Nonperturbative Approximation Research Articles

Low-energy electron scattering from CO. III. Analytic method for outer region in frame-transformation theory

For pt.III see Phys. Rev. A, vol.16, p.80-108 (1977). A non-perturbative approximation, suitable for electron scattering from both weakly and strongly polar molecular, has been proposed. The coupling in the scattering equations due to the presence of the non-central electron-dipole interaction is removed by merely shifting the relative positions of the rotational levels but without either making them degenerate or neglecting their energies. This method has been used to simplify application of the frame-transformation theory (FT) to electron scattering from polar systems. The problem in the outer region of the FT can now be solved without numerical integration and the solution written in a form suited for an R-matrix treatment of the molecular core. The proposed procedure achieves very high level of efficiency and economy of calculation without much sacrifice of accuracy. This method has been applied to electron scattering from weakly polar carbon monoxide. The integrated cross sections for various rotational transitions have been calculated from 0.005 to 10.0 eV and compared with previously obtained accurate results. The usefulness of this method for electron scattering from strongly polar neutral and ionic systems has been discussed.

Unitarisation of the eikonal-Born series method for electron-atom and positron-atom collisions

A systematic study is made of the unitarisation method which the authors recently proposed for the eikonal-Born series approximation, in the case of fast electron-atom and positron-atom collisions. The authors' unitarisation procedure is based on a generalisation of the potential scattering eikonal expansion of Wallace to the multiparticle case, suitably modified to take the long-range polarisation effects and absorption effects at small angles into account correctly. They also develop a new non-perturbative approximation for the exchange scattering amplitude, which is appropriate to use with the authors' unitarised EBS direct scattering amplitude for electron-atom collisions. The authors' then apply their unitarised EBS method to the elastic scattering of electrons and positrons by atomic hydrogen in the energy range 100-400 eV. A detailed comparison is made with other related theoretical methods and with recent absolute measurements of (e--H) elastic differential cross sections. Total (integrated) elastic and total (complete) cross sections for both electron and positron impact are also discussed.

The scattering of spin-1 particles by quantum gravitational bubbles

Quantum gravitational bubbles may be used to obtain a non-perturbative approximation to the path integral of quantum gravity. There are three basic types of bubblesCP 2,S 2 ×S 2, andK3, and in this paper the propagation of elementary spin-1 particles inCP 2 is investigated. To date information about the propagation of particles other than scalars has been obtained by making approximations to the basic bubble types. The work presented here represents the first exact calculation. It is found that spin-1 particles scatter very strongly, particularly at low energies, which is at odds with both physical observation and the earlier work on this subject. Possible explanations for this discrepancy are offered.

Unitarisation of the eikonal-Born series method for electron- and positron-atom collisions

A systematic study is made of the unitarisation method which the authors recently proposed for the eikonal-Born series approximation, in the case of fast electron-atom and positron-atom collisions. The authors' unitarisation procedure is based on a generalisation of the potential scattering eikonal expansion of Wallace to the multiparticle case, suitably modified to take the long-range polarisation effects and absorption effects at small angles into account correctly. They also develop a new non-perturbative approximation for the exchange scattering amplitude, which is appropriate to use with the authors' unitarised EBS direct scattering amplitude for electron-atom collisions. The authors' then apply their unitarised EBS method to the elastic scattering of electrons and positrons by atomic hydrogen in the energy range 100-400 eV. A detailed comparison is made with other related theoretical methods and with recent absolute measurements of (e--H) elastic differential cross sections. Total (integrated) elastic and total (complete) cross sections for both electron and positron impact are also discussed.

Comparison of nonperturbative variational approximations to the scattering amplitude for a Yukawa potential

Variational methods for calculating approximations to the scattering amplitude of a spinless particle from a Yukawa potential are compared. These arise from the variation of functionals that depend on trial forms of the Heitler operator K or the transition operator T. The trial forms are nonperturbative, and the variational principle is applied in the free particle and partial wave basis to obtain independent approximations. In addition, no artificial separability constraint is imposed on the momentum space representation of either the trial T or K. The full free particle Green’s function is utilized without basis set truncation in all of the calculations. The different approximations provide a choice of technique in the event that one or the other variational method is unstable. The results at two representative energies are in good agreement with an exact calculation.

Nonperturbative calculation of the 2-photon transverse resonance in radio-frequency spectroscopy

A new analytic nonperturbative approximation procedure is used to calculate the frequency shift of the first transverse resonance in the 2-level atomic system. The accuracy of our result as tested for the vanishing constant magnetic field H 0 against the known exact result is within 1 ppm.

Overlapping-divergence-free skeleton expansion in non-Abelian gauge theories

We derive a skeleton expansion in Yang-Mills theory, which is completely free of overlapping divergences. This skeleton expansion provides a generalization of the Schwinger-Dyson equations of quantum electrodynamics to non-Abelian gauge theories. It yields a simple procedure for calculating renormalized perturbation theory integrals without using regulators and could serve as a starting point for nonperturbative approximations. In carrying out this analysis, we show that the Yang-Mills vector self-energy function and the Yang-Mills vector three-point function are uniquely determined by the Ward-Takahashi identity in terms of the other three basic vertex functions.

