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

Journal of physics. A, Proceedings of the Physical Society. General

A two-particle Green function self-energy expansion

R Lee

Published Jul 1, 1972• 7 citations
Publisher

Abstract

The two-particle self-energy operator is derived. The method is an extension of the Kadanoff and Baym method for the one-particle self-energy operator. The self-energy is then expanded to illustrate the similarities and differences with the one-particle case. An example is used to study the structure of the expansion. In conclusion a brief discussion of the one-particle method to derive the two-particle Green function is given.

DOI
10.1088/0305-4470/5/7/006

Related papers

Two-particle approximation for the quantum third virial coefficient: Comparison of two approaches

R. E. Johnson, Noel K. Pope, A. J. Barrett

Two distinct approaches have been followed to produce an approximate expression for the third virial coefficient of a quantum system in terms of the pertinent two-particle quantities (standard T matrix elements and bound state eigenvalues and eigenstates): One approach is based on the T approximation for the self-energy in the theory of temperature Green functions, and the other uses the diagrams which represent the various terms in the multiple-scattering expansion of the three-particle T operator. It is shown that the result given by the former method is reproduced exactly by the contribution of a small number of selected diagrams. The Green function method requires the evaluation of the Kadanoff and Baym generalized T matrix to one order in the activity higher than its familiar low-density limit. The common result can be written entirely in terms of the lowest order self-energy and should give a reasonable approximation to the third virial coefficient for systems with short-range forces.

Journal of Mathematical Physics•Apr 1, 1974•1 citation

Extended Two-Particle Green’s Functions and Optical Potentials for Two-Particle Scattering by Many-Body Targets

Joachim Brand, Lorenz S. Cederbaum

An extension of the fermionic particleparticle propagator is presented that possesses similar algebraic properties to the single-particle Green’s function. In particular, this extended two-particle Green’s function satisfies Dyson’s equation and its self energy has the same analytic structure as the the self energy of the single-particle Green’s function. For the case of a system interacting with one-particle potentials only, the two-particle self energy takes on a particularly simple form, just like the common self energy does. The new two-particle self energy also serves as a well behaved optical potential for the elastic scattering of a two-particle projectile by a many-body target. Due to its analytic structure, the two-particle self energy avoids divergences that appear with effective potentials derived by other means. 1996 Academic Press, Inc. I.

•Jan 1, 1996•7 citations

Green functions in the renormalized many-body perturbation theory

V. Janiš

I review the way the many-body Green functions are used to renormalize the perturbation theory of correlated fermions. The Green functions are introduced to implement systematically dynamical corrections to the static mean-field theory. The renormalizations enter the perturbation theory via self-consistent evaluations of one-particle and two-particle Green functions. The one-particle self-consistency is discussed within the Baym-Kadanoff construction. We point to an inherent dichotomy of the Baym-Kadanoff approach in the relation between one and two-particle Green functions. They are the Schwinger-Dyson equation, directly derived from the generating Luttinger-Ward functional, and the Ward identity. The latter imposes a severe restriction on the two-particle irreducible vertex and is not guaranteed in the Baym-Kadanoff approach. No approximate theory is capable to obey both relations. I further discuss the two-particle approach with two-particle renormalizations that offers a way to reconcile the Schwinger-Dyson equation and the Ward identity. The two-particle irreducible vertex from the critical Bethe-Salpeter equation becomes a generating functional in the two-particle renormalization of the perturbation theory. The irreducible vertex is used in the Ward identity to determine the order parameter beyond the critical point and in the Bethe-Salpeter equation representing the vertex of the Schwinger-Dyson equation to determine the spectral self-energy in such a way that the thermodynamic critical behavior from both equations is qualitatively the same. Finally, I discuss a static mean-field-like approximation with a renormalized effective interaction determined self-consistently that leads to a qualitatively correct description of strong electron correlations in impurity and lattice models.

arXiv:2109.04231•Sep 9, 2021•0 citations

Proper incorporation of the self-adjoint extension method to the Green function formalism: one-dimensional -function potential case

D K Park

The one-dimensional -function potential is discussed in the framework of the Green function formalism without invoking perturbation expansion. It is shown that the energy-dependent Green function for this case is crucially dependent on the boundary conditions which are provided by the self-adjoint extension method. The most general Green function which contains four real self-adjoint extension parameters is constructed. The relation between the bare coupling constant and the self-adjoint extension parameter is also derived.

