Quick Review of Field Theories

Abstract

This chapter describes the gauge-field theoretical approach to physical problems in general. The concept of symmetry in physics is reviewed. The discussion starts with geometrical symmetry and extends to symmetry in physical theory, the postulate that physical theories should not depend on the choice of coordinate-system. The local and global transformations are discussed in conjunction with symmetry. Here the local transformation is the case where the transformation matrix is coordinate dependent whereas the global transformation is the case when the transformation is coordinate independent. In general, a globally symmetric theories is not necessarily locally symmetric because of the additional term associated with differentiation of the transformation matrix. To recover local symmetry, it is necessary to replace the usual derivatives with covariant derivatives. The gauge term is introduced as part of the covariant derivatives. Once a gauge is found, the field equations can be derived with the use of Lagrangian formalism. The chapter concludes with showing electrodynamics as an example of a gauge theory where the electromagnetic field acts as a gauge field to make quantum dynamics locally symmetric.