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
14 - Langreth Rules: Convolutions and Products
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
In the theory of the contour-ordered Green’s functions, one encounters convolutions and products. The task of this chapter is to obtain the corresponding expressions in terms of the real-time functions. This task is accomplished in terms of the so-called Langreth–Wilkins rules, which are here discussed in detail for convolutions as well as for particle–hole-type and particle–particle-type products. A preliminary introduction to what is referred to as the Keldysh space is also provided.
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- Many-Body Green's Functions for Time-Dependent Problems , pp. 96 - 115Publisher: Cambridge University PressPrint publication year: 2026
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