Introduction

In the previous chapter, we have described free particles and fields in termsof Lagrangian, equations of motion, their solutions and quantization.However, in the physical world, particles and fields are visualized by theirinteractions with other particles and fields or among themselves. Forexample, the simple processes of Compton scattering, photoelectric effect,and Coulomb scattering involving photons and electrons, are well known. Allthe known elementary particles interact with each other through the fourfundamental interactions, that is, electromagnetic, weak, strong, andgravitational, through the exchange of gauge fields. Since the particlesthemselves can be described in terms of fields, all the physical processesgoverned by the four fundamental interactions are examples of various typesof fields in interaction with each other including self interaction, allowedby the general principles of physics. These interactions are described by aninteraction Lagrangian Lint(x),to be included along with the free LagrangianLfree(x), described inChapter 2 for a quantum description of the evolution of physical systems.The interaction Lagrangians can be obtained by using the symmetry propertiesof the physical system defined by certain transformations called local gaugetransformations and imposing the invariance of the free Lagrangian underthese 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 Lagrangiansinvolving spin 0, spin and spin 1 particles and illustrate the generalprinciples to write them. We use the example of electromagnetic interactionto demonstrate the general method of the relativistic perturbation theory tofind out the solution of the equations of motion of fields in the presenceof the interaction LagrangianLint(x). It is assumed thatthe strength of the interaction Lagrangian can be quantified by a parameterwhich is small, so that perturbation theory can be applied. This is normallythe situation in the case of electromagnetic and weak interactions; in alimited range of kinetic variables, it is also true in the case of stronginteractions. The relativistic perturbation theory has been very useful indescribing physical processes.