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Journal of Mathematical Physics

A computable expansion for multiparticle propagators

David J. Galas, C. A. Jensen, Harry L. Sahlin

Published Apr 1, 1973• 0 citations
Publisher

Abstract

A formulation of the propagator for a quantum system is constructed to simplify computation for multiparticle systems. The propagator is manifested in terms of any complete set on the Hilbert space of physical states of the system. In the short-time limit a particularly simple matrix expression is obtained using an energy eigenstate basis. This formulation is shown to contain all orders of perturbation theory in an approximate form. Identical particle exchange is handled in a straightforward manner in which only the number of terms, but not the complexity, of expressions is affected. The number of terms is held to manageable proportions by approximating multiple-particle exchange with repeated operations containing only two-particle exchange. Three-particle exchange is represented exactly by this approximation in the limit of infinitesimal time steps. An argument is presented that indicates that this is also true for an arbitrary particle number.

DOI
10.1063/1.1666348

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A formulation of the propagator for a quantum system is constructed to simplify computation for multiparticle systems. The propagator is manifested in terms of any complete set on the Hilbert space of physical states of the system. In the short-time limit a particularly simple matrix expression is obtained using an energy eigenstate basis. This formulation is shown to contain all orders of perturbation theory in an approximate form. Identical particle exchange is handled in a straightforward manner in which only the number of terms, but not the complexity, of expressions is affected. The number of terms is held to manageable proportions by approximating multiple-particle exchange with repeated operations containing only two-particle exchange. Three-particle exchange is represented exactly by this approximation in the limit of infinitesimal time steps. An argument is presented that indicates that this is also true for an arbitrary particle number.
A new type of ``intermediate'' state, \ensuremath{\Vert}\ensuremath{\Psi}${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathit{J}}$〉, is introduced in the representation of many-body propagators and related resolvent matrix elements. A simple algebraic procedure is presented for constructing the unitary transformation matrix Q that relates the intermediate states and the exact energy eigenstates, \ensuremath{\Vert}${\mathrm{\ensuremath{\Psi}}}_{\mathit{n}}$〉, of the interacting system. Here the starting point is the matrix X of generalized spectroscopic amplitudes 〈${\mathrm{\ensuremath{\Psi}}}_{\mathit{n}}$\ensuremath{\Vert}C${\mathrm{^}}_{\mathit{J}}$\ensuremath{\Vert}${\mathrm{\ensuremath{\Psi}}}_{0}^{\mathit{N}}$〉, where C${\mathrm{^}}_{\mathit{J}}$ denotes a physical excitation operator and \ensuremath{\Vert}${\mathrm{\ensuremath{\Psi}}}_{0}^{\mathit{N}}$〉 is the N-electron ground state. A block QR decomposition [G. H. Golub and C. F. von Loan, MatrixB Computations (Johns Hopkins University, Baltimore, 1989)] of X according to the equation X=Q F allows one to determine explicit expressions for the subblocks of Q and the intermediate propagator representations constituted by a nondiagonal effective interaction matrix C and an effective spectroscopic matrix f. These effective quantities f and C are formulated entirely in terms of ground-state density-matrix elements and related energy expectation values. The relevance of the intermediate representations as a means for deriving computational schemes is based on the regularity and compactness of the perturbation expansions for the effective matrices f and C. These basic properties also establish that the intermediate representations are closed-form versions of the algebraic-diagrammatic construction approximation schemes derived previously as a reformulation of the original diagrammatic propagator perturbation series. The intermediate representations represent a link between algebraic and diagrammatic approaches in the field of propagator methods and are expected to be useful also in the development of nonperturbative approximations.
The recently developed (energy) sudden representation of the Schrödinger propagator is examined in conjunction with the path integral method. The perturbation expansion of the propagator in this representation (with the sudden approximation as the zeroth order term) is easily understood using Feynman diagrams. The zeroth order term is a form involving only sudden paths (i.e., paths resulting from the sudden approximation). Each higher order term is a series of sudden paths which are connected to one another due to the nonsudden effects of the dynamics. The sudden representation, and in particular its perturbation expansion, is also used to study the bath effects in many body systems which are expressed in the form of influence functionals in the path integral formalism. In our treatment, the bath coordinates are assumed to be the slow coordinates. The zeroth order (sudden approximation) influence functionals are expressed in a very simple form; they involve only an ordinary integral over the bath coordinates. Finally, to illustrate the approach, a harmonic oscillator coupled linearly to an unspecified primary system is used as a simple model bath. The reduction of the exact influence functional to the sudden approximation form is also considered.
