Sum Of Diagrams Research Articles

On the electrical conductivity of disordered systems: The kernel of the Bethe-Salpeter equation

The macroscopic conductivity tensor of a disordered system may be expressed in terms of operators { G( z 1) J μ G( z 2)}, for which in a suitable momentum representation, an integral equation can be derived. In this paper the kernel of this equation is evaluated exactly by a diagram summation technique. The results of Ashcroft and Schaich can be obtained using an appropriate approximation having an obvious physical interpretation.

Diagram Technique for the Green Function of a Heisenberg Ferromagnet with Arbitrary Spin

AbstractThe diagram technique previously developed by one of the authors for the Green's function of a Heisenberg ferromagnet with spin 1/2 is generalized to arbitrary spin. The summation of diagrams is the main problem in the present paper. The diagrams split into two classes; one of them may be summed with help of Dyson's equation, the other cannot be treated as a contribution to the mass operator. The latter class of diagrams is found to be the series expansion of 〈Sz〉 and 〈S S〉. The resulting spin wave energy corresponds in successive approximations to the results of MFA, of Tyablikov's decoupling, of Dyson, and of the decoupling due to Mubayi and Lange, and Kenan, respectively. The low temperature magnetization is calculated directly from the Green's function and agrees with Callen's result in the highest approximation worked out.

To the theory of the generalized collision integral. II

The method of nonstationary diagram summation, previously developed [1], is applied to constructing the collision integral for a classical plasma, in which ring diagrams with prior ladder renormalization are included, which is equivalent to including screening of a long-range potential and finite momentum transfer in binary collisions. For a spatially homogeneous plasma it is shown that in the collision integral obtained the collision cross section does not contain the characteristic Coulomb divergences.

Electron-Positron Annihilation from the Light Cone

We show how to use light-cone expansions and the Callan-Symanzik equation to calculate some structure functions of nonforward virtual Compton amplitudes in renormalizable field theories. By making an analytic continuation of the discontinuity we arrive at integral equations for ${e}^{+}{e}^{\ensuremath{-}}$ annihilation structure functions. These are equivalent to equations previously found by Gribov and Lipatov by direct summation of diagrams.

On Different Perturbation Theoretical Approaches to the Heisenberg Ferromagnet with Spin 1/2

AbstractThe results of the drone‐fermion representation (DFR) in application to the Heisenberg ferromagnet with spin 1/2 are compared with the corresponding results obtained by means of Pauli operators. The expressions calculated in DFR by Spencer for the perturbational expansion of the one‐particle Green function are shown to coincide with the corresponding expressions arising in the calculations using Pauli operators. In spite of that, different ways of summation of diagrams lead to different results for physically interesting quantities; these differences can be explained explicitely as resulting from certain neglections made in the use of DFR.

High-energy summation of ladder diagrams in nonpolynomial field theories

The leading high-energy behavior of the sum of ladder diagrams is discussed in local and nonlocal field theories of nonpolynomial type. It is shown that the complete amplitude satisfies an integral equation of Volterra type at high-energy and in several cases it possesses Regge-behaved upper bounds. This high-energy behavior is confirmed also model calculations.

On the choice of form factors in non-local quantum electrodynamics

The photon Green function in the approximation of principal logarithmic diagrams is investigated in local quantum electrodynamics. It is supposed that the result of the summation of principal diagrams leads to a non-local theory. Conditions under which the obtained Green function is unambiguous are stated. It turns out that any diagrams of the perturbation theory series, calculated by means of the partially summed photon Green function, are convergent. Within the frame of the procedure proposed, the value for elementary length is obtained:

Interaction of Randomly Phased Vibrational Waves via the Collision Dominated Solid State Plasma

Abstract A diagrammatic technic is presented which is appropriate to study the nonlinear response of the collision dominated solid state plasma to randomly phased ac fields. By summation of diagrams, a nonlinear integral equation is established for the propagator S′ which is directly related to the nonlinear response function. Using the equation for S′, an equivalence is derived between a model solution in the theory of nonlinear coherent ultrasound amplification and the interaction theory of a great number of randomly phased vibrational modes, confined to a small frequency band.

