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Cambridge University Press eBooks

Non-equilibrium Diagrammatics: Feynman Rules

Giancarlo Calvanese Strinati

Published Feb 6, 2026• 0 citations
Publisher

Abstract

This chapter applies the Wick’s theorem to the contour single-particle Green’s function. The corresponding average is represented in the interaction picture, with no need to specify at the outset the kind of contour that is used. This procedure is summarized in a set of Feynman diagrammatic rules, which are reported schematically. Here, only the normal phase is considered, while, in Part II, the Feynman rules are extended to superfluid Fermi systems.

DOI
10.1017/9781009411509.011

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This chapter applies the Wick’s theorem to the contour single-particle Green’s function. The corresponding average is represented in the interaction picture, with no need to specify at the outset the kind of contour that is used. This procedure is summarized in a set of Feynman diagrammatic rules, which are reported schematically. Here, only the normal phase is considered, while, in Part II, the Feynman rules are extended to superfluid Fermi systems.
At present, the only general method available for gaining knowledge from the fundamental principles about the dynamics of a system is the perturbative study. According to Feynman, as described in Chapter 4, instead of formulating quantum theory in terms of operators, the canonical formulation, for calculational purposes quantum dynamics can conveniently be formulated in terms of a few simple stenographic rules, the Feynman rules for propagators and interaction vertices. In Chapters 4 and 5, we showed how to arrive at the Feynman rules of diagrammatic perturbation theory for non-equilibrium states starting from the Hamiltonian defining the theory. The feature of non-equilibrium states, originally carried by the dynamical indices, could be expressed in terms of two simple universal vertex rules for the RAK-components of the matrix Green's functions. We are thus well acquainted with diagrammatics even for the description of non-equilibrium situations. However, for the situations studied using the quantum kinetic equations in Chapters 7 and 8, only the Dyson equation was needed, i.e. the self-energy, the 2-state one-particle irreducible amputated Green's function. No need for higher-order vertex functions was required, and the full flourishing diagrammatics was not put into action. In this chapter we shall proceed the other way around. We shall show that the diagrammatics of a physical theory, including the description of non-equilibrium states, can be obtained by simply stating quantum dynamics, the superposition principle, as the two exclusive options for a particle: to interact or not to interact ! From this simple Shakespearean approach we shall construct the Feynman diagrammatics of non-equilibrium dynamics.
This chapter considers a general form of the Wick’s theorem, which leads to a perturbation expansion of the (contour) single- and two-particle Green’s functions, which are expressed in terms of the contour time-ordering operator. The strategy for proving the Wick’s theorem is similar to that adopted within the Matsubara formalism for the Green’s functions at finite temperature and relies on the Gibbs form of the statistical operator in the interaction picture. An extension to superfluid Bose and Fermi systems is also considered.
Wick's theorem is combined with a cluster composition to derive a generalized statistical Wick's theorem for a non-equilibrium system with initial correlations. This enables a diagrammatic time-dependent perturbation theory which is a simple extension of the usual one to be developed for time-path causal statistical Green functions and their matrix equivalents. This provides a complete non-equilibrium theory. Reduction of diagrams and equations of motion for the Green functions are also discussed.
The contour-ordered Green's function considered in the previous chapter was ideal for discussing general closed time path properties such as the perturbative diagrammatic structure for non-equilibrium states. However, the contour-ordered Green's function lacks physical transparency and does not appeal to intuition. We need a different approach, which brings quantities back to real time. To accomplish this we introduce a representation where forward and return parts of the closed time path are ordered by numbers, specifying the position of a contour time by two indices, i = 1, 2. Next is the diagrammatic perturbation theory in the real-time technique then formulated in a fashion where the aspects of non-equilibrium states emerge in the physically most appealing way. In particular, we shall construct the representation where spectral properties and quantum statistics show up on a different footing in the diagrams. Lastly, we consider the connection to the imaginary-time treatment of non-equilibrium states, and establish its equivalence to the real-time approach propounded in this chapter. Real-time matrix representation To let our physical intuition come into play; we need to get from contour times back to real times. This is achieved by labeling the forward and return contours of the closed time path, depicted in Figure 4.5, by numbers, specifying the position of a contour time by an index. The forward contour we therefore label c 1 and the return contour c 2 , i.e. a contour time variable gets tagged by the label 1 or 2 specifying its belonging to forward or return contour, respectively.
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.
The complete rule of the Feynman-diagram expansion of a nonequilibrium process is given by using the path-integral technique. At an initial time tI{\mathit{t}}_{\mathit{I}}tI​, the system is in equilibrium and is brought into the nonequilibrium state by a time-dependent Hamiltonian. All the interactions, including the initial correlation, are taken into account by a diagrammatic expansion. Besides the well-known equilibrium propagators and the conventional 2\ifmmode\times\else\texttimes\fi{}2 nonequilibrium propagator matrix, the propagator becomes a 3\ifmmode\times\else\texttimes\fi{}3 matrix that contains extra elements depending explicitly on tI{\mathit{t}}_{\mathit{I}}tI​. Both the Feynman rule of the coordinate representation and the coherent-state representation are presented. It is shown that the propagator matrix is expressed compactly by the contour representation. The classical limit is also discussed.