It is not possible to obtain an analytical solution for the field equation with the field itself in the interaction term. A perturbation theory was developed to obtain approximate solutions step by step. The interaction between a charged fermion and the photon is in the form where A is the electromagnetic vector potential and is the field for the fermion, which appears on both side of the Green's function solution as shown in Eq.(8). An iteration procedure yields a sum of integrals as shown in the formula below:

where K is the Green's function. In this form the unknown field on the left hand side is now expressed in terms of all known quantities on the right hand side. The first term is the free field solution, and the integration is over all the space-time x', x'', x''', ... Note that each of the following term is multiplied by the power of e, from e1, to e2, ... Since e=1/137 for the electromagnetic interaction, computation on a few terms would be sufficient to obtain a result with acceptable accuracy.

Another formulism is the S-matrix theory, which was very popular many years ago. It is the transition probability expressed in an expansion as the result of the iteration procedure on the transition operator, which transforms the system from an initial state (at time negative infinity) to a final state (at time positive infinity) as shown in the formula below.

where HI involves the interaction fields (integrating over all space and multiplied by the coupling strength) and t1 > t2 > ... > tn-1. In this picture the fields obey the free field equations, the interaction enters via HI. It was thought that since we cannot measure the fields directly, so we should not talk about it, while we do measure S-matrix elements, so this is what we should be mindful about. It is now realized that analyzing the S-matrix alone is not sufficient, information on the quantum fields is also necessary. It can be shown that the various orders in the S-matrix are related to the petrubation expansion in Eq.(10).

Let us take the nucleon-pion system as an example of S-matrix application:

where Eq.(12) is the free field equation for the pion, and Eq.(13) is the free field equation for the nucleon (the Dirac equation). Expressing in terms of the field itself, it can be shown that the quantization rules in Eq.(4) become:

where {a,b} = ab + ba is the anticommunition expression, and the quantities on the right-hand side are the Green's functions for the pion and nucleon respectively. Since the interaction is go the nth order term in the S-matrix expansion Eq.(11) has the explicit form:

where the symbol N is the normal-order operator, which shifts all the creation operators to the left (to avoid infinite vacuum energy), while T is the time-order operator, which re-arranges the fields so that the one associated with later time is on the left (to take care of the integration limits in Eq.(11)). .