arXiv:hep-th/9512097•Journal of Physics A Mathematical and General•Oct 7, 1996•19 citations

Perturbation Expansion for Real-Time Green's Functions

Richard A. Craig

The development of the time-translation operators in a matrix element of an arbitrary operator is examined. It is noted that we may interpret time as evolving from some remotely early time (t0) to a time in the far future (t∝) and then back to (t0). Using this interpretation, a perturbation expansion is developed for Green's functions defined along this path and a separation of the two-particle interaction terms into self-energy parts and single-particle Green's function terms is justified for quantities on this path. A connection is established between the real-time Green's functions and the Green's function defined along the path, thereby yielding a perturbation expansion for the real-time functions and a justification of the separation of the interaction terms in the equations of motion for the real-time quantities. The transport equations of Kadanoff and Baym are derived without resorting to an analytic continuation from imaginary times and without the correction terms of Fujita.

Journal of Mathematical Physics•Apr 1, 1968•178 citations

Green's functions and self-energy approximations

Allan Griffin, Tetsuro Nikuni, E. Zaremba

In Chapter 3, we introduced a simple but reasonable approximation for the nonequilibrium dynamics of a Bose-condensed gas based on a generalized GP equation coupled to a kinetic equation. In Chapters 4–7, we turn to the question of how to derive such coupled equations for the condensate and noncondensate components in a way that gives a deeper understanding of the ZNG theory. Chapters 4–7 involve an introduction to Green's functions and field theoretic techniques for nonequilibrium problems. These provide the natural language and formalism to deal with the many subtle aspects of a Bose-condensed gas at finite temperatures. These four chapters are fairly technical. This chapter is mainly based on Kadanoff and Baym (1962) and Imamović-Tomasović (2001). Readers who are not interested in these questions can go straight to Chapter 8, which begins the discussion of applications of the ZNG coupled equations given in Chapter 3. Overview of Green's function approach To derive a microscopic theory of the nonequilibrium behaviour of a dilute weakly interacting Bose-condensed gas at finite temperatures, there are several different approaches available in the literature. We will use the wellknown Kadanoff–Baym (KB) nonequilibrium Green's function method. The generalization of this formalism to a Bose-condensed system was first considered by Kane and Kadanoff (1965), whose goal was to derive the Landau two-fluid hydrodynamic equations for a system with a Bose broken symmetry. The general problem consists of how to calculate the nonequilibrium response of a system induced by an external (space- and time-dependent) perturbation. In response to such an external perturbation, many interesting physical phenomena appear, including the excitation of collective modes and various transport processes.

