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Journal of Mathematical Physics

Perturbation Expansion for Real-Time Green's Functions

Richard A. Craig

Published Apr 1, 1968• 178 citations
Publisher

Abstract

The development of the time-translation operators in a matrix element of an arbitrary operator is examined. It is noted that we may interpret time as evolving from some remotely early time (t0) to a time in the far future (t∝) and then back to (t0). Using this interpretation, a perturbation expansion is developed for Green's functions defined along this path and a separation of the two-particle interaction terms into self-energy parts and single-particle Green's function terms is justified for quantities on this path. A connection is established between the real-time Green's functions and the Green's function defined along the path, thereby yielding a perturbation expansion for the real-time functions and a justification of the separation of the interaction terms in the equations of motion for the real-time quantities. The transport equations of Kadanoff and Baym are derived without resorting to an analytic continuation from imaginary times and without the correction terms of Fujita.

DOI
10.1063/1.1664616

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The development of the time-translation operators in a matrix element of an arbitrary operator is examined. It is noted that we may interpret time as evolving from some remotely early time (t0) to a time in the far future (t∝) and then back to (t0). Using this interpretation, a perturbation expansion is developed for Green's functions defined along this path and a separation of the two-particle interaction terms into self-energy parts and single-particle Green's function terms is justified for quantities on this path. A connection is established between the real-time Green's functions and the Green's function defined along the path, thereby yielding a perturbation expansion for the real-time functions and a justification of the separation of the interaction terms in the equations of motion for the real-time quantities. The transport equations of Kadanoff and Baym are derived without resorting to an analytic continuation from imaginary times and without the correction terms of Fujita.
In Chapter 4, we introduced the Kadanoff–Baym equations of motion for the imaginary-time nonequilibrium Green's functions for a Bose gas, as given by (4.59) and (4.60). In this chapter, we will use the generalization of these equations of motion to find the equivalent equations of motion for the real-time Green's functions. These can be written in a natural way in the form of a kinetic equation. Using a simple Hartree–Fock approximation, we show how the coupled equations for the condensate and thermal cloud given in Chapter 3 emerge naturally from the Kadanoff–Baym (KB) formalism. This chapter is based on Imamović-Tomasović and Griffin (2001) and Imamović-Tomasović (2001), building on the pioneering work of Kane and Kadanoff (1965). In this chapter and Chapter 7 we review the KB formalism. However, we also encourage the reader to read the original account given by Kadanoff and Baym (1962). The goals and accomplishments of their seminal book are beautifully captured by the following quote from p. 138: Our rather elaborate Green's function arguments provide a means of describing transport phenomena in a self-contained way, starting from a dynamical approximation, i.e. an approximation for G 2 ( U ) in terms of G 1 ( U ). These calculations require no extra assumptions. The theory provides at the same time a description of the transport processes that occur and a determination of the quantities which appear in the transport equations. A closely related way of treating the nonequilibrium dynamics of a Bose-condensed gas is based on the two-particle irreducible (2PI) effective action together with the Schwinger–Keldysh closed-time path formalism. Berges (2004) gives a detailed review of this approach, which allows one to derive the nonequilibrium action on the basis of controllable approximations.
We present a Green's-function formalism generalized to treat arbitrary time-dependent systems initially prepared with an arbitrary density matrix. On the basis of a simultaneous expansion of the usual real-time development operator and the density matrix, we define a Green's function ordered along an extended contour. In this way we are able to treat real-time many-particle Green's functions and initial correlations on the same footing. Both can be decomposed by Wick's theorem. Thus we may utilize familiar diagrammatic analysis for evaluating the Green's function, and finally arrive at a generalized Dyson's equation. The Green's function can be represented by a (3\ifmmode\times\else\texttimes\fi{}3) matrix, which includes the Green's functions used in the Feynman, Matsubara, and Keldysh theories, if the corresponding statistical average is taken. From a matrix representation of Dyson's equation, a basic set of five coupled equations is derived, which explicitly shows the corrections to the Keldysh theory due to an arbitrary initial many-particle density matrix.
Abstract Chapter 09 Nonequilibrium Green’s functions (NEGF), including coupled-correlated (C) single- and multi-particle Green’s functions, are defined as averages weighted with the time-development operator U(t0+τ,t0). Linear conductivity is exhibited as a two-particle equilibrium Green’s function (Kubo-type formulation). Admitting particle sources (S:η,η+) and non-conservation of number, the non-equilibrium multi-particle Green’s functions are constructed with numbers of creation and annihilation operators that may differ, and they may be derived as variational derivatives with respect to sources η,η+ of a generating functional eW=TrU(t0+τ,t0)CS/TrU(t0+τ,t0)C. (In the non-interacting case this yields the n-particle Green’s function as a permanent/determinant of single-particle Green’s functions.) These variational relations yield a symmetric set of multi-particle Green’s function equations. Cumulants and the Linked Cluster Theorem are discussed and the Random Phase Approximation (RPA) is derived variationally. Schwinger’s variational differential formulation of perturbation theories for the Green’s function, self-energy, vertex operator, and also shielded potential perturbation theory, are reviewed. The Langreth Algebra arises from analytic continuation of integration of products of Green’s functions in imaginary time to the real-time axis with time-ordering along the integration contour in the complex time plane. An account of the Generalized Kadanoff-Baym Ansatz is presented.
The equation of motion for nonequilibrium Green functions is derived within the framework of the Schwinger and Keldysh formalism of perturbation expansion. For nonequilibrium distribution Green functions, the equation of motion derived from quantum mechanics contains undefined singularities, whose explicit form depends on the specific initial or boundary condition. In the present work, the exact expression of singular terms is found in the equation of motion from the time-looped perturbation theory in which the adiabatic initial condition is implied. Unlike the usual Dyson perturbation formalism or the well known Kadanoff-Baym equation of motion, our resulting equation can be adopted directly for calculations without the graphical analysis, which depends on the specific form of the Hamiltonian. On the basis of this equation of motion, the procedure of a nonperturbative solution is outlined and potential applications are briefly discussed.
We have developed a time-propagation scheme for the Kadanoff-Baym equations for general inhomogeneous systems. These equations describe the time evolution of the nonequilibrium Green function for interacting many-body systems in the presence of time-dependent external fields. The external fields are treated nonperturbatively whereas the many-body interactions are incorporated perturbatively using Phi-derivable self-energy approximations that guarantee the satisfaction of the macroscopic conservation laws of the system. These approximations are discussed in detail for the time-dependent Hartree-Fock, the second Born, and the GW approximation.