Keldysh space

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... 

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

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,... 

All three functions belong to the Keldysh space. For any function in the Keldysh space, we define the greater and the lesser functions on the real-time axis as... 

For retarded and advanced functions in the Keldysh space, we can write... 

The external potential is included in these equations. The self-energy is defined as a function or operator in the Keldysh space according to Eq. (20). obeys the Kubo-Martin-Schwinger boundary conditions (16). The ( )-projections of the Kadanoff-Baym equations can written as... 

The Keldysh function space is rather complicated because it has two realtime branches and one imaginary branch. Multiplication rules for operators with time arguments from different branches were proved by Langreth [41] and Wagner [48-50]. 

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. 

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