The Feynman rules

We now anrilyzo tht. expansion of the product in powers of, 6c Elie individual terms of this expansion can be represented by Feynman diagrams for each polymer chain we draw a straight line, and we represent the iiiUu- 

Some labeled diagrams for a three-chaiu system 

We iutrodnee some simplifying notation. Wc call polymer liner the full line reprc senting a polymer. Interaction linc s gt nemlly kiiowm as Vertices. The endpoints of the polymer liiu s and tlic points where vertices are attached 

A priori the Sj range over a complicated domain, reflecting the fact that the r are constrained to volume Q. To overcome this problem we note that ultimately we are interested in the thermodynamic limit oc, so that tht linear size of the container L can be taken to be large compared 

Th( factors (47T ) cancel factors 4kiF) associated with the rj- integrals 

The integral (3.33) is evaluated by introducing the segment vectors Sj = Tj — 1 so = To- This yields 

We next carry through the r-integrations associated with two interacting segments. The relevant labeled substructure of a diagram is shown in Fig. 1.6. It yields 

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To describe the fully compactified model, with Euclidean coordinates, say Xi, restricted to segments of length Li (i = 1,2,. D) and the field tp(x) satisfying anti-periodic (bag model) boundary conditions, the Feynman rules should be modified following the Matsubara prescription... 

The organization of this chapter is as follows. In Sect. 5.1 we present the basic formalism and work out the Feynman rules for the grand canonical ensemble. Diagrammatic representations valid in the thermodynamic limit are derived for both thermodjmamic quantities and correlation functions. The proof of the Linked Cluster Theorem is given in Appendix A 5.1. Section 5.2... 

It is useful to modify the Feynman rules to account for the chemical potential. For grand canonical diagrams we close each polymer line by crosses (Fig. 5.1). and we add the following rule. 

The organization of this chapter is as follows. In Sect. 7.1 wo carefully define the continuous chain limit and we introduce the appropriate modification of the Feynman rules. We. then establish the two parameter scheme by dimensional analysis. Section 7.2 is devoted to the question whether the continuous chain limit exists. The analysis is presented on the diagrammatic level. It exploits the field theoretic representation, which also is derived on the level of diagrams. All the analysis is based on the cluster expansion. Extension to the loop expansion is not difficult, but will not be considered, since it is not needed in the sequel. 

We thus find the following modification of the Feynman rules as given in Sect. d.2. 

These rules axe identical to the Feynman rules of a special Euclidean field theory, extensively used in the theory of critical phenomena. (See, for instance [Aml84].) This formulation therefore is known as the field theoretic representation of polymer theory. We will elaborate on the relation to field theory in Appendix A 7.1. 

Here r = ri2 and En = En (l— 0), so that the Feynman rules for the integration over u) variables are assumed. All sums run over the complete Dirac spectrum for the electron in the field of the nucleus. The expressions for the counterterms... 

The standard approach to the calculation of the propagators (A.4, A.5) is perturbation theory with respect to the electron-electron coupling constant a = e /(hc) on the basis of the interaction picture. Technically this results in an expansion of expectation values of interacting field operators in powers of expectation values of the free (or noninteracting) field operators i o and Ag. The structure of this expansion can be summarised in a set of formal rules, the Feynman rules. For instance for the electron propagator one obtains ... 

By convention all the topologically similar vertices are given the same sign in the Feynman rules. The overall sign of a diagram has to be determined by comparing the order of the fermion operator in the diagram with their normal order in the S operator. 

The factor (—1) in (A2.4.5) does not come from the Feynman rules. It comes from comparing the order of the fermion operators underlying the two diagrams. Symbolically we have ... 

The Appendix has been much enlarged. It now contains a more detailed specification of the Feynman rules for electroweak theory and QCD, and a discussion of the relations between [Pg.532]

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