Green’s Functions for Fermions

Abstract

For simplicity of notation we have so far in this book discussed only the path integrals and generating functions for scalar fields. All this formalism can also be easily generalized to vector fields which obey the Proca equation (4.26) and thus fulfill component by component the Klein-Gordon equation. For fermion fields, however, there is a problem. The main idea in using path integrals is to express quantum mechanical transition amplitudes by integrals over classical fields; the values of these fields at the discrete coordinate-sites were taken to be commuting numbers. Such a formalism can, however, not “know” about the Pauli principle. For example, with the formalism developed so far, a fermion could be propagated to a point in configuration space which is already occupied. In nature, however, this propagation is Pauli-forbidden.