MASS OPERATOR OF THE INTERACTING ELECTRONS IN FORMALISM OF PERMUTATION GROUP THEORY

Double permutation (DP) method is developed here for designing Feynman diagrams for mass operator (MO) of interacting electrons in high orders of perturbation theory (PT). The derived expression allows finding the Young diagrams for the class of permutations corresponding to disconnect Feynman diagrams. The classification of DPs, carried out before, allows to identify the permutations corresponding to disconnected, singly connected (improper) diagrams and to derive expressions for intolerant cycles of permutations. Ordering the nonprimed digits in natural order in the cycles of DP, we avoid the permutations, corresponding to the Feynman diagrams of the same topology because of other numbering of nodes. Thus, the numbers of considered permutations is sufficiently reduced: (from 24 to 6 and from 720 to 42) in the second and the third orders of PT. All 414 expressions (diagrams) for MO in the fourth order of PT were designed using this method. The use of group theory allows us to conclude that no more Feynman diagrams can be designed. The developed method can be used as algorithm for Feynman diagrams designing for MO of interacting electrons (one sort fermions) in higher orders of PT.