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Journal of Mathematical Physics

Interaction-set scattering equations

Kenneth L. Kowalski, A. Picklesimer

Published Feb 1, 1983• 3 citations
Publisher

Abstract

The generalization of the pair-labeled Rosenberg equations for many-particle scattering are found in the case where there are arbitrary multiparticle interactions. These are called interaction-set equations because they involve auxiliary transition operators which are labeled by the same set of partitions which characterizes the various connectivities of the interparticle interactions. The technique which we employ also provides the analogous extension of a recently proposed set of connected-kernel multiple scattering equations for the Watson-type transition operators. Further, the structure of the interaction-set equations leads to the identification of an entire class of interaction-set connected-kernel scattering integral equations, each of which is based upon a distinct choice of unperturbed Green’s function and its associated connectivity structure. The generalized Rosenberg equations and the connected-kernel Watson-type multiple scattering equations, which are limiting members of this class, correspond to the choice of the N-free-particle and two-cluster-channel unperturbed Green’s functions, respectively.

DOI
10.1063/1.525704

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The generalization of the pair-labeled Rosenberg equations for many-particle scattering are found in the case where there are arbitrary multiparticle interactions. These are called interaction-set equations because they involve auxiliary transition operators which are labeled by the same set of partitions which characterizes the various connectivities of the interparticle interactions. The technique which we employ also provides the analogous extension of a recently proposed set of connected-kernel multiple scattering equations for the Watson-type transition operators. Further, the structure of the interaction-set equations leads to the identification of an entire class of interaction-set connected-kernel scattering integral equations, each of which is based upon a distinct choice of unperturbed Green’s function and its associated connectivity structure. The generalized Rosenberg equations and the connected-kernel Watson-type multiple scattering equations, which are limiting members of this class, correspond to the choice of the N-free-particle and two-cluster-channel unperturbed Green’s functions, respectively.
We consider the nonrelativistic N-body scattering problem for a system of particles in which some subsets of the particles are identical. We demonstrate how the particle identity can be included in a general class of linear integral equations for scattering operators or components of scattering operators. The Yakubovskii, Yakubovskii–Narodestkii, Rosenberg, and Bencze–Redish–Sloan equations are included in this class. Algebraic methods are used which rely on the properties of the symmetry group of the system. Operators depending only on physically distinguishable labels are introduced and linear integral equations for them are derived. This procedure maximally reduces the number of coupled equations while retaining the connectivity properties of the original equations.
Integral equations of the Watson-Faddeev type, developed in the context of few-body nonrelativistic scattering theory, are shown to play an important role in the structure analysis of many-body Green's functions and in variational techniques for the approximate determination of these Green's functions. Such equations sum up an infinite chain of few-body correlations. Those correlations which are not of this form, i.e., the intrinsic many-body correlations, are accounted for by the appearance of an effective (or "renormalized") interaction, as can be seen by direct transformation of the exact Martin-Schwinger equations for the Green's functions. For a given choice of effective interaction, the few-body correlation functions must be determined self-consistently. The Schwinger variational principle, originally developed for linear integral equations, is nevertheless the key element in a variational formulation of the nonlinear equations encountered here. For simplicity we confine our attention to equations which determine the two-body Green's function, although the entire procedure can be generalized to higher orders.
This is the first of a series of papers dealing with many-particle systems from a unified, nonperturbative point of view. It contains derivations and discussions of various field-theoretical techniques which will be applied in subsequent papers. In a short introduction the general method of approach is summarized, and its relationship to other field-theoretic problems indicated. In the second section the macroscopic properties of the spectra of many-particle systems are described. Asymptotic evaluations are performed which characterize these macroscopic features in terms of intensive parameters, and the relationship of these parameters to thermodynamics is discussed. The special characteristics of the ground state are shown to follow as a limiting case of the asymptotic evaluations. The third section is devoted to the time-dependent field correlation functions, or Green's functions, which describe the microscopic behavior of a multiparticle system. These functions are defined, and related to intensive macroscopic variables when the energy and number of particles are large. Spectral representations and other properties of various one-particle Green's functions are derived. In the fourth section the treatment of non-equilibrium processes is considered. As a particular example, the electromagnetic properties of a system are expressed in terms of the special two-particle Green's function which describes current correlation. The discussion yields specifically a fluctuation-dissipation theorem, a sum rule for conductivity, and certain dispersion relations. The fifth section deals with the differential equations which determine the Green's functions. The boundary conditions that characterize the Green's function equations are exhibited without reference to adiabatic decoupling. A method for solving the equations approximately, by treating the correlations among successively larger numbers of particles, is considered. The first approximation in this sequence is shown to yield a generalized Hartree-like equation. A related, but rigorous, identity for the single-particle Green's function is then derived. A second approximation, which takes certain two-particle correlations into account, is shown to produce various additional effects: The interaction between particles is altered in a manner characterized by the intensive macroscopic parameters, and the modification and spread of the energy-momentum relation come into play. In the final section compact formal expressions for the Green's functions and other physical quantities are derived. Alternative equations and systematic approximations for the Green's functions are obtained.
Abstract Starting with the equation of motion for the field operator ψ(x,t) of an interacting many-particle system, the n-particle Green’s function (Gn) equation of motion is developed, with interparticle interactions generating an infinite chain of equations coupling it to (n+1)- and (n−1)-particle Green’s functions (Gn+1 and Gn−1, respectively). Particularly important are the one-particle Green’s function equation with its coupling to the two-particle Green’s function and the two-particle Green’s function equation with its coupling to the three-particle Green’s function. To develop solutions, it is necessary to introduce non-correlation decoupling procedures involving the Hartree and Hartree-Fock approximations for G2 in the G1 equation; and a similar factorization “ansatz” for G3 in the G2 equation, resulting in the Sum of Ladder Diagrams integral equation for G2, with multiple Born iterates and finite collisional lifetimes. Similar treatment of the G11-equation for the joint propagation of one-electron and one-hole subject to mutual Coulomb attraction leads to bound electron-hole exciton states having a discrete hydrogen like spectrum of energy eigenstates. Its role in single-particle propagation is also discussed in terms of one-electron self-energy Σ and the T-matrix
The equations of motion for the general many-time causal Green's functions for a fermion system are iterated and are shown not to lead to unlinked graphs, which is a general proof of the linked cluster theorem. An explicit expression is obtained for the perturbation expansion of an arbitrary Green's functions which is applied to the one- and two-particle Green's functions. By connecting lines systematically in a set of diagrams obtained from the equations of motion, the usual topologically different linked graphs and rules are generated.