Feynman rules

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 methods illustrated here can be used to evaluate all quantities of interest in the excluded volume problem. Another example (A ) will be considered after we derive the general Feynman rules. 

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... 

Feynman rules for the Green function. In the Furry picture, in addition to the standard Feynman rules in the energy representation (see [24,13]), the following vertices and lines appear (we assume that the Coulomb gauge is used)... 

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 ... 

Transcription of the diagrammar into explicit equations using standard Feynman rules (within the notation chosen here) yields... 

According to Feynman rules of QED, this energy shift is given by... 

Whit these (Feynman) rules the connected expressions are calculated following the (the modified Wick rules) prescriptions of Section 3.3.2, here particularized as ... 

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