point is a given nonperturbative zeroth order Green's function, obtained by a suitable “relativistic dressing” of the nonrelativistic Green's function for the Schrödinger equation with Coulomb potential, which embodies the Coulombic bound states and is known. The comparison with the complete Green's function as given by standard perturbative QED gives a perturbative kernel which is then used for the expansion of the QED Green's function in terms of the given non-perturbative zeroth order Green's function.     

A successive scattering methodology with application to two cylinders in the Fresnel region of each other

We present an efficient approach to compute the second-order scattering of an electromagnetic wave by two discrete scatterers in proximity to each other. Such a two-body system represents the simplest canonical arrangement to address near-field volume scattering phenomena in microwave remote sensing models of vegetation. Using an analytical wave-based approach, a successive scattering methodology is employed to derive the first interaction term in multiple scattering by two arbitrary scatterers in terms of their transition operators. The general formulation is applied to find the second-order bistatic scattering amplitude for a pair of finite length thin cylinders at arbitrary interaction distances using the exact Green's function. To improve computational efficiency, the solution is then specialized to the Fresnel region. These second-order bistatic scattering amplitude results are in agreement with the exact Green's function model when the scatterers are in the Fresnel region of each other. Additionally, it is demonstrated that using the far field approximation in the Fresnel region can yield significant deviations from the exact results. The Fresnel model, unlike the far field approximation, accurately predicts the scattering amplitude peak values and null locations, and is suited to fast solutions in realistic canopy simulations.     

Scattering of a scalar wave from a two-dimensional randomly rough Neumann surface

Abstract

We present a reciprocity and unitarity preserving formulation of the scattering of a scalar plane wave from a two-dimensional, randomly rough surface on which the Neumann boundary condition is satisfied. The theory is formulated on the basis of the Rayleigh hypothesis in terms of a single-particle Green's function G(q_‖|k_‖) for the surface electromagnetic waves that exist at the surface due to its roughness, where k_‖ and q_‖ are the projections on the mean scattering plane of the wave vectors of the incident and scattered waves, respectively. The specular scattering is expressed in terms of the average of this Green's function over the ensemble of realizations of the surface profile function (G(q_‖|k_‖)). The Dyson equation satisfied by (G(q_‖|k_‖)) is presented, and the properties of the solution are discussed, with particular attention to the proper self-energy in terms of which the averaged Green's function is expressed. The diffuse scattering is expressed in terms of the ensemble average of a two-particle Green's function, which is the product of two single-particle Green's functions. The Bethe-Salpeter equation satisfied by the averaged two-particle Green's function is presented, and properties of its solution are discussed. In the small roughness limit, and with the irreducible vertex function approximated by the sum of the contribution from the maximally-crossed diagrams, which represent the coherent interference between all time-reversed scattering sequences, the solution of the Bethe-Salpeter equation predicts the presence of enhanced backscattering in the angular dependence of the intensity of the waves scattered diffusely.     

Transformation of Gorkov's equation for type II superconductors into transport-like equations

The stationary behavior of type II superconductors is completely described by Gorkov's equations for a set of four Green's functions, supplemented by two self-consistency equations for gap parameterΔ(r) and vector potentialA(r). Expanding all quantities as usual at the Fermi surface and averaging over impurity positions, this set of equations is transformed into a simpler set for integrated Green's functions (which still contain much more information than is needed in most cases). The resulting equations, when linearized, yield essentially Lüders' transport equation for de Gennes' correlation function. The full equations contain all the known results and provide a promising starting point for numerical calculations. The thermodynamic potential is constructed as a functional of the integrated Green's functions and the mean fieldsΔ andA and a variational principle is formulated which uses this functional. Finally, paramagnetic scatterers are included (in Born approximation) as an example for possible generalizations of the new equations.     

A boundary integral method to compute Green's functions for the radiative transport equation

The Green's function for the time-independent radiative transport equation in the whole space can be computed as an expansion in plane wave solutions. Plane wave solutions are a general class of solutions for the radiative transport equation. Because plane wave solutions are not known analytically in general, we calculate them numerically using the discrete ordinate method. We use the whole space Green's function to derive boundary integral equations. Through the solution of the boundary integral equations, we compute the Green's function for bounded domains. In particular we compute the Green's function for the half space, the slab, and the two-layered half space. The boundary conditions used here are in their most general form. Hence, this theory can be applied to boundaries with any kind of reflection and transmission law.     

