Elimination of Ghosts in Propagators

Within the general framework of perturbation theory a method for calculating modified propagators in terms of proper Feynman diagrams is derived. This method differs from previous approaches in that one insists that the propagator have the correct analytical behavior as a function of $p^2$. As a result one gets an expression for the propagator which is similar to a conventional term-by-term perturbation theory expansion except that it is only necessary to consider proper diagrams and that the iteration of the proper diagrams is represented by a damping factor. As an example, the meson propagator for a pseudoscalar meson coupled to nucleons with a pseudoscalar coupling is approximated by considering only the lowest order proper diagram, a nucleon-antinucleon bubble. The resulting expression for the propagator has the following interesting properties: (1) by construction it has the proper analytical behavior as a function of $p^2$, (2) the result has a singularity at $g^2=0\mathrm{}$ when considered as a function of $g^2$, and (3) the wave function renormalization is finite. These three properties are intimately connected and when this connection is realized it is easy to understand why the usual methods of expressing propagators in terms of proper Feynman diagrams leads to ghosts. It is the purpose of this paper to understand this connection and to indicate how it is possible to take into account consistently the iteration of proper Feynman diagrams without ever having ghosts appear. It is also found that an asymptotic expansion valid in the region $g^2=0\mathrm{}$ is possible and that this asymptotic expansion is identical with the perturbation theory series.