Rigorous study of the gap equation for an inhomogeneous superconducting state nearTc

A rigorous analytic study of the self-consistent gap equation (symobolically $\ensuremath{\Delta}={\mathcal{F}}_{T}[\ensuremath{\Delta}]$), for an inhomogeneous superconducting state, is presented in the Bogoliubov formulation. The gap function $\ensuremath{\Delta}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ is taken to simulate a planar normal-superconducting phase boundary: $\ensuremath{\Delta}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})={\ensuremath{\Delta}}_{\ensuremath{\infty}}tanh(\frac{\ensuremath{\alpha}{\ensuremath{\Delta}}_{\ensuremath{\infty}}z}{{v}_{F}})\ensuremath{\bigominus}(z)$, where ${\ensuremath{\Delta}}_{\ensuremath{\infty}}(T)$ is the equilibrium gap, ${v}_{F}$ is the Fermi velocity, and $\ensuremath{\bigominus}(z)$ is a unit step function. First a special space integral of the gap equation $\ensuremath{\propto}\ensuremath{\int}{0+}^{\ensuremath{\infty}}({\mathcal{F}}_{T}\ensuremath{-}\ensuremath{\Delta})(\frac{d\ensuremath{\Delta}}{\mathrm{dz}})dz$ is evaluated essentially exactly, except for a nonperturbative WKBJ approximation used in solving the Bogoliubov-de Gennes equations. It is then expanded near the transition temperature ${T}_{c}$ in power of ${\ensuremath{\Delta}}_{\ensuremath{\infty}}\ensuremath{\propto}{(1\ensuremath{-}\frac{T}{{T}_{c}})}^{\frac{1}{2}}$, demonstrating an exact cancellation of a subseries of "anomalous-order" terms. The leading surviving term is found to agree in order, but not in magnitude, with the Ginzburg-Landau-Gor'kov (GLG) approximation. The discrepancy is found to be linked to the slope discontinuity in our chosen $\ensuremath{\Delta}$. A contour-integral technique in a complex-energy plane is then devised to evaluate the local value of ${\mathcal{F}}_{T}\ensuremath{-}\ensuremath{\Delta}$ exactly. Our result reveals that near ${T}_{c}$ this method can reproduce the GLG result essentially everywhere, except within a BCS coherence length [not $\ensuremath{\xi}(T)$!] from a singularity in $\ensuremath{\Delta}$, where ${\mathcal{F}}_{T}\ensuremath{-}\ensuremath{\Delta}$ can have a singular contribution with an "anomalous" local magnitude, not expected from the GLG approach. This anomalous term precisely accounts for the discrepancy found in the special integral of the gap equation as mentioned above, and likely explains the ultimate origin of the anomalous terms found in the free energy of an isolated vortex line by Cleary.

Nonperturbative path-integral approximations

A nonperturbative path-integral approximation scheme is introduced heuristically, and then interpreted from the point of view of Feynman path integrals. The utility and the properties of the technique are explored by application to three widely different problems. When applied to the problem of a bound charged particle in an electromagnetic field, the method yields the momentum-translation approximation plus corrections. For high-energy scattering, the technique gives immediately the standard eikonal results, and also suggests corrections. For a charged free particle in an electromagnetic field, the method in simplest form leads to an approximation to the exact Volkov solution, and correction terms yield precisely the Volkov solution.

Linear response theory of the driving forces for electromigration

Abstract Using linear response theory, we have obtained a response function for the force exerted on an impurity in the presence of an electric field. The response function measures the total force including both the local screening of the applied field and the dynamic polarization due to the “electron wind” (the electron-drag effect). Because of low frequency divergences, the response function cannot be evaluated by perturbation theory. We propose a non-perturbative approximation scheme whereby the force response in each order of perturbation theory is taken to be proportional to the electronic current in the next lower order. The constant of proportionality is calculated to lowest order in the density or strength of the scatterers. The summation of these terms for the force response yields expressions for the electron-drag force which, in the appropriate models, agree with the formulas of Huntington, Fiks, Bosvieux and Friedel, and Sorbello. The advantage of the force response function is that it permits calculation for cases which appear difficult to treat with the earlier theories. As an illustration, we analyze the case of surface electromigration within the linear response model.

Multiplicative stochastic processes, Fokker-Planck equations, and a possible dynamical mechanism for critical behavior

A derivation of the Fokker-Planck equations for additive stochastic processes is given which involves treating the continuity equation in the configuration space representation of the additive stochastic process as a multiplicative stochastic process. The average of the continuity equation becomes the Fokker-Planck equation. A presentation of the ``multiplicative stochastic, Markov approximation'' follows. This approximation is applied to the analysis of the dynamics of a heavy particle in a molecular fluid as described by Hamilton's equations. The nonperturbative approximation technique leads to the Fokker-Planck equation for simple Brownian motion. As part of the analysis, ``intrinsic diffusion'' is discovered and used to show ergodicity for the autocorrelation formula which appears during the Brownian motion calculation. An account of how these methods might be used to study the dynamical origins of critical behavior is given.

