Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Cambridge University Press eBooks

Converting Contour-Time to Real-Time Arguments

Giancarlo Calvanese Strinati

Published Feb 6, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1017/9781009411509.014

Related papers

Kubo–Martin–Schwinger Boundary Conditions

Giancarlo Calvanese Strinati

This chapter considers the boundary conditions on the time variables z ₁ and z ₂ for the contour single-particle Green’s function, which run over a generic contour in the complex-time z -plane. Different contours of interest are then specified. The boundary conditions for the integral form of the Dyson equation are also considered.

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

The Kadanoff–Baym Equations

Giancarlo Calvanese Strinati

This chapter converts the Dyson equation for the contour single-particle Green’s function to real-time variables. The corresponding equations for the Matsubara, right Keldysh, left Keldysh, lesser, greater, retarded, advanced, and Keldysh components are explicitly considered. A connection is also established with the original Kadanoff–Baym equations.

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

Contour-Ordered Green’s Functions in the Nambu Representation

Giancarlo Calvanese Strinati

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.

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

Real-time formalism

Jørgen Rammer

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.

Cambridge University Press eBooks•Jul 19, 2007•0 citations

Miscellany and Addenda to Part II

Giancarlo Calvanese Strinati

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.

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

Non-equilibrium Dyson Equations

Giancarlo Calvanese Strinati

This chapter considers some of the items discussed in the previous chapters and cast them in a more formal way so as to adapt them for future developments. In this way, the integro-differential form of the Dyson equation for the contour single-particle Green’s function (as well as its integral counterpart) is obtained, which play an important role in the following chapters for capturing the dynamical evolution of the physical system.

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