Green function retarded

Using Eqs. (D44) and (D50), the retarded self-energies for electron and phonon Green functions (retarded) coming from the electron-phonon coupling are obtained as... 

To keep the calculation tractable, we will only consider the weak-coupling limit Ti,T2 t), in this case the advanced (and retarded) Green functions do not depend on the Ts, and for the model defined by equation (6) they are given as... 

Once again, Heras [13] gave another formulation of Helmholtz theorem now based on Green s retarded function G = 8(f — ts — R/c)/R for a value B5(r5. ts) function of space and retarded time ts = t — R/c. Heras obtained the following equation ... 

The retarded single-particle matrix Green function GR(e) is determined by the equation... 

One can ask what is the advantage to use the more complex two-time Green functions instead of density matrices There are several reasons. First of all, NGF give, as we shall see below, a clear description of both density of states and distribution of particles over this states. Then, the equations of motion including interactions and the influence of environment can be obtained with the help of a diagrammatic technique, and (very important) all diagrammatic results of equilibrium theory can be easily incorporated. Retardation effects are conveniently taken into account by two-time Green functions. And,. .. finally, one can always go back to the density matrix when necessary. 

Now consider the simplest possible example - retarded Green function for free particles (fermions). 

Eqs. (394) prompts a way to include further many-particle effects into the Dyson equation, Eq. (394b), by replacing the Dyson-first-order retarded Green function G with the EOM G. Then one obtains already the correct results to describe CB while keeping the Hartree-like self-energy. 

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... 

To further proceed, let us now introduce two kinds of retarded thermal Green functions related to the central chain Gj (t) and to the backbones Pji t), respectively (taking ft = 1) ... 

We are interested in Gcc(E), i.e., in the retarded Green function of the central scattering region. Explicitly, from H(E) GR(E) = I, we have the following system ... 

In linear response theory the optical activity is obtained from the part of the generalized susceptibility involving the temporal correlations of the electric and magnetic polarization fields46,47. For a system such as a normal fluid, described by a statistical operator that is invariant under space and time translations, the appropriate retarded Green function is,... 

The retarded Green function is directly related to the generalized susceptibility... 

Expanding t and g in terms of Pauli matrices oc and the unit matrix we obtain four coupled one-dimensional integral equations for the components t (cp, ff) of the matrix t, which contain the energy integrated normal and anomalous retarded Green functions... 

The retarded and advanced Green functions can be expressed in terms of the electron propagator matrix that is supposed to be known from numerical calculations [55, 56]. Usually, it can be found, for example, from GAUSSIAN program [22] whose output provides all necessary information for the... 

To find retarded and advanced Green function matrices, it is necessary to... 

Key features of both the non-interacting and the interacting Fermi liquid theory can be illustrated through the retarded one-particle Green function, which can be defined from the following Fourier transform... 

Fyj ( )= 7 r[rfG (aj)r G (ffl)] is the transmission function, describing electron tunneling ability between lead-/ and lead-a via QDs, and w) is the Fermi distribution function in lead- a In the expression of, the retarded Green functions are written as... 

In developing the quantum amplitude for an optical process, it is necessary to determine the matrix elements of the time evolution operator, and to this end it is frequently expedient to invoke an expansion in terms of operators rather than the embedded time integrals of Eq. (28). The method of resolvent operators, which affords a framework for both perturbative and nonperturbative analysis [28-30], proceeds through the introduction of a retarded Green function, K+(t) = t/(f,O)0(f), together with its advanced counterpart, K (t) U(t,0)0( t),... 

Here D0 is the bare phonon Green function while 1, is given by Equ. (2). As usual, the phonon mediated electron-electron interaction is retarded in time. Following Ref.6) we express the renormalized phonon Green function D... 

This statement implies that not only the Coulomb interaction is included in Er and Exc but also the (retarded) Breit interaction. It thus points at the fact that a consistent and complete discussion of many-electron systems and consequently of RDFT must start from quantum electrodynamics (QED). RDFT necessruily has to reflect the various features of QED, both on the formal level and in the derivation of explicit functionals. The most important differences to the noiu-elativistic situation arise from the presence of infinite zero point energies and ultraviolet divergencies. In addition, finite vacuum corrections (vacuum polarization, Casimir energy) show up in both fundamental quantities of RDFT, the four current and the total energy. These issues have to be dealt with by a suitable renormalization procedure which ultimately relies on the renormalization of the vacuum Greens functions of QED. The first attempt to take... 

We shall make use of Eqs. (9) and (10) in discussing the retarded and advanced Green functions in Appendix 14D. Superoperator algebra was surveyed in Ref. [49]. 

The present model [37-39] ignores electron-electron interactions. These may be treated using the GW technique [56-58] formulated in terms of the superoperators and extended to non-equilibrium situations. All non-equilibrium observables can be obtained from a single generating functional in terms of left and right operators. The retarded (advance) Green function that describes the forward (backward) motion of the system particle can also be calculated in terms of the basic Green functions. Gap (see Appendix 14D). 

The Dyson equations for the retarded Green function (see Appendix 14D, Eq. (D48)) in frequency (energy) can be expressed in the matrix form as... 

In this appendix, we define the retarded and advance Green s functions and the corresponding self-energies and relate them to the basic Green functions and selfenergies obtained in Appendix 14C. The Liouville space retarded (Gr) and advance (GJ Green functions are defined as... 

Expansions in terms of continuum wave functions As is well known, the time evolution of the decaying wave solution given by Eq. (47) may also be calculated by expanding the retarded Green function in terms of the complete set of continuum wave functions of the problem, the so-called physical wave solutions r) to obtain... 

If the superposition of point-source solutions is determined by a Green function , then by analogy with Wheeler-Feynman absorber theory the total interaction results from a symmetrical combination of retarded and advanced Green functions. On dehning these Green functions to be compatible with space-time geometry (Riemann tensor) the interaction is shown to be consistent with Einstein s field equations. 

There is clear now that the Green function formalism do not automatically solve the initial Schrodinger problem, but replaces it with a more general one when also the causality of events counts. The passage from the retarded to advanced Green function equation can be easily made though the previously stipulated recipe. 

From the need the two expressions be equivalent we find the momentum-energy retarded free Green function as ... 

While noting this distance appears at the denominator of the space averaged (retarded) Green function, we can employ once more the asymptotic contribution ... 

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