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Cambridge University Press eBooks

Contour-Ordered Green’s Functions in the Nambu Representation

Giancarlo Calvanese Strinati

Published Feb 6, 2026• 0 citations
Publisher

Abstract

This chapter utilizes the Nambu pseudo-spinor field operators for the superfluid phase, to reformulate in terms of them the closed-time-path Green’s functions, the ensuing nonequilibrium Dyson equations, the conversion of contour-time to real-time arguments, and the Langreth rules. In this way, the results obtained previously within a mean-field decoupling are framed in a more general context, which will later make it possible to include beyond-mean-field effects for the superfluid phase.

DOI
10.1017/9781009411509.025

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Similar to Chapter 20 of Part I, this chapter considers the treatment of a few topics, which are relevant to the general purposes of the book but whose inclusion in previous chapters would have diverted the discussion of the main topics of interest therein. Specifically, it considers the Dyson equations for the contour single-particle Green’s function in the Nambu representation, the relative strength of different frequency terms in the derivation of the time-dependent Gross–Pitaevskii equation, the detailed calculation of an integral occurring in the derivation of the time-dependent Ginzburg–Landau equation, the irrelevance of the reference time t ₀ for the convolutions entering the Kadanoff–Baym equations at equilibrium, and the average energy of the system expressed in terms of the lesser Green’s function even in nonequilibrium situations.

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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.

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The nonequilibrium diagrammatics and the Dyson equations contain integrations over the time variables that run over a generic contour. In both cases, the time variables run first forward and then backward along the ordinary time axis. For computational purposes, it is then required to convert these time integrals into ordinary time integrals. To this end, it is first necessary to single out all possible combinations of the pair of time variables in the contour single-particle Green’s function. This is what is done in the present chapter.

Cambridge University Press eBooks•Feb 6, 2026•0 citations

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Cambridge University Press eBooks•Feb 6, 2026•0 citations
This chapter utilizes the Nambu pseudo-spinor field operators for the superfluid phase, to reformulate in terms of them the closed-time-path Green’s functions, the ensuing nonequilibrium Dyson equations, the conversion of contour-time to real-time arguments, and the Langreth rules. In this way, the results obtained previously within a mean-field decoupling are framed in a more general context, which will later make it possible to include beyond-mean-field effects for the superfluid phase.
Similar to Chapter 20 of Part I, this chapter considers the treatment of a few topics, which are relevant to the general purposes of the book but whose inclusion in previous chapters would have diverted the discussion of the main topics of interest therein. Specifically, it considers the Dyson equations for the contour single-particle Green’s function in the Nambu representation, the relative strength of different frequency terms in the derivation of the time-dependent Gross–Pitaevskii equation, the detailed calculation of an integral occurring in the derivation of the time-dependent Ginzburg–Landau equation, the irrelevance of the reference time t ₀ for the convolutions entering the Kadanoff–Baym equations at equilibrium, and the average energy of the system expressed in terms of the lesser Green’s function even in nonequilibrium situations.
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.
The nonequilibrium diagrammatics and the Dyson equations contain integrations over the time variables that run over a generic contour. In both cases, the time variables run first forward and then backward along the ordinary time axis. For computational purposes, it is then required to convert these time integrals into ordinary time integrals. To this end, it is first necessary to single out all possible combinations of the pair of time variables in the contour single-particle Green’s function. This is what is done in the present chapter.
This chapter introduces the Nambu representation for the pseudo-spinor fields and expresses the system Hamiltonian in terms of them. In this way, the anomalous single-particle Green’s function is made to match the form of the single-particle Green’s functions treated in Part I, where an even number of creation and destruction operators appear. On physical grounds, this approach exploits the fact that opposite-spin fermions are coupled in pairs. The special role played by the Hartree–Fock self-energy for a superfluid Fermi system is duly emphasized.
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.