Particle lines

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. 

Figure 1.16 Imaging of the folding and stretching of particle lines when an electrokinetic perturbation is applied. The bright line indicates the interface between the particle solution and pure water [48] (by courtesy of Springer-Verlag).

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...

I ECE peak (experimental), 2 column with porous particles lines experimental, points calculated... 

The moments have physical meaning. The zeroth order moment (iq is the total number of particles per unit volume (or mass, depending on the basis). The first-order moment fii is the total length of the particles per unit volume, with the particles lined up along the characteristic length. The second-order moment is proportional to the total surface area, and the third-order moment is proportional to the total volume. Many physical characteristics of the particles such as the number-mean crystal size, weight-mean crystal size, the variance of the distribution function, and the coefficient of variation also can be represented in terms of the lower order moments of the distribution. 

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,... 

Each pair of unique, external hole or particle lines introduces a permutation function, P(pq) (as defined earlier in Eq. [154]), to ensure antisymmetry of the final expression. 

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.)... 

Fig. 5. (a) External hole line, (b) external particle line, (c) internal hole line, and (d) internal particle line. 

If two particle lines begin at the same vertex and end at the same vertex, cp = 2. 

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. 

Upward arrows represent particle lines and downward hole lines. 

Fig. 5.21. Feynman graphs for the first-order correction to the energy (a) shows the basic interaction, while (b), (c), (d) and (e) show different matrix elements. Lines drawn upwards and labelled k and t correspond to particle lines or occupied excited states, while lines drawn downwards represent a hole or the absence of an electron from the state a (after H.P. Kelly [241]).

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