In the previous chapter, we have described free particles and fields in terms of Lagrangian, equations of motion, their solutions and quantization. However, in the physical world, particles and fields are visualized by their interactions with other particles and fields or among themselves. For example, the simple processes of Compton scattering, photoelectric effect, and Coulomb scattering involving photons and electrons, are well known. All the known elementary particles interact with each other through the four fundamental interactions, that is, electromagnetic, weak, strong, and gravitational, through the exchange of gauge fields. Since the particles themselves can be described in terms of fields, all the physical processes governed by the four fundamental interactions are examples of various types of fields in interaction with each other including self interaction, allowed by the general principles of physics. These interactions are described by an interaction Lagrangian Lint(x), to be included along with the free Lagrangian Lfree(x), described in Chapter 2 for a quantum description of the evolution of physical systems. The interaction Lagrangians can be obtained by using the symmetry properties of the physical system defined by certain transformations called local gauge transformations and imposing the invariance of the free Lagrangian under these transformations. These will be discussed in some detail in Chapter 8, in the case of electromagnetic, weak, and strong interactions of scalar, vector, and spin particles.
In this chapter, we give some simple examples of interaction Lagrangians involving spin 0, spin and spin 1 particles and illustrate the general principles to write them. We use the example of electromagnetic interaction to demonstrate the general method of the relativistic perturbation theory to find out the solution of the equations of motion of fields in the presence of the interaction Lagrangian Lint(x). It is assumed that the strength of the interaction Lagrangian can be quantified by a parameter which is small, so that perturbation theory can be applied. This is normally the situation in the case of electromagnetic and weak interactions; in a limited range of kinetic variables, it is also true in the case of strong interactions. The relativistic perturbation theory has been very useful in describing physical processes.