Diagrams particle line

The ultraviolet divergence is generated by the diagrams with insertions of two anomalous magnetic moments in the heavy particle line. This should be expected since quantum electrodynamics of elementary particles with nonvanishing anomalous magnetic moments is nonrenormalizable. 

We see that the whole effect of the infinite summation (91) appeared in the so-called denominator shift [compare expressions (81) and (95)]. The denominator shift just presented, corresponds to putting the interaction line (91) between two particle lines. For this reason we talk about the diagonal particle-particle ladder diagram. 

Figure 1 Diagram elements (a) one-electron operator, (b) two-electron operator, (c) particle line, (d) hole line...

Label all directed lines with appropriate indices. By the convention we have used so far, hole lines would be labeled with i, j, k, I,. .. and particle lines with a, b, c, d,. . . . Therefore, for the diagram abovewe label the... 

Therefore, the two energy diagrams are equivalent, because the two hole lines and the two particle lines from the T2 diagram both connect to the same diagram fragment ... 

Unlike the diagram in Eq. [165], this diagram contains a pair of equivalent vertices since both Tj fragments are connected to the same interaction line in exactly the same manner (each by a hole line and a particle line), a prefactor of Vi is multiplied into the final expression. Generally speaking, if there are n equivalent vertices in the diagram, they contribute a prefactor of Hn to the final expression. 

However, the first two of these fragments can connect to the T diagram in only one index—via either a single hole line or particle line—thus leaving an additional line extending below the Tj interaction line in the final diagram, for example,... 

Similarly, the external particle lines in the rightmost diagram must be permuted in its algebraic expression ... 

Careful inspection, however, reveals that the diagrams are equivalent because one can be produced from the other by permutation of the hole or particle lines on the T2 fragment. (This equivalence can also be proven algebraically, and the reader is encouraged to carry this analysis out independently.)... 

If simultaneous interchange of the particle lines and the hole lines yields a topologically equivalent diagram and the hole and particle lines are not independently interchangeable, cPh = 2. 

Now the energies of the particle lines linked through a do not vary during this interaction, whence the associated propagators exp (— , ,.) remain unchanged. Consequently, the time 4 at which a takes place only appears in the limits of the integration over the ordered times. Let us then consider the family of the diagrams... 

The basic elements of the Brandow diagrams are shown in Figure 3. These are a one-electron matrix element, a two-electron matrix element, a particle line represented by a line with an upward directed arrow and a hole line represented by a line with a downward directed arrow. Particle lines represent the particle lines created above the Fermi level when an electron is excited whilst hole lines represent the hole which are simultaneously created below the Fermi level. A time-dependent physical interpretation of the diagram may be given. An example of such an interpretation is given in Figure 4. 

Figure 3 Basic elements of the diagrams introduced by Brandow (a) particle line (b) hole line (c) one-electron interaction (d) two-electron interaction...

Restricting f to only some simple fermion-like products of electron field operators will generate only certain types of diagrams that can then be summed to all orders. For instance, self-energy diagrams in third order of the ring and ladder types, which can easily be generalized in any order. It is notable that between consecutive interaction lines, there only occur one hole line and two particle lines or vice versa [see Fig. 9.1]. 

Exercise 6.1 Write down and evaluate all fifth-order diagrams that have the property that an imaginary horizontal line crosses only one hole and one particle line. Show that the sum of such diagrams is... 

We now generalize our previous development to obtain a diagrammatic representation of RS perturbation theory as applied to an N-state system. Consider the problem of finding the perturbation expansion for the lowest eigenvalue of such a system. Here we still have only one hole state, 1>, but there are now N - 1 particle states n>, n = 2, 3,..., AT. We draw the same set of diagrams as before. However, now we can label the particle lines with any index n. For example, the diagram... 

These expressions are identical to our previous results for the second- and third-order energies (Eqs. (6.12) and (6.15)) when i = 1. What if we want the perturbation expansion for some state i, which is not necessarily the lowest What do the diagrams look like One can easily verify that we get the same answers as before if we label our hole lines by the index i and the particle lines by the indices m, n, k,..., which can take on the values 1,2,..., i - 1, I + 1,..., N. Thus we now have a complete diagrammatic representation of RS perturbation theory, which is applicable to any perturbation and any zeroth-order state. 

In Section 62, we introduced a completely general diagrammatic repre sentation of RS perturbation theory. To adapt this to handle orbital perturbations we take the downward and upwaM lines to represent hole and particle spin orbitals, respectively, and the dots to correspond to the one-particle perturbation v. Ilien we draw the same set of diagrams as before, labeling the hole lines by indices a,b,...and the particle lines by indices r, s, Thus we have... 

Finally, we mention that if one were to sum all double excitation diagrams (an imaginary horizontal line crosses only two hole and two particle lines for such diagrams) one would obtain the doubly excited MBPT (D-MBPT(oo)) of Bartlett and coworkers. This approximation is equivalent to the linear CCA discussed in Subsection 5.2.3. Thus, the appropriate correlation energy is (see Eq. (5.65))... 

If the interaction term is small, in the sense that it is proportional to a small coupling constant, a perturbative approach may work. The successive terms are usually shown graphically as Feynman diagrams—a line representing the firee propagation of a particle, a vertex the interaction between particles, the structure of which is controlled by the form of the interaction term in the Lagrangian. 

Liquidus is the solubility curve for liquid particle. So in our Interpretation, the Hquidus curve is in a temperature-concentration diagram, the line connecting the temperatures at which freezing is just started for various compositions of a starting Hquid phase. ... 

Figure 6 The two nonfactorizable Q-term ri2 amplitude (Brandow-type) diagrams that involve five-electron integrals. Note the cyclic connection through tour double-arrowed particle lines of four R- and, g-type operators, These diagrams vanish by virtue of the standard approximation...

Particle lines are directed from left to right. Hole lines go from right to left. In the Goldstone diagrammatic convention, the diagrams... 

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