Chapter 9 - Nonperturbative Results

Abstract

This chapter explains the derivation of the Goldstone theorem using the effective action-generating functional. In the chapter, the method of effective potential is used to determine the conditions under which the scalar fields may develop a nonzero vacuum expectation value. The chapter discusses the examination of spontaneously broken symmetry for the case of a complex scalar field coupled to an abelian gauge field to demonstrate the Higgs–Kibble mechanism. Its extension to the nonabelian case of the Glashow–Salam–Weinberg model is also discussed in the chapter. The chapter also demonstrates the chiral anomaly using path integral techniques and discusses it as an example of an index theorem. In the chapter, the path integral is used to discuss the role of classical solutions to the equations of motion and it is also applied to solitons and to the solutions of the Euclidean equations of motion known as instantons. The chapter also discusses the applications of the effective potential to demonstrate the dynamical breakdown of symmetry and the restoration of symmetry at finite temperature.