Diagram summation for resonance broadening calculations

It is shown that results for resonance broadening of spectral lines derived from the statistical theory of line broadening can also be obtained approximately by selective summation of diagrams.

Pair effects in random lattices; a generalized C.P.A. and its equivalence to diagramatic expansions

A generalization of the coherent potential approximation that is exactly equivalent to self-consistent diagrammatic summation techniques is discussed. Explicit expressions for the pair case are given.

High orders corrections to the Van der Waals–London forces. II. Interaction of two molecules with isotropic polarizabilities

AbstractThe infinite summation of intermolecular ring diagrams is extended to any pair of interacting molecules with isotropic polarizabilities. The calculation no longer requires the use of a minimal basis set. The polarizabilities α,α′ may be factorized at all orders, and an expression is derived, which gives modified Van der Waals energies. For heavy atoms the series converges if αα′/R6 is smaller than 1, R being the interatomic distance. This inequality is easily satisfied for the Van der Waals distances, and in practice the correction due to the high orders of the perturbation expansion remains weak. The role of the EPV diagrams and the connection with a configuration interaction treatment are discussed.

Asymptotic Behavior and the Possibility of Duality without Ghosts in Feynman-Diagram Models

Elastic scattering of two spinless particles with equal or unequal masses is considered in Feynman-diagram models of the Van Hove type with the usual couplings and with arbitrary spectra of masses, spins, and coupling constants, including infinitely large masses and spins. It is shown that if the amplitude defined by analytic continuation of the sum of all lowest-order diagrams with $s$-channel poles has no unphysical or $u$-channel singularities, then for fixed $t$ this amplitude is not bounded by any power of $s$ as $|s|\ensuremath{\rightarrow}\ensuremath{\infty}$ in any infinitely multiply connected domain which excludes only neighborhoods of the poles. Since a dual Born term with no $u$-channel poles can be represented entirely by the sum of $s$-channel pole diagrams (or by the sum of $t$-channel pole diagrams), this bad asymptotic behavior cannot be canceled in the Born approximation in a dual multiparticle theory. Regge asymptotic behavior is also supposed to be associated with duality, and consequently one sees that Feynman-diagram models of the Van Hove type cannot exhibit duality. The result is independent of the order of summation of diagrams. It depends crucially on the requirement that coupling constants be real.

LOCALIZED SPIN FLUCTUATIONS IN METALS

Abstract : The concepts and results of several theories of localized spin fluctuations are contrasted. The random phase approximation--qualitatively correct for weak exchange coupling U--overemphasizes the collective nature of the fluctuations and in dilute alloys leads to an unphysical instability for U = or > U sub c. For U approximately equal to U sub c, nonlinear interactions between spin fluctuations suppress the instability. These anharmonic effects have been treated through the functional average and the diagram summation schemes, and they lead to a smooth variation of the system properties as U increases through U sub c. A complementary treatment in which one studies the time-evolution of spin fluctuations is also considered. (Author)

Composite States for Fourfermion Interactions by the Method of Broken Symmetries

Composite states of quasiparticles are studied in the frame of Bopp's quantum theory of elementary particles in a lattice space. By solving the homogeneous Bethe-Salpeter equations in the combined chain and ladder approximation the theory is found to contain a massless pseudoscalar particle (the Goldstone boson), a scalar particle of mass 2 m and an axialvector particle of mass 2 m - d, d > 0, where m is the mass of the quasiparticles and d a small quantity depending on the number of lattice points (or equivalently the cutoff). The method of summation of chain and ladder diagrams by means of the Fierz formula is treated in some detail. Analogies with the model of Nambu and Jona-Lasinio are pointed out. Finally some remarks on the scattering problem are added

Hartree-Fock Theory and theΔTransformation in Quantum-Statistical Mechanics. I. Normal Fermi Systems

Hartree-Fock theory is used in conjunction with the quantum Ursell expansion to obtain the grand potential $f(\ensuremath{\beta}, g, \ensuremath{\Omega})$ and the momentum distribution $〈n(k)〉$ for a Fermi fluid. It is shown that the perturbation expansion of the quasiparticle Ursell functions thus obtained yields a formal expansion of the grand potential which is easily related to the theory of de Dominicis. After the summation of certain ladder diagrams in the perturbation expansion, $f(\ensuremath{\beta}, g, \ensuremath{\Omega})$ and $〈n(k)〉$ take on forms identical to those obtained in the $\ensuremath{\Lambda}$-transformed theory derived by Mohling, RamaRao, and Shea. It is therefore possible to identify the Hartree-Fock single-particle energy with the quasiparticle energy of those authors' work.

