Dyson-like equation

The single particle propagator [20] G(x,x E) is an energy-dependent (or time-dependent) function of two compound space spin variables. It satisfies the Dyson-like equation... 

Keywords Field theory, ionic systems, local density fluctuations, Dyson-like equation,... 

In the next section we first introduce the main ingredients of a FT by mimicking as far as possible what is done in QFT. To be illustrative, in Sec. 3 we show how it is possible to derive the virial expression for the pressure in a FT. In Sec. 4 we show that we have at our disposal some relations between field-field correlation functions that are obtained from symmetry arguments or from the fact that fields are dummy variables. In Sec. 5 on two examples, we derive some results starting from Dyson-like equations. Finally, in Sec. 6, leaving the purely microscopic level we introduce a model showing the existence of a demixion transition at a metal-solution interface using a mean field approximation. Brief conclusions are presented in Sec. 7. 

We consider the case of a 1 1 electrolyte near a hard wall, the solvent is a dielectric continuum that fills all the space and therefore there is no-image potential. Here we have to introduce two fields pi(r) with i I or — or a combination of them such as the density s(r) = p+(r) + p-(r) and the charge density q(r) = p+(r)—p (r). It is easy to write the Dyson-like equations [17] for this system. If the interaction potential is only the Coulomb potential, we have,... 

It is also interesting to investigate how the introduction of an impurity on a site influences the scattering path operator. This means that t goes to t on one site. The result [39] is that r obeys a Dyson-like equation ... 

The size of the f a) subspace is quite large for reasonable basis sets. For N electrons and a spin orbital basis of rank K, there are K — N) K — 2)N/2 elements in the orthogonal complement space. The solution of the Dyson-like equation would require the inversion of a matrix of this dimension for every value of the energy parameter. This approximation of the self-energy first introduced by J. Linderberg and Y. Ohrn,, has been discussed by G. D. Purvis III, and Y. Ohrn, and J. Schirmer and L. S. Cederbaum . A simplified version of this approach is the so-called Diagonal 2p-h TDA , which becomes... 

The self-energy can be obtained from a self-consistent set of Dyson-like equations known as Hedin s equations ... 

It is productive to calculate the sp propagator from a Schrddinger-like equation. This is accomplished by relating the propagation of the dressed particle to propagation in a simple potential well in terms of the Dyson equation. 

Alternatively, by means of the Dyson equation and the second-order truncation approximation, taking Hartree-like self-energies r = U ns) (= E M), we can also get the retarded GFs as follows... 

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. 

Closing Section 3.1.1, we should like to derive the inverse Dyson equation in the diagonal approximation (equation 15), because this is not known by all chemists. One can start from the one-particle Green s matrix equation (Dyson... 

We also want to mention that a Dyson-equation approach for propagators like the polarisation and the particle-pcurticle propagator has been formulated and used to derive a self-consistent extension of the RPA, also called cluster-Hartree-Fock approximation, that has been applied in the fields of plasma and nuclear physics [12-14], This formalism, however, has similar problems like Feshbach s theory and does not yield a universal, well-behaved optical potential because the two-particle space has to be restricted in order to make the approach well-defined [14]. 

The connection between the appealing and useful optical potentials and GFs is not restricted to the one-particle GF. Very recently, an exact optical potential for the scattering of a projectile consisting of two particles has been derived. For this purpose it has been necessary to extend the two-particle GF in such a manner that it obeys a Dyson equation similar to that in equation (8). The extension of theory to describe more complicated projectiles via higher members of the GF hierarchy seems likely. 

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