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Cambridge University Press eBooks

The Contour Idea for Time-Dependent Averages: Forward and Backward Branches

Giancarlo Calvanese Strinati

Published Feb 6, 2026• 0 citations
Publisher

Abstract

This chapter introduces the contour Schwinger–Keldysh method for time-dependent averages, in light of its relevance to nonequilibrium processes. A key feature of this approach is that it leaves open the possibility that no state of a system in the future can be identified with any of its states in the past. This method is here illustrated in detail with reference to time-dependent quantum averages, whereby for definiteness the system is initially prepared at the reference time t ₀ in a definite quantum state.

DOI
10.1017/9781009411509.007

Related papers

Connection between the Schwinger–Keldysh Closed-Contour Approach and the Lindblad Master Equation

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This chapter explores to what extent the closed-contour Schwinger–Keldysh approach and the Lindblad Master equation can be connected with each other. Here, the connection with the Schwinger–Keldysh closed-contour approach does not involve the full machinery of the Green’s functions method, but rather refers directly to the time evolution of the many-body density matrix, which contains a forward evolution operator from the reference time t ₀ to the measuring time t and a backward evolution operator from t back to t ₀. The key approximations to derive the Lindblad Master equation are specified in detail. As an example, a two-level system coupled to a phonon bath is explicitly considered.

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

Many-Body Green's Functions for Time-Dependent Problems

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Quantum many-body systems are a central feature of condensed matter physics, relevant to important, modern research areas such as ultrafast light-matter interactions and quantum information. This book offers detailed coverage of the contour Green's function formalism – an approach that can be successfully applied to solve the quantum many-body and time-dependent problems present within such systems. Divided into three parts, the text provides a structured overview of the relevant theoretical and practical tools, with specific focus on the Schwinger-Keldysh formalism. Part I introduces the mathematical frameworks that make use of Green's functions in normal phase states. Part II covers fermionic superfluid phases with discussion of topics such as the BCS-BEC crossover and superconducting systems. Part III deals with the application of the Schwinger-Keldysh formalism to various topics of experimental interest. Graduate students and researchers will benefit from the book's comprehensive treatment of the subject matter and its novel arrangement of topics.

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

Time-Dependent Quantum and Ensemble Averages: Initial Preparation of the System

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This chapter considers the initial preparation of the many-particle system, whose control is achieved before the reference time t ₀ when the time-dependent perturbation begins to act on the system. After t ₀, the system is let to evolve in time according to the full time-dependent Hamiltonian. The initial control can be either full or partial. Full control signifies that at t ₀ the system is prepared in a definite “pure” quantum state (like the ground state), while partial control signifies that initially the system is only known to be in a “mixture” of states with given probabilities, such that the information on the phases of the superposition is lost. These two cases are here treated separately.

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

Dynamics for a Correlated Initial State and Various Kinds of Contours in the Complex Time Plane

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This chapter extends the treatment of the previous chapters to the case when the system is initially prepared in an ensemble average. This requires adding a “vertical track” to the oriented contour. Alternative formalisms are considered in this context, depending on the way the vertical track is dealt with. The relationship between transient phenomena and the adiabatic assumption is also considered.

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

Converting Contour-Time to Real-Time Arguments

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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.

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Nonequilibrium Quantum Field Theory

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This chapter examines the Schwinger-Keldysh technique, applying path integrals, and the functional method of 2PI effective actions—with the latter serving to construct powerful symmetry-preserving nonperturbative approximations for equilibrium as well as nonequilibrium descriptions of strongly interacting quantum many-body systems. It provides a rigorous mathematical basis for exploring the quantum mechanical basis of thermalization and decoherence. Mesoscopic systems with quantum processes, in particular, play a key role. The domain of nonequilibrium mesoscopic systems is flooded with open questions some of which are solved but still others have provisional or incomplete solutions. A correlation function encodes the correlation between the field operators at different times and is an inference to all thermodynamic quantities. The beauty of the Keldysh time path is the definition of the Green's function, which plays an analogous role in the nonequilibrium formalism similar to the role the causal Green's function plays in the equilibrium theory.

CRC Press eBooks•Jun 20, 2019•0 citations
This chapter introduces the contour Schwinger–Keldysh method for time-dependent averages, in light of its relevance to nonequilibrium processes. A key feature of this approach is that it leaves open the possibility that no state of a system in the future can be identified with any of its states in the past. This method is here illustrated in detail with reference to time-dependent quantum averages, whereby for definiteness the system is initially prepared at the reference time t ₀ in a definite quantum state.
This chapter explores to what extent the closed-contour Schwinger–Keldysh approach and the Lindblad Master equation can be connected with each other. Here, the connection with the Schwinger–Keldysh closed-contour approach does not involve the full machinery of the Green’s functions method, but rather refers directly to the time evolution of the many-body density matrix, which contains a forward evolution operator from the reference time t ₀ to the measuring time t and a backward evolution operator from t back to t ₀. The key approximations to derive the Lindblad Master equation are specified in detail. As an example, a two-level system coupled to a phonon bath is explicitly considered.
Quantum many-body systems are a central feature of condensed matter physics, relevant to important, modern research areas such as ultrafast light-matter interactions and quantum information. This book offers detailed coverage of the contour Green's function formalism – an approach that can be successfully applied to solve the quantum many-body and time-dependent problems present within such systems. Divided into three parts, the text provides a structured overview of the relevant theoretical and practical tools, with specific focus on the Schwinger-Keldysh formalism. Part I introduces the mathematical frameworks that make use of Green's functions in normal phase states. Part II covers fermionic superfluid phases with discussion of topics such as the BCS-BEC crossover and superconducting systems. Part III deals with the application of the Schwinger-Keldysh formalism to various topics of experimental interest. Graduate students and researchers will benefit from the book's comprehensive treatment of the subject matter and its novel arrangement of topics.
This chapter considers the initial preparation of the many-particle system, whose control is achieved before the reference time t ₀ when the time-dependent perturbation begins to act on the system. After t ₀, the system is let to evolve in time according to the full time-dependent Hamiltonian. The initial control can be either full or partial. Full control signifies that at t ₀ the system is prepared in a definite “pure” quantum state (like the ground state), while partial control signifies that initially the system is only known to be in a “mixture” of states with given probabilities, such that the information on the phases of the superposition is lost. These two cases are here treated separately.
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 examines the Schwinger-Keldysh technique, applying path integrals, and the functional method of 2PI effective actions—with the latter serving to construct powerful symmetry-preserving nonperturbative approximations for equilibrium as well as nonequilibrium descriptions of strongly interacting quantum many-body systems. It provides a rigorous mathematical basis for exploring the quantum mechanical basis of thermalization and decoherence. Mesoscopic systems with quantum processes, in particular, play a key role. The domain of nonequilibrium mesoscopic systems is flooded with open questions some of which are solved but still others have provisional or incomplete solutions. A correlation function encodes the correlation between the field operators at different times and is an inference to all thermodynamic quantities. The beauty of the Keldysh time path is the definition of the Green's function, which plays an analogous role in the nonequilibrium formalism similar to the role the causal Green's function plays in the equilibrium theory.