Green’s connected function

Green s connected functions and the vertex functions play an important role in field theory. 

Green s connected function Gc, with connected graphs to correspond to it, exhibits interesting and important properties. 

It is specially proved in the field theory that if F J is the generating functional of Green s connected functions... 

It follows from the above equations that if two coordinates of a line are distant from each other. Green s connected functions tend to zero. 

The functions in the right-hand sides are the cumulants of the statistical distribution of the system fields. In their turn, Green s functions are expressed via Green s connected ones by means of... 

In this section we explicitly demonstrate how the antiresonant line shapes characteristic of configuration interaction between a discrete BO state and BO continuum can be obtained from the Green s function formalism. We restrict attention to the case of one discrete BO state interacting with one BO continuum. We shall assume that the ground state is connected to both the discrete BO state and the BO continuum by nonvanishing dipole matrix elements. 

It would be very useful if there were an equation of motion for the electron density or for the reduced density matrices corresponding to a pure state of a many-electron system from which these quantities could directly be determined. Unfortunately this is not the case. The quantity closest to the one-matrix that has an equation of motion from which it can be determined by well-defined approximations is the one-electron Green s function or electron propagator. We explore its connection to the electron density in the next section. 

In summary, we have derived formulas for the Weyl-Titchmarsh m-function, where the imaginary part serves as a spectral function of the differential equation in question. Before we look at the full m-function, we will see how it works in connection with the spectral resolution of the associated Green s function... 

Figure 5.26 Diagrammatic representation of the SM e-ph interaction in the QD Green s function expansion. There are two types of diagrams the phonon Green s function connects two SM Green s functions separated by the QD Green s function (the diagram above the time line), and the phonon Green s function is located within the same semiconductor electron Green s function (two dashed lines below the time line).

Equation (5.13) can be arrived at by the following considerations. Recall that the spectral density D(E) of a given operator H is connected to its Green s function G( ) by the relation... 

The technique of combining CPA concepts with a continued fraction analysis of the Green s function should also be invaluable in connection with the study of disorder with augmented space techniques. Progress is expected when memory function methods are systematically used in the mentioned areas of investigation. 

Expression (13.80) is called WKBJ (ray theoretic) Green s function, because it is associated with the names of several physicists, G. Wentzel, H. A. Kramers, L. Brillouin, and H. Jeffreys, who independently introduced this approximation in connection with the solution of different physical problems (Morse and Feshbach, 1953). I would also recommend an excellent book by Bleistein et al. (2001), where the interested reader can find a more thorough mathematical analysis of the WKBJ approximation. 

It should finally be observed that, in the literature, the kernels of the Hamiltonian resolvent R(z) and the Liouvillian superresolvent R(z) are often referred to as Green s functions, and this means that resolvent techniques are often referred to as Green-function methods. Some of these methods are now so well developed and powerful that they are applied to the very large biomolecules in connection with the problem of the mechanism of chemical carcinogenesis.8... 

It can be seen that the Green s function connects the value of a wave function at point r and time t to its value at a point r and time t. In fact, we can get the wave function at time t from the wave function at a time t = 0 as [39] ... 

Laplace-de Gennes transform). This correspondence bet ween Green s functions and partition functions remains valid for their cumulants (also called connected parts). Thus, eqn (11.3.6) applies also to connected Green s functions and connected partition functions (the fact that they are connected is denoted by the absence of the index G). 

Moreover, it is clear that such a relation also implies that the connected Green s functions are the Laplace transforms of connected partition functions (without index G). We call it the Laplace-de Gennes transformation. 

The Green s functions can be expanded in powers of B and represented by diagrams. Connected diagrams correspond to connected Green s functions 30(ri,. . ., rp jp a) which are the cumulants of the general Green s... 

More generally, the ordered Green s functions are defined by diagrams (see Fig. 11.2) in which the open solid lines always connect r, to r2, r3 to r4,.. . r2N-1 to r2M Thus, for N < n the calculation does not depend on the indices but only on n. It is therefore trivial to generalize the ordered Green s function to the case N Ss n, by giving a diagrammatic definition of these functions. 

As always, the connected Green s functions are the cumulants of the Green s functions, and consequently their generating function bG B] is given by... 

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