A simple derivation of the finite wavelength, high frequency dielectric constant of a classical Coulomb plasma

The lowest order collisional contribution to the frequency and wave vector dependent dielectric constant of a classical plasma is obtained through a simple diagram summation similar to the well known summation leading to the Vlassov polarizability. The connection with the infinite wavelength results is explicitly given.

On the integral equations for the nonrelativistic scattering amplitudes

A method of derivation of integral equations of the Fredholm type for N-particle scattering amplitudes based on time-dependent theory and the second quantisation formalism is proposed. The method corresponds to the nonrelativistic diagram summation technique introduced by authors earlier for obtaining the equations for the case N=3 and N=4.

Diagram Approach to the Theory of Collisionless Plasma Turbulence

From the Vlasov equation written as a nonlinear equation for the distribution function ƒ, a perturbation expansion of the fluctuating part of ƒ, ƒ̃̃=ƒ — 〈ƒ〉, is carried out in terms of the linear solution using a diagram representation similar to that of Wyld. From this a diagram expansion is obtained for the correlation function U=〈ƒ̃ƒ̃̃〉 and similar diagram series for two auxiliary functions, the averaged linear response function G=〈Ĝ〉 and the generalized vertex function Γ. Approximate asymptotic equations for U imply the summation of appropriate infinite subsets of diagrams. To derive such approximations two criteria are applied: 1) Consistency with the basic conservation laws (particle number, momentum, energy); 2) Selection of diagrams analogous to those leading to good approximations in Kraichnan’s random oscillator problem, the principal results of which are regained. Contrary to Wyld’s summation procedure diagram summation are not carried out directly in the expansion of U, but in the expansions of the functions 〈ƒ̃ƒ̃ƒ̃̃〉 and 〈ƒ̃̃ Ĝ〉, which appear on the right hand side of the differential equations for U and G. This automatically guaranties the validity of the conservation laws. The first approximation we discuss, which is equivalent to the quasi-Gaussian approximation well known from hydrodynamic turbulence, leads to the kinetic wave equation of weak turbulence. The next approximation consists of a coupled set of equations for U and G which are identical with the equations of the symmetrical random coupling model of Orszag and Kraichnan. The third approximation derived implies the generalized vertex function Γ. The first and the third of these approximations cannot be obtained in Wyld’s summation scheme. On the other hand it does not seem possible to derive the third approximation for the nonlinear Vlasov equation using Kraichnan’s diagram technic.

Deuteron decay in the field of the nucleus

The amplitude of the A+d→n+p+A reaction was obtained on the basis of the diagram summation method for the particular configurations of the particle in the final state, when (i) the neutron and proton have a small relative energy, or (ii) one of the nucleons and the nucleus have a small relative energy. It was shown that from the analysis of the experimental data of the reaction A+d→n+p+A and by using the obtained formula the value of the spin-dependent and spin-independent parts of the nucleon-nucleus scattering amplitude can be found.

The de Haas-van Alphen effect in the presence of a random array of scatterers

The Green function method developed by Edwards is applied to an electron gas in a magnetic field interacting with a random array of scatterers. The Green function in the presence of the scatterers is represented as a sum of diagrams. The set of these important when the scattering interaction and the density of scatterers is not too large is summed. The density of states calculated from this Green function is compared with that obtained from a simple relaxation time assumption. The new density of states is used to calculate the susceptibility in the same way as was done by Dingle. It is found that the effect of the scatterers is to shift the de Haas-van Alphen oscillations as well as to reduce their amplitude.