Keldysh Green functions

The Pauli matrices a (r) operate in the Nambu (Keldysh) space. The counting current I x) is to be found from the quantum kinetic equations [15] for the 4x4 matrix Keldysh-Green function G in the mesoscopic normal region of the interferometer confined between the reservoirs,... 

The calculation of the integrand in Eq. (6) is performed as follows. The Keldysh-Green function Gr(x) in the normal reservoir is traceless in the Keldysh space and therefore it can be expanded over the Pauli matrices r as... 

APPENDIX 14A SUPEROPERATOR EXPRESSIONS EOR THE KELDYSH GREEN FUNCTIONS... 

The effect of thermal pion fluctuations on the specific heat and the neutrino emissivity of neutron stars was discussed in [27, 28] together with other in-medium effects, see also reviews [29, 30], Neutron pair breaking and formation (PBF) neutrino process on the neutral current was studied in [31, 32] for the hadron matter. Also ref. [32] added the proton PBF process in the hadron matter and correlation processes, and ref. [33] included quark PBF processes in quark matter. PBF processes were studied by two different methods with the help of Bogolubov transformation for the fermion wave function [31, 33] and within Schwinger-Kadanoff-Baym-Keldysh formalism for nonequilibrium normal and anomalous fermion Green functions [32, 28, 29],... 

In this section, we describe our model, and give a brief, self-contained account on the equations of the non-equilibrium Green function formalism. This is closely related to the electron and particle-hole propagators, which have been at the heart of Jens electronic structure research [7,8]. For more detailed and more general analysis, see some of the many excellent references [9-15]. We restrict ourselves to the study of stationary transport, and work in energy representation. We assume the existence of a well-defined self-energy. The aim is to solve the Dyson and the Keldysh equations for the electronic Green functions ... 

Motivated by the growing interest in high-order correlation functions, we develop in the present Paper a systematic approach to full statistics of charge transport in Andreev interferometers. We adopt several simplifying assumptions, which enables us to present an analytical solution for the CGF and, without a loss of generality, to clearly demonstrate essential features of coherent effects in the current statistics in NS structures. Our approach is based on the extended Keldysh-Green technique [13, 14], in which the CGF is determined by the equation... 

In these review we consider different methods. The density matrix can be determined from the master equation. For Green functions the EOM method or Keldysh method can be applied. Traditionally, the density matrix is used in the case of very weak system-to-lead coupling, while the NGF methods are more successful in the description of strong and intermediate coupling to the leads. The convenience of one or other method is determined essentially by the type of interaction. Our aim is to combine the advantages of both methods. 

Now we are able to define contour or contour-ordered Green function - the useful tool of Keldysh diagrammatic technique. The definition is similar to the previous one... 

The problem can be generally resolved by using the EOM on the Schwinger-Keldysh time contour. Contour-ordered Green function is defined as... 

The lesser (retarded, advanced) Green function matrix of a nonequilibrium molecule G<(ll A> = GAjft A> can be found from the Dyson-Keldysh equations in the integral form... 

Kohn-Sham Hamiltonian within Keldysh Green s functions. 126... 

The operators P and obey the usual equal time anticommutation relations. The time-dependence of the field operators appearing here is due to the Heisenberg representation in the L-space. In view of the foregoing development which parallels the traditional Schrodinger quantum theory we may recast the above Green function in terms of the interaction representation in L-space. This leads to the appearance of the S-matrix defined only for real times. We will now indicate the connection of the above to the closed-time path formulation of Schwinger [27] and Keldysh [28] in H-space. Equation (82) can be explicitly... 

We will now develop the transport equations in L-space from the above Green functions. Following the Keldysh approach in //-space, the transport equations for non-equilibrium plasmas and radiation have been given by DuBois [29]. A similar transport equation for a system of ions may be found in Kwok [30], which is based on the Green function associated with ion positions. In a separate paper [31], we will derive the appropriate transport equations for the coupled system of electrons, ions, and electromagnetic fields. 

The electric conductance of a molecular junction is calculated by recasting the Keldysh formalism in Liouville space. Dyson equations for non-equilibrium many-body Green functions (NEGF) are derived directly in real (physical) time. The various NEGFs appear naturally in the theory as time-ordered products of superoperators, while the Keldysh forward/backward time loop is avoided. 

Following Keldysh, we shall rearrange the superoperator Green functions in a 2 X 2 matrix G,... 

Keldysh space proportional to the Nambu matrix Green s function g,... 

Keywords Density functional theory (DFT) Green s functions Keldysh non-equilibrium Green s functions (NEGF) linear combination of atomic orbitals (LCAO) tunnel junction metal-fullerene-metal junction density of states (DOS) transmission function Landauer formula renormalized molecular levels (RMLs) I-V curves. 

For equilibrium problems, the imaginary part of the retarded Green s function determines the charge density p [46]. For transport problems, p(r) can still be computed by Green s functions — the Keldysh non-equilibrium Green s functions [51]. The formula is [51] ... 

For such nonequilibrium processes, the direct mapping of the electron propagator methods to calculations of electric current becomes inapplicable because of the time-dependent nature of electric current in both phenomena. A time-dependent problem requires the further development of the theory of Green s functions to electron dynamics in which e-e correlation effects are taken into account. Such methodology already exists in physics in which many-body ideas have been developed for time-dependent problems. This theory is based on nonequilibrium Green s or Keldysh functions [2,5, 6, 40-46]. 

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