One of the useful and practical methods for solving quantum-mechanical many-body systems is to recast the full problem into a form of the effective interaction acting within a model space of tractable size. Many of the effective-interaction theories in nuclear physics have been formulated by use of the so-called $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box introduced by Kuo et al. It has been one of the central problems how to calculate the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box accurately and efficiently. We first show that, introducing new basis states, the Hamiltonian is transformed to a block-tridiagonal form in terms of submatrices with small dimension. With this transformed Hamiltonian, we next prove that the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box can be expressed in two ways: One is in the form of a continued fraction and the other is a simple series expansion up to second order with respect to renormalized vertices and propagators. This procedure ensures derivation of an exact $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box, if the calculation converges as the dimension of the Hilbert space tends to infinity. The $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box given in this study corresponds to a nonperturbative solution for the energy-dependent effective interaction which is often referred to as the Bloch-Horowitz or the Feshbach form. By applying the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Z}$-box approach based on the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box proposed previously, we introduce a graphical method for solving the eigenvalue problem of the Hamiltonian. The present approach has a possibility of resolving many of the difficulties encountered in the effective-interaction theory.
A purely algebraic approach to Green's functions is presented. The new formulation allows the determination of a closed-form equation for Green's functions which depends only on the ground state of the system. By a straightforward perturbation expansion of this state an interesting approximation scheme is obtained which is nonperturbative in the final states (e.g., excited or ionic states). This method, which can be used for various propagators, is explicitly applied to the particle-particle propagator carrying out the scheme up to third order. The approach is discussed in detail and briefly compared with the well-established algebraic diagrammatic construction approximation scheme.
Two distinct approaches have been followed to produce an approximate expression for the third virial coefficient of a quantum system in terms of the pertinent two-particle quantities (standard T matrix elements and bound state eigenvalues and eigenstates): One approach is based on the T approximation for the self-energy in the theory of temperature Green functions, and the other uses the diagrams which represent the various terms in the multiple-scattering expansion of the three-particle T operator. It is shown that the result given by the former method is reproduced exactly by the contribution of a small number of selected diagrams. The Green function method requires the evaluation of the Kadanoff and Baym generalized T matrix to one order in the activity higher than its familiar low-density limit. The common result can be written entirely in terms of the lowest order self-energy and should give a reasonable approximation to the third virial coefficient for systems with short-range forces.
Due to recent advances in quantum information, as well as in mesoscopic and nanoscale physics, the interest in the theory of open systems and decoherence has significantly increased. In this paper, we present an interesting approach to solving a time-convolutionless equation of motion for the open system reduced density matrix beyond the limit of weak coupling with the environment. Our approach is based on identifying an effective, memory-containing interaction in the equations of motion for the representation submatrices of the evolution operator (these submatices are written in a special basis, adapted for the ``partial-trace-free'' approach, in the system+environment\mathrm{system}+\mathrm{environment}system+environment Liouville space). We then identify the ``memory dressing,'' a quantity crucial for solving the equation of motion for the reduced density matrix, which separates the effective from the real physical interaction. The memory dressing obeys a self-contained nonlinear equation of motion, which we solve exactly. The solution can be represented in a diagrammatic fashion after introducing an ``information exchange propagator,'' a quantity that describes the transfer of information to and from the system, so the cumulative effect of the information exchange results in the memory dressing. In the case of weak system-environment coupling, we present the expansion of the reduced density matrix in terms of the physical interaction up to the third order. However, our approach is capable of going beyond the weak-coupling limit, and we show how short-time behavior of an open system can be analyzed for arbitrary coupling strength. We illustrate the approach with a simple numerical example of single-particle level broadening for a two-particle interacting system on short time scales. Furthermore, we point out a way to identify the structure of decoherence-free subspaces using the present approach.