Graph Scheme for the Fermion Green's Function Based on the Functional Integral

摘要

A recently introduced new diagram technique for the evaluation of Green's function is extended to the case of Fermi fields. A Hilbert space of anticommuting ''classical'' variables (so-called Grassmann variables) with involution has been used. The 2-point function is a certain functional integral over this space. The basic idea is to expand the exponential of the derivative part of the Lagrangean. The resulting sum is treated within a lattice space formalism. The full interaction is contained in some trivial integrals over Grassmann variables. All infinite dimensional integrals are carried out explicitly. Although the mathematical manipulations are quite different from the usual ones the resulting graphs are very similar to those of the Bose case. The full graph scheme may be summed explicitly in the case of the free Fermion. A generalization of the graph scheme to n-point functions is straightforward. At a later stage we hope to treat the Thirring model and the interaction of several fields. The formalism is an alternative to the expansion of the exponential of the coupling term which would yield the old-fashioned perturbation theory. (Atomindex citation 09:392719)