Keldysh formalism

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 introduce the Keldysh formalism that allows for calculations of different quantum correlation functions necessary for various time-dependent problems. In our description, we follow the reviews of Danielewitcz [42] and van Leeuwen and Dahlen [46, 47]. We embark our discussion on the description of evolution of a system that is originally in statistical equilibrium. We are interested in an expectation value of an operator... 

Note that we incorporate the factor 2 for spin degeneracy, and we always include it without reference in this chapter. The Keldysh formalism leads to the following Keldysh-Ladanoff-Baym (KKB) equation ... 

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. 

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 ... 

In normal systems, the matrices G and / are traceless in the Keldysh space and therefore they can be expressed through 3-vectors with the components diagonal in the Nambu space, G = gr, I = It, where r is the vector of the matrices r, and g2 = 1. Since the lhs of Eq. (4) turns to zero in normal systems, the formal solution of Eq. (4) for the matrix current density IN in each segment of the wire can be easily obtained,... 

Note, that the expressions for the retarded and lesser functions, described above, can be obtained in a more formal way by the EOM method formulated on the Keldysh contour. 

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