hypothesis. However self interaction never fades away and, besides many other effects, it causes a difference between the Lagrangian mass and the physical mass of the boson. This fact must be taken into account by rewriting the equation of motion as follows: This equation can be solved formally using the retarded Green's function of the Klein–Gordon operator : allowing us to split interaction from asymptotic behaviour. The solution is: The factor is a normalization factor that will come handy later, the field is a solution of the homogeneous equation associated with the equation of motion: and hence is a free field which describes an incoming unperturbed wave, while the last term of the solution gives the perturbation of the wave due to interaction. The field is indeed the in field we were seeking, as it describes the asymptotic behaviour of the interacting field as , though this statement will be made more precise later. It is a free scalar field so it can be expanded in plane waves: where: The inverse function for the coefficients in terms of the field can be easily obtained and put in the elegant form: where: The Fourier coefficients satisfy the algebra of creation and annihilation operators: and they can be used to build in states in the usual way: The relation between the interacting field and the in field is not very simple to use, and the presence of the retarded Green's function tempts us to write something like: implicitly making the assumption that all interactions become negligible when particles are far away from each other. Yet the current contains also self interactions like those producing the mass shift from to . These interactions do not fade away as particles drift apart, so much care must be used in establishing asymptotic relations between the interacting field and the in field. The correct prescription, as developed by Lehmann, Symanzik and Zimmermann, requires two normalizable states and , and a normalizable solution of the Klein–Gordon equation . With these pieces one can state a correct and useful but very weak asymptotic relation: The second member is indeed independent of time as can be shown by differentiating and remembering that both and satisfy the Klein–Gordon equation. With appropriate changes the same steps can be followed to construct an out field that builds out states. In particular the definition of the out field is: where is the advanced