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

The Kadanoff–Baym Equations

Giancarlo Calvanese Strinati

Published Feb 6, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1017/9781009411509.016

Related papers

Real-Frequency Green’s Functions from the Kadanoff–Baym Equations in the Equilibrium Case

Giancarlo Calvanese Strinati

This chapter considers the closed-time-path Green’s functions approach when specified to equilibrium situations and shows that it offers an alternative to the more standard Matsubara plus analytic continuation procedure for obtaining physical quantities directly in real frequency. In this case, the number of independent components of the single-particle Green’s function (as well as of the related self-energy) reduces considerably, thereby making it easier to solve the Kadanoff–Baym equations. The fluctuation–dissipation theorem and the single-particle spectral function (with its related sum rule) are also considered.

Cambridge University Press eBooks•Feb 6, 2026•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

State-of-the-Art Numerical Methods

Giancarlo Calvanese Strinati

This chapter gives a brief survey about the “time-stepping procedure” and the “predictor-corrector scheme” for solving the Kadanoff–Baym equations with two ( t and t ′) time variables. In this respect, the solutions of the Kadanoff–Baym equations for the greater and lesser Green’s functions are combined with each other in the positive quadrant of the t − t ′ plane, together with the solution of the lesser Green’s function along the time diagonal where t = t ′. The “generalized Kadanoff–Baym ansatz”, which aims at somewhat simplifying the solution of the Kadanoff–Baym equations themselves, is also introduced and derived in detail.

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

Converting Contour-Time to Real-Time Arguments

Giancarlo Calvanese Strinati

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

Expansions of nonequilibrium Green’s functions

Mathias Wagner

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.

Physical review. B, Condensed matter•Sep 15, 1991•163 citations

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
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.
This chapter considers the closed-time-path Green’s functions approach when specified to equilibrium situations and shows that it offers an alternative to the more standard Matsubara plus analytic continuation procedure for obtaining physical quantities directly in real frequency. In this case, the number of independent components of the single-particle Green’s function (as well as of the related self-energy) reduces considerably, thereby making it easier to solve the Kadanoff–Baym equations. The fluctuation–dissipation theorem and the single-particle spectral function (with its related sum rule) are also considered.
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.
This chapter gives a brief survey about the “time-stepping procedure” and the “predictor-corrector scheme” for solving the Kadanoff–Baym equations with two ( t and t ′) time variables. In this respect, the solutions of the Kadanoff–Baym equations for the greater and lesser Green’s functions are combined with each other in the positive quadrant of the t − t ′ plane, together with the solution of the lesser Green’s function along the time diagonal where t = t ′. The “generalized Kadanoff–Baym ansatz”, which aims at somewhat simplifying the solution of the Kadanoff–Baym equations themselves, is also introduced and derived in detail.
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.
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.