Renormalized perturbation series

The analytic properties of A (z) can be studied conveniently by using the renormalized perturbation series (RPS) expansion (Anderson (1958) Watson (1957)) for this quantity,... 

It is a statistical-mechanical theory of solutions to express the solvation free energy as a functional of distribution functions. Traditionally, the theory of solutions is formulated with a diagrammatic approach [13], in which an approximation is provided in a two-step procedure. In the first step, the free energy and/or distribution function is expanded with respect to the solute-solvent interaction potential function or its related function as an infinite, perturbation series. In the second step, a renormalization scheme is applied a set of functions are defined through partial summation of the series and are employed for substitution to make the infinite series more tractable. An approximation is typically introduced by neglecting diagrams of ill character. 

However, very interesting results were obtained by applying analytic renormalization methods to perturbation series. These techniques initiated by Wilson himself, were developed by many physicists, and, among them, Brezin, Le Guillou, and Zinn-Justin played an important role. 

The renormalization constants Zj, Z2, and 8m have to be understood as functions of the finite physical charge e and mass m of the electrons, which will have to be constructed order by order in the perturbation series. In other words The original fields and parameters in mren are no longer interpreted as the correct physical fields and parameters, but rather as bare, unrenormalized quantities. 

The diagrammatic representation of the centroid density enhances one s ability to approximately evaluate the full perturbation series [3]. For example, one can focus on a class of diagrams with the same topological characteristics. The sum of such a class results in a compact analytical expression that includes infinite terms in the summation. A very useful technique in such cases is the renormalization of diagrams [57,58]. This procedure can be applied to the vertices to define the effective potential theory diagrammatically [3, 21-23] and, in doing so, an accurate approximation to the centroid density [3]. 

As one approaches the critical dimension (dc = 4 in the case of ( -theory and dc = 6 for 0 -theory), from below, the terms in perturbation series for vertex functions develop poles m e = dc — d, which correspond to logarithmic ultraviolet divergences. In the dimensional renormalization scheme, the coupling constants of the given model Aq and renormalization factors and Z 2 are written as series in the renormalized couplings A, which for k couplings are defined as follows ... 

Figure 5.6. Diagrammatic representation of the perturbation series of the renormalized unperturbed distribution function Go - notched line is Go, wavy lines are short subchains (Oono and Freed, 1981a) [Reprinted with permiawiion from Y.Oono, K.K.Kreed. J. Chem. Phys. 75 (1981) 993-1008. Copyright 1981 American Institute of Physics]...

Thus, with the aid of the renormalized distribution function Go (see Figure 5.6) and the renormalization parameter R.,(v2) (see Figure 5.10), the perturbation series (see Figure 5.5) is representable as the diagrams shown in Figure 5.11. 

One final approach to the development of a regular perturbation series is worth examining, and that is the renormalization perturbation theory of Sadlej et al. (1994, 1995). This approach follows the line of direct perturbation theory, but the formal perturbation is different. Writing the Dirac equation with 1 /c extracted from the small component according to (17.60) and replacing l/c2 in the metric with we have... 

In order to prepare the discussion of the relativistic generalization of the HK-theorem in Section 3 we finally consider the renormalization procedure for inhomogeneous systems. As the underlying renormalization program of vacuum QED is formulated within a perturbative framework (see Appendix B) we assume the perturbing potential to be sufficiently weak to allow a power series expansion of all relevant quantities with respect to V. In particular, this allows an explicit derivation of the counterterms required for the field theoretical version of the KS equations, i.e. for the four current and kinetic energy of noninteracting particles. 

Equations (87) and (88) represent cluster series for si and g in which the perturbation potential has been eliminated in favor of a renormalized potential The size of the individual graphs, and hence the rate of convergence of the series, depends on the value of the density, the range and strength of the ho bond, and the range and strength of the bond. 

The renormalized terms of the interetctioii R, vt) arc obtained from the row of diagrams (see Equation 27), which must have only two interaction points if all the wavy lines contract into a point (Figure 5.10). In this scaling procedure of structural elements Go and the new renormalized series of perturbation takes the form... 

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