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Cambridge University Press eBooks

Langreth Rules: Convolutions and Products

Giancarlo Calvanese Strinati

Published Feb 6, 2026• 0 citations
Publisher

Abstract

In the theory of the contour-ordered Green’s functions, one encounters convolutions and products. The task of this chapter is to obtain the corresponding expressions in terms of the real-time functions. This task is accomplished in terms of the so-called Langreth–Wilkins rules, which are here discussed in detail for convolutions as well as for particle–hole-type and particle–particle-type products. A preliminary introduction to what is referred to as the Keldysh space is also provided.

DOI
10.1017/9781009411509.015

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In the theory of the contour-ordered Green’s functions, one encounters convolutions and products. The task of this chapter is to obtain the corresponding expressions in terms of the real-time functions. This task is accomplished in terms of the so-called Langreth–Wilkins rules, which are here discussed in detail for convolutions as well as for particle–hole-type and particle–particle-type products. A preliminary introduction to what is referred to as the Keldysh space is also provided.
Abstract In non-equilibrium many-body perturbation theory, Langreth rules are an efficient way to extract real-time equations from contour ones. However, the standard rules are not applicable in cases that do not reduce to simple convolutions and multiplications. We introduce a procedure for extracting real-time equations from general multi-argument contour functions with an arbitrary number of arguments. This is done for both the standard Keldysh contour, as well as the extended contour with a vertical track that allows for general initial states. This amounts to the generalization of the standard Langreth rules to much more general situations. These rules involve multi-argument retarded functions as key ingredients, for which we derive intuitive graphical rules. We apply our diagrammatic recipe to derive Langreth rules for the so-called double triangle structure and the general vertex function, relevant for the study of vertex corrections beyond the GW approximation.
This chapter considers the product of operators in the Heisenberg representation and express it in terms of the contour time-ordering operator. Since the relative order in which the operators enter the quantum average matters, this order has to be specified in detail. This procedure leads to considering the single- and two-particle Green’s functions, where the product contains, respectively, two and four field operators, which are at the core of the diagrammatic many-particle theory to be developed in what follows.
This paper gives an introduction to the Keldysh formalism, with emphasis on its usefulness in time-dependent density functional theory. In the first part we introduce the Keldysh contour and the one-particle Green function defined on this contour. We then discuss how to combine and manipulate functions with time-arguments on the contour. The effects of electron-electron interaction can be taken systematically into account, as we illustrate by propagating the Kadanoff-Baym equations for the second-order self-energy approximation. It is important to use conserving approximations such that the evolution of the electron density, momentum, and total energy agrees with the macroscopic conservation laws. One of the main topics in this paper is the non-interacting Green function, which is the relevant quantity for time-dependent density functional theory. We discuss this Green function in detail, and show how the Keldysh contour in a simple way allows us to derive the time-dependent Kohn-Sham potential from an action functional. The formalism in a similar way leads to response functions that obey the causality principle. To illustrate these points, we discuss the time-dependent optimized effective potential equations.
In this chapter we get acquainted with the one-particle Green's function G , or simply the Green's function. The chapter is divided in three parts. In the first part (Section 6.1) we illustrate what kind of physical information can be extracted from the different Keldysh components of G . The aim of this first part is to introduce some general concepts without being too formal. In the second part (Section 6.2) we calculate the noninteracting Green's function. Finally in the third part (Sections 6.3 and 6.4) we consider the interacting Green's function and derive several exact properties. We also discuss other physical (and measurable) quantities that can be calculated from G and that are relevant to the analysis of the following chapters. What can we learn from G ? We start our overview with a preliminary discussion on the different character of the space– spin and time dependence in G (1; 2). In the Dirac formalism the time-dependent wavefunction Ψ(x, t ) of a single particle is the inner product between the position–spin ket |x〉 and the time evolved ket |Ψ( t )〉. In other words, the wavefunction Ψ(x, t ) is the representation of the ket |Ψ( t )〉 in the position–spin basis.