Soluções clássicas na teoria quântica de campos e suas implicações

Abstract

In this Masters dissertation, topological solitons will be investigated in the classical theory of fields, more specifically, kinks, vortices, monopoles and instantons. These objects emerge in the theory after a symmetry breaking pattern, in which we are able to create a non-trivial map between the vacuum manifold and the spacial manifold analysed at the infinity. Firstly, the kink solution will be found in the $\lamda\phi^4$ and sine-Gordon model, his classical energy will be derived and calculated and, after that, the appropriate quantum corrections, for the masses of these structures, will be done. Following a natural process, more degrees of freedom will be added to the system, and it will lead us to global vortices, endowed with an infinite energy. To solve this issue, one uses Derrick's theorem, in which a gauge term is added to the Lagrangian, permitting one to find a topological soliton with finite energy, also known as Nielsen-Olesen vortex. Inspired by Dirac's monopole, the same previous process will be applied to derive the 't Hooft-Polyakov monopole solution, that possesses a finite energy configuration. However, the orders of magnitude are way above what modern colliders can detect. We will continue searching for this particle in the Glashow-Weinberg-Salam theory, also known as Cho-Maison monopole, and an extension in the hipercharge sector $U(1)_Y$ will be done, in order to estimate this particle's mass. Lastly, we will conclude this dissertation addressing the instantons, topological defects that arise in the pure Yang-Mills theory, and it will be shown how this structure solve the $U(1)-$problem in QCD.