Book contents
- Frontmatter
- Dedication
- Contents
- Author’s Note
- Part INormal Phase
- 1Introduction
- 2The Schrödinger and Heisenberg Representations: Time-Evolution Operator
- 3Splitting the Hamiltonian: Heisenberg and Interaction Pictures
- 4Time-Dependent Quantum and Ensemble Averages: Initial Preparation of the System
- 5Quantum Averages over the Ground State and the Gell–Mann–Low Theorem
- 6The Contour Idea for Time-Dependent Averages: Forward and Backward Branches
- 7Closed-Time-Path Green’s Functions
- 8Dynamics for a Correlated Initial State and Various Kinds of Contours in the Complex Time Plane
- 9Perturbation Theory: Wick’s Theorem for Strings of Operators Ordered along a Contour
- 10Non-equilibrium Diagrammatics: Feynman Rules
- 11Non-equilibrium Dyson Equations
- 12Kubo–Martin–Schwinger Boundary Conditions
- 13Converting Contour-Time to Real-Time Arguments
- 14Langreth Rules: Convolutions and Products
- 15The Kadanoff–Baym Equations
- 16The t-Matrix Approximation in the Normal Phase
- 17Contour Diagrammatic Structure in Terms of Functional Derivatives
- 18Beyond Linear-Response Theory
- 19Time-Dependent Hartree–Fock Approximation and Mean-Field Decoupling
- 20Miscellany and Addenda to Part I
- Part IIFermionic Superfluid Phase
- Part IIIApplications
- Bibliography
- Index
13 - Converting Contour-Time to Real-Time Arguments
from Part I - Normal Phase
Published online by Cambridge University Press: 06 February 2026
Book contents
- Frontmatter
- Dedication
- Contents
- Author’s Note
- Part INormal Phase
- 1Introduction
- 2The Schrödinger and Heisenberg Representations: Time-Evolution Operator
- 3Splitting the Hamiltonian: Heisenberg and Interaction Pictures
- 4Time-Dependent Quantum and Ensemble Averages: Initial Preparation of the System
- 5Quantum Averages over the Ground State and the Gell–Mann–Low Theorem
- 6The Contour Idea for Time-Dependent Averages: Forward and Backward Branches
- 7Closed-Time-Path Green’s Functions
- 8Dynamics for a Correlated Initial State and Various Kinds of Contours in the Complex Time Plane
- 9Perturbation Theory: Wick’s Theorem for Strings of Operators Ordered along a Contour
- 10Non-equilibrium Diagrammatics: Feynman Rules
- 11Non-equilibrium Dyson Equations
- 12Kubo–Martin–Schwinger Boundary Conditions
- 13Converting Contour-Time to Real-Time Arguments
- 14Langreth Rules: Convolutions and Products
- 15The Kadanoff–Baym Equations
- 16The t-Matrix Approximation in the Normal Phase
- 17Contour Diagrammatic Structure in Terms of Functional Derivatives
- 18Beyond Linear-Response Theory
- 19Time-Dependent Hartree–Fock Approximation and Mean-Field Decoupling
- 20Miscellany and Addenda to Part I
- Part IIFermionic Superfluid Phase
- Part IIIApplications
- Bibliography
- Index
Summary
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.
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- Many-Body Green's Functions for Time-Dependent Problems , pp. 83 - 95Publisher: Cambridge University PressPrint publication year: 2026
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