Nonperturbative approximation schemes for the effective interaction in nuclei

Several methods are investigated to deal with the singularities of the effective interaction which cause the perturbation series to diverge. Some of these methods, which are found to be quite useful, are based upon a semiphenomenological treatment of the “intruder” states. One method, based upon the use of Padé approximants, seems to be free of such semiphenomenological input data and appears to yield a particular effective interaction which has as eigenvalues those levels of the system for which the corresponding eigenfunctions have maximum overlap with the model space. The methods are investigated with the help of simple examples. Those methods which are found to be useful are applied to the 0 + and 2 + levels of 18O.

Relativistic quantum field theory and the Hartree-Fock method

The possibility of applying non-perturbative approximation techniques, which have been developed for non-relativistic situations, to relativistic quantum field theoretic problems, is considered. The nuclear force problem is discussed.

Magnetic Dipole Moment of the Electron in the Ladder Approximation

In order to calculate the magnetic dipole moment of the electron in a nonperturbative manner, the vertex function of quantum electrodynamics is studied in the ladder approximation. The vertex function is written as the sum of form factors and coupled, linear integral equations are derived for these form factors. After converting the system of integral equations into differential equations, the form factors are expressed by means of an infinite series. By using the leading term in the series solution, an approximation for the magnetic dipole moment of the electron is obtained. This nonperturbative approximation is considered in the limit of a large coupling constant, $g$, and the electric dipole moment, ${\ensuremath{\mu}}_{E}$, of a magnetic monopole with mass $M$ is found to be ${\ensuremath{\mu}}_{E}\ensuremath{\sim}\frac{[\sqrt{2}{(4\ensuremath{\pi})}^{2}{g}^{\ensuremath{-}\frac{5}{2}}]g}{2M}$. Assuming the neutron to be a bound state of magnetic monopoles, the electric dipole moment of the neutron is estimated as being ${10}^{\ensuremath{-}18}$ e cm.

Vertex theory of many-fermion systems

A general theory of the weak-response of many-fermion systems is formulated in terms of the longitudinal vertex function, Γ. By including effects of exchange the vertex theory embeds the conventional dielectric theory, as expressed by the momentum-energy space factorization Γ = ϵ −1 Γ . Here Γ is the proper vertex function and ϵ −1 is the generalized dielectric constant. The integral equation for Γ is rederived via weak-response theory, then factored into separate equations for Γ and ϵ −1. Basic properties of Γ, particularly gauge-invariance, are discussed. A nonperturbative approximation, called the υ′ approximation, is introduced in which the interaction kernel of the integral equation for Γ is replaced by a suitably chosen energy-independent effective interaction υ′. The υ′ approximation is shown to be a generalization of the time dependent Hartree-Fock approximation. By using the υ′ approximation as an input to a more exact theory expressions are obtained for the quasi-particle lifetime and the coupling of the quasi particle to the collective mode. Some new physical consequences of the theory are (a) the momentum dependence of the quasi-particle scattering amplitude in a weak external potential, (b) the existence of an intrinsic longitudinal coupling constant renormalization associated with the static long wavelength limit of Γ , and (c) the formal occurrence of a “spurious” pole of Γ coupled with a zero of ϵ −1 for a sufficiently strong repulsive interaction υ′. The last bears a resemblance to a phenomenon recently noted in elementary-particle theory. A preliminary note on some of this work employing a simplified nongauge-invariant version of the υ′ approximation was published earlier. Detailed results on the solution for Γ in the υ′ approximation and in lowest order perturbation theory will be published in a second paper.

Calculation of nucleon form factors through the Green function approximation scheme

The photon-pion-nucleon system is studied by a non-perturbative approximation method based on the truncation of covariant Green functions in field theory. The prescription of spectral representations is used to incorporate correct analyticity requirements. The anomalous magnetic moments and mean square charge radii of the nucleon calculated by this approximation method are an improvement over perturbation theory. The results are compared with those of dispersion theory.

Higher Random-Phase Approximations and the Theory of the Electron Gas

Using the example of the degenerate electron gas, it is shown how the operator equations of motion for a manyparticle system may be exploited to generate systematically a sequence of nonperturbative approximations of which some version of the random phase approximation is the first. Some questions left unsettled by previous attempts in this direction are resolved by close attention to the structure of the spectrum of the system. The solution of the equations is carried only so far as to make contact with previously substantiated results. Finally, a rigorous proof is given that the plasmon frequency approaches the classical plasmon frequency in the longwavelength limit. (auth)

Many-body propagators of a self-coupled spinor field in Edwards-Lieb’s approximation

The continuous integral representation of the complete, many-body propagators of a self-coupled spinor field is deduced and investigated. The continuous integrals are calculated by means ofEdwards—Lieb’s non-perturbative approximation method. In first approximation, the physical one-body propagator agrees with the propagator of a free, dressed particle and the physical two-body propagator corresponds to the one obtained in the “closed-loop chain” approximation ofAbrikosov et al.