Cambridge University Press eBooks•Feb 19, 2009•0 citations
The two-particle self-energy operator is derived. The method is an extension of the Kadanoff and Baym method for the one-particle self-energy operator. The self-energy is then expanded to illustrate the similarities and differences with the one-particle case. An example is used to study the structure of the expansion. In conclusion a brief discussion of the one-particle method to derive the two-particle Green function is given.
Two distinct approaches have been followed to produce an approximate expression for the third virial coefficient of a quantum system in terms of the pertinent two-particle quantities (standard T matrix elements and bound state eigenvalues and eigenstates): One approach is based on the T approximation for the self-energy in the theory of temperature Green functions, and the other uses the diagrams which represent the various terms in the multiple-scattering expansion of the three-particle T operator. It is shown that the result given by the former method is reproduced exactly by the contribution of a small number of selected diagrams. The Green function method requires the evaluation of the Kadanoff and Baym generalized T matrix to one order in the activity higher than its familiar low-density limit. The common result can be written entirely in terms of the lowest order self-energy and should give a reasonable approximation to the third virial coefficient for systems with short-range forces.
An extension of the fermionic particleparticle propagator is presented that possesses similar algebraic properties to the single-particle Green’s function. In particular, this extended two-particle Green’s function satisfies Dyson’s equation and its self energy has the same analytic structure as the the self energy of the single-particle Green’s function. For the case of a system interacting with one-particle potentials only, the two-particle self energy takes on a particularly simple form, just like the common self energy does. The new two-particle self energy also serves as a well behaved optical potential for the elastic scattering of a two-particle projectile by a many-body target. Due to its analytic structure, the two-particle self energy avoids divergences that appear with effective potentials derived by other means. 1996 Academic Press, Inc. I.
I review the way the many-body Green functions are used to renormalize the perturbation theory of correlated fermions. The Green functions are introduced to implement systematically dynamical corrections to the static mean-field theory. The renormalizations enter the perturbation theory via self-consistent evaluations of one-particle and two-particle Green functions. The one-particle self-consistency is discussed within the Baym-Kadanoff construction. We point to an inherent dichotomy of the Baym-Kadanoff approach in the relation between one and two-particle Green functions. They are the Schwinger-Dyson equation, directly derived from the generating Luttinger-Ward functional, and the Ward identity. The latter imposes a severe restriction on the two-particle irreducible vertex and is not guaranteed in the Baym-Kadanoff approach. No approximate theory is capable to obey both relations. I further discuss the two-particle approach with two-particle renormalizations that offers a way to reconcile the Schwinger-Dyson equation and the Ward identity. The two-particle irreducible vertex from the critical Bethe-Salpeter equation becomes a generating functional in the two-particle renormalization of the perturbation theory. The irreducible vertex is used in the Ward identity to determine the order parameter beyond the critical point and in the Bethe-Salpeter equation representing the vertex of the Schwinger-Dyson equation to determine the spectral self-energy in such a way that the thermodynamic critical behavior from both equations is qualitatively the same. Finally, I discuss a static mean-field-like approximation with a renormalized effective interaction determined self-consistently that leads to a qualitatively correct description of strong electron correlations in impurity and lattice models.
The one-dimensional -function potential is discussed in the framework of the Green function formalism without invoking perturbation expansion. It is shown that the energy-dependent Green function for this case is crucially dependent on the boundary conditions which are provided by the self-adjoint extension method. The most general Green function which contains four real self-adjoint extension parameters is constructed. The relation between the bare coupling constant and the self-adjoint extension parameter is also derived.
The development of the time-translation operators in a matrix element of an arbitrary operator is examined. It is noted that we may interpret time as evolving from some remotely early time (t0) to a time in the far future (t∝) and then back to (t0). Using this interpretation, a perturbation expansion is developed for Green's functions defined along this path and a separation of the two-particle interaction terms into self-energy parts and single-particle Green's function terms is justified for quantities on this path. A connection is established between the real-time Green's functions and the Green's function defined along the path, thereby yielding a perturbation expansion for the real-time functions and a justification of the separation of the interaction terms in the equations of motion for the real-time quantities. The transport equations of Kadanoff and Baym are derived without resorting to an analytic continuation from imaginary times and without the correction terms of Fujita.
In Chapter 3, we introduced a simple but reasonable approximation for the nonequilibrium dynamics of a Bose-condensed gas based on a generalized GP equation coupled to a kinetic equation. In Chapters 4–7, we turn to the question of how to derive such coupled equations for the condensate and noncondensate components in a way that gives a deeper understanding of the ZNG theory. Chapters 4–7 involve an introduction to Green's functions and field theoretic techniques for nonequilibrium problems. These provide the natural language and formalism to deal with the many subtle aspects of a Bose-condensed gas at finite temperatures. These four chapters are fairly technical. This chapter is mainly based on Kadanoff and Baym (1962) and Imamović-Tomasović (2001). Readers who are not interested in these questions can go straight to Chapter 8, which begins the discussion of applications of the ZNG coupled equations given in Chapter 3. Overview of Green's function approach To derive a microscopic theory of the nonequilibrium behaviour of a dilute weakly interacting Bose-condensed gas at finite temperatures, there are several different approaches available in the literature. We will use the wellknown Kadanoff–Baym (KB) nonequilibrium Green's function method. The generalization of this formalism to a Bose-condensed system was first considered by Kane and Kadanoff (1965), whose goal was to derive the Landau two-fluid hydrodynamic equations for a system with a Bose broken symmetry. The general problem consists of how to calculate the nonequilibrium response of a system induced by an external (space- and time-dependent) perturbation. In response to such an external perturbation, many interesting physical phenomena appear, including the excitation of collective modes and various transport processes.