Relativistische funktionale Methode zur Behandlung stark verkoppelter Fermionenfelder†

Abstract

A practical method for the calculation of GREEN’S functions from the Lagrangian of a fermionfield is developed (in close analogy to 3), which is working in a LORENTZ-invariant manner and allows to treat also strong couplings (or nonlinearities). The essential point is the use of the values for functional integrals over the fermion-field calculated in lattice space. It is considered, how the anticommutativity of the fermion-field appears in the result. There are two controls in our calculations: First the result of the integration must fulfill some functional differential equations. Secondly the result should coincide in one trivial, well-known example (free field) with the well-known exact result. Our result fits both controls. Explicitly we present here only the general aspects of the problem and apply them (in order to illustrate and to controle the scheme) to the calculation of the 2-point-function of a free fermion-field, in which the mass-term has been split artificially into two parts, one of them is handled together with the free Lagrangian, the other one is treated as an “interaction”. The use of certain graphs also here proves to be very useful. The application to more interesting cases is in preparation.