Green's electron function in a quantized plane wave field

It is shown that the Green's function of an electron that interacts with a quantized plane wave can be expressed in terms of the corresponding Green's function of a scalar particle. By using the known expression for the Green's function of a scalar particle, an integral representation is found with respect to the intrinsic time for the Green's electron function in a quantized plane wave of arbitrary form.     

Exact Green's function for rectangular potentials and its application to quasi-bound states

In this work we calculate the exact Green's function for arbitrary rectangular potentials. Specifically we focus on Green's function for rectangular quantum wells enlarging the knowledge of exact solutions for Green's functions and also generalizing and resuming results in the literature. The exact formula has the form of a sum over paths and always can be cast into a closed analytic expression. From the poles and residues of the Green's function the bound states eigenenergies and eigenfunctions with the correct normalization constant are obtained. In order to show the versatility of the method, an application of the Green's function approach to extract information of quasi-bound states in rectangular barriers, where the standard analysis of quantum amplitudes fail, is presented.     

Electromagnetic volume scattering in the half-space: a moment-method analysis without Sommerfeld integrals or their approximations

Abstract

Traditionally, in moment-method analyses of electromagnetic scattering, the elements of the impedance matrix are calculated as convolutions of the basis elements with the appropriate dyadic Green's function. However, for scattering in the half-space, the vertical and azimuthal copolar terms of the Green's function require evaluation of Sommerfeld integrals which are computationally burdensome. In this paper, it is shown that, in populating the impedance matrix for the half-space problem, evaluation of Sommerfeld integrals is, in fact, not necessary. For monochromatic excitation, the plane-wave expansion of the scattered field constitutes a Fourier transform, in the horizontal plane, of a vector spectral function. This vector function results from the convolution, in the vertical dimension, of the respective angular spectra of the Green's function and the equivalent current. On application of the moment method, through the Weyl identity, the impedance-matrix elements corresponding to the singular terms of the Green's function are convolutions in the horizontal plane of spherical potentials, and Fourier transforms of scalar spectral functions. These scalar functions are derived from the basis elements and, with a judicious choice of basis, they are well behaved and of compact support, and consequently their Fourier transforms can be computed as FFTs.     

Theory of the free full-circle arc

A theoretical investigation of the full-circle arc located between two planes is presented. The circular arc shape is due to an applied magnetic field. The basic equations for conservations of mass, momentum, energy, and charge, as well as Maxwell's equations and the equation of state lead to a coupled set of partial differential equations. By means of Green's formula, this set is transformed into a set of integral equations. Using the analytically known Green's function, the system may be solved by an iteration procedure. For a simplified arc model, the quantities of interest are computed: The temperature distribution, the mass flow field, and the external magnetic field necessary to maintain this arc configuration.     

Slave bosons in radial gauge: A bridge between path integral and Hamiltonian language

We establish a correspondence between the resummation of world lines and the diagonalization of the Hamiltonian for a strongly correlated electronic system. For this purpose, we analyze the functional integrals for the partition function and the correlation functions invoking a slave boson representation in the radial gauge. We show in the spinless case that the Green's function of the physical electron and the projected Green's function of the pseudofermion coincide. Correlation and Green's functions in the spinful case involve a complex entanglement of the world lines which, however, can be obtained through a strikingly simple extension of the spinless scheme. As a toy model we investigate the two-site cluster of the single impurity Anderson model which yields analytical results. All expectation values and dynamical correlation functions are obtained from the exact calculation of the relevant functional integrals. The hole density, the hole auto-correlation function and the Green's function are computed, and a comparison between spinless and spin 1/2 systems provides insight into the role of the radial slave boson field. In particular, the exact expectation value of the radial slave boson field is finite in both cases, and it is not related to a Bose condensate.     

An application of second order approximations for the Green's function

In a previous paper there was developed an approximation theory for the Green's functions which gives approximations consistent with the conservation laws of the Hamiltonian. We have chosen a concept of reduction which is different and which leads to other approximations for the particle-hole Green's function than the usual concept of reduction. The particle-hole Green's function is the function $$\langle \Psi _0 |\tau \{ a_1^ + (0)a_2 (0)a_3^ + (t)a_4 (t)\} |\Psi _0 \rangle $$ where ¦Ψ_o〉 is the real ground state and τ stands for the time ordered products of the operators. Now, in this paper we want to give an example of an application of this theory. We present a second order approximation since the first order approximation is the well known Random Phase Approximation.     

Harmonic Green's functions for flexural waves in semi-infinite plates with arbitrary boundary conditions and high-frequency approximation for convex polygonal plates

This paper provides a method for obtaining the harmonic Green's function for flexural waves in semi-infinite plates with arbitrary boundary conditions and a high frequency approximation of the Green's function in the case of convex polygonal plates, by using a generalised image source method. The classical image source method consists in describing the response of a point-driven polygonal plate as a superposition of contributions from the original source and virtual sources located outside of the plate, which represent successive reflections on the boundaries. The proposed approach extends the image source method to plates including boundaries that induce coupling between propagating and evanescent components of the field and on which reflection depends on the angle of incidence. This is achieved by writing the original source as a Fourier transform representing a continuous sum of propagating and evanescent plane waves incident on the boundaries. Thus, the image source contributions arise as continuous sums of reflected plane waves. For semi-infinite plates, the exact Green's function is obtained for an arbitrary set of boundary conditions. For polygonal plates, a high-frequency approximation of the Green's function is obtained by neglecting evanescent waves for the second and subsequent reflections on the edges. The method is compared to exact and finite element solutions and evaluated in terms of its frequency range of applicability.     

Approximation of Green's functions

Special symmetries of the Green's functions of a non-relativistic many fermion-system and conservation laws, expressible by hermitian generators, are formulated as relations for a Green's function operator. Approximations for the Green's functions, defined as partial summations of the perturbation expansion, and consistent with the symmetries and conservation laws are presented.     

Many‐body Green's function theory for thin ferromagnetic films: exact treatment of the single‐ion anisotropy

A theory for the magnetization of ferromagnetic films is formulated within the framework of many‐body Green's function theory which considers all components of the magnetization. The model Hamiltonian includes a Heisenberg term, an external magnetic field, a second‐ and fourth‐order uniaxial single‐ion anisotropy, and the magnetic dipole‐dipole coupling. The single‐ion anisotropy terms can be treated exactlyby introducing higher‐order Green's functions and subsequently taking advantage of relations between products of spin operators which leads to an automatic closure of the hierarchy of the equations of motion for the Green's functions with respect to the anisotropy terms. This is an improvement on the method of our previous work, which treated the corresponding terms only approximately by decoupling them at the level of the lowest‐order Green's functions. RPA‐like approximations are used to decouple the exchange interaction terms in both the low‐order and higher‐order Green's functions. As a first numerical example we apply the theory to a monolayer for spin S = 1 in order to demonstrate the superiority of the present treatment of the anisotropy terms over the previous approximate decouplings.     

On the perturbation series for radiative effects

We present a generalization of the recently developed first-order perturbation approximation for the computation of radiative effects such as layer heating rates and surface fluxes. The basis for this generalization is Dyson's equation for Green's function or the inverse transport operation.     

Properties of advanced coupled-cluster Green's function

ABSTRACT

We present an extension of the analysis previously applied to the retarded part of the coupled cluster (CC) Green's function to its advanced part. In analogy to our earlier studies for the retarded part, we demonstrate that the advanced CC Green's function is expressed in terms of connected diagrams only, which is a direct consequence of algebraic form of equations satisfied by CC amplitudes. We also demonstrate that ω-derivatives of the advanced CC Green's function can be calculated analytically and can be expressed in terms of connected diagrams only. We analyse the structure of connected diagrams and the role of intermediate operators which satisfy electron affinity equation-of-motion CC-type conditions.     

Zur Greenschen Funktion optisch anisotroper Medien

Green's Function of Optically Anisotropic Media The time-fouriertransformed dyadic Green's function is calculated in the far-field-approximation for optically anisotropic media. To this end the time- and space-fouriertransformed Green's tensor-function is represented by dyadic products of the eigenvectors of the homogeneous Fresnel's equation, and the transformation back into space is done in the asymptotic limit by the stationary phase integration method. As an application the radiation field of an electric dipole in an optically anisotropic medium is evaluated. All results are discussed in the case of uniaxial crystals.     

Localized magnetic excitations of coupled impurities in a transverse Ising ferromagnet

A Green's function formalism is used to calculate the spectrum of excitations of two neighboring impurities implanted in a semi-infinite ferromagnetic. The equations of motion for the Green's functions are determined in the framework of the Ising model in a transverse field and results are given for the effect of the exchange coupling, position and orientation of the impurities on the spectra of localized spin wave modes.