Non-Linear Abelian Scenarios and Yang-Mills Theory

Yang-Mills Theories [1, 2, 3, 4] play an important hole on the description of both strong and weak interactions, being one of the most successful pillars of Fundamental Interactions. Nowadays electromagnetic and gravitational interactions can also be systematically described by the mechanisms of this class of field theories [5, 6, 7]. Here we develop a different approach to derive from classical Maxwellian electrodynamics a genuine classical Yang-Mills theory by using Noether Theorem and by introducing non-linearities to the Maxwellian regime. We start from a global symmetry and end up with a genuine Yang-Mills model having a local symmetry as the ultimate symmetry of the problem. In section 2, we briefly review Maxwell’s Electrodynamics in covariant notation and also explore briefly the concept of internal degrees of freedom and their correspondence to a colored electrodynamics. In section 3 we discuss a non-linear Yang-Mills model that describes an electromagnetic theory extension that can lead to an unification with the electroweak theory. In section 4, we discuss the formulation of general non-linear Yang-Mills models by introducing a recursive procedure from which we introduce gradually the non-linearities to the Lagrangian of the theory and see how the physical quantities transform according to the present symmetries. In section 5, we derive the final Lagrangian of the theory and also introduce the concept of Lie-algebra-valued fields. We also extend our model to the case of a more general -group. In section 6, we discuss the symmetries present in our problem and show how the initial global symmetry of the theory becomes local and the more general symmetry of the Yang-Mills model.