Topological Objects in Field Theory
In the previous chapters, we studied the quantization of small excitations above the vacuum state. The latter, a spacetime-independent field configuration, was the simplest solution of the classical equations of motion. In this chapter, we shall call such solutions trivial. The nontrivial solutions are other field configurations which obey the equation of motion. Depending on the context, they can describe either objects with finite energy or processes (such as tunneling). The stability of such objects is guaranteed by conservation laws, which in many cases have a topological origin. This explains the title of this chapter. We consider in this book only theories with a weak coupling constant. Let us denote the coupling by a and let the mass of the field quanta be m. Then the typical mass of a classical object is as large as m/ a » m and the typical action of a processes is l/a» 1. Some of the topological objects have been observed in real systems (kinks and vortices), other are presently purely theoretical constructions (magnetic monopoles). There are reasons to believe that QeD has a nontrivial vacuum structure due to fluctuations which are dominated to some accuracy by topological objects. In this chapter we consider a few topological objects in successively increasing number of space dimensions and make some remarks about